Citation |

- Permanent Link:
- https://ufdc.ufl.edu/UFE0024873/00001
## Material Information- Title:
- Three Essays on School Choice the Case of Open Enrollment Programs
- Creator:
- Ozek, Umut
- Place of Publication:
- [Gainesville, Fla.]
- Publisher:
- University of Florida
- Publication Date:
- 2009
- Language:
- english
- Physical Description:
- 1 online resource (116 p.)
## Thesis/Dissertation Information- Degree:
- Doctorate ( Ph.D.)
- Degree Grantor:
- University of Florida
- Degree Disciplines:
- Economics
- Committee Chair:
- Figlio, David N.
- Committee Members:
- Romano, Richard E.
Kenny, Lawrence W. Zsembik, Barbara A. - Graduation Date:
- 8/8/2009
## Subjects- Subjects / Keywords:
- High school students ( jstor )
Lotteries ( jstor ) Mathematics ( jstor ) Open enrollment ( jstor ) Public schools ( jstor ) School choice ( jstor ) School districts ( jstor ) Schools ( jstor ) Students ( jstor ) Test scores ( jstor ) Economics -- Dissertations, Academic -- UF boston, choice, enrollment, mechanism, open, school Pinellas County ( local ) - Genre:
- Electronic Thesis or Dissertation
born-digital ( sobekcm ) Economics thesis, Ph.D.
## Notes- Abstract:
- Open enrollment programs such as inter-district and intra-district school choice, which allow parents to send their children to public schools outside of the neighborhood they reside, have become increasingly popular in the United States. This study analyzes open enrollment programs along two dimensions. Chapter 2 and Chapter 3 examine the theoretical aspect by evaluating different assignment mechanisms that are commonly used to implement these programs. Chapter 4, on the other hand, looks into the empirical aspect of such programs by analyzing their impact on households? public school choice behavior and student outcomes. The findings presented in this study might provide useful insights in the design of new public school choice programs. Chapter 2 of this study evaluates the Nash equilibrium characteristics of the Boston mechanism (BM), a commonly employed means of determining public school assignments of students under open enrollment, under the informational circumstances that arise in major school districts using this mechanism in the U.S. The results suggest significant departures from the current body of knowledge in the literature, implying that the applicability of the earlier findings on the equilibrium properties of the BM in these school districts is rather limited. Chapter 3 shows that two of the ?strategy-proof? alternatives proposed in the recent literature as replacements for the BM fail to satisfy a standard 14th Amendment equal protection requirement, which is satisfied by the BM. These findings may help to explain the persistence of the BM in many school districts despite its serious flaws discussed in the recent economics literature. Chapter 4 analyzes households? response to the introduction of intra-district school choice and examines the impact of exercising this form of school choice on student test scores in Pinellas County Public Schools. The results indicate that households react strongly to the incentives created by such programs. However, I find that those who opt out of their default public schools often perform significantly worse on standardized tests than similar students who stay behind. The results further suggest that the impact of opting out varies significantly with respect to grade and socio-economic status of the students. ( en )
- General Note:
- In the series University of Florida Digital Collections.
- General Note:
- Includes vita.
- Bibliography:
- Includes bibliographical references.
- Source of Description:
- Description based on online resource; title from PDF title page.
- Source of Description:
- This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
- Thesis:
- Thesis (Ph.D.)--University of Florida, 2009.
- Local:
- Adviser: Figlio, David N.
- Statement of Responsibility:
- by Umut Ozek.
## Record Information- Source Institution:
- University of Florida
- Holding Location:
- University of Florida
- Rights Management:
- Copyright Ozek, Umut. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
- Resource Identifier:
- 489208708 ( OCLC )
- Classification:
- LD1780 2009 ( lcc )
## UFDC Membership |

Downloads |

## This item has the following downloads: |

Full Text |

PAGE 1 THREE E SSAYS ON SCHOOL CHOICE: THE CASE OF OPEN ENROLLMENT PROGRAMS By UMUT OZEK A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLOR IDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2009 1 PAGE 2 2009 Umut Ozek 2 PAGE 3 To m y lovely wife and family 3 PAGE 4 ACKNOWL EDGMENTS I would like to thank David Figlio, Richard Romano, Lawrence Kenny, Jonathan Hamilton, Damon Clark, Steven Slutsky and Da vid Sappington for useful comments, Burak zek for excellent research assistance, Pinellas County Public Schools for providing the data, and my lovely wife and family for inspiration and endless support. 4 PAGE 5 TABLE OF CONTENTS page ACKNOWLEDGMENTS...............................................................................................................4 LIST OF TABLES................................................................................................................. ..........7 LIST OF FIGURES.........................................................................................................................8 ABSTRACT.....................................................................................................................................9 CHAPTER 1 INTRODUCTION................................................................................................................. .11 2 GAMES OF SCHOOL CHOICE UNDER UNCERTAINTY...............................................14 Introduction................................................................................................................... ..........14 Literature Review.............................................................................................................. .....17 Public School Assignment Problem and the Two Assignment Mechanisms.........................20 Boston Mechanism..........................................................................................................21 Gale-Shapley Deferred Acceptance Mechanism.............................................................23 Results.....................................................................................................................................24 Stability in Equilibrium...................................................................................................24 Pareto Efficiency and Transition to the GS-DA Mechanism..........................................28 Transition in the Presence of Sincere Students...............................................................30 Conclusions.............................................................................................................................33 3 EQUAL TREATMENT AS A MEANS OF EVALUATING PUBLIC SCHOOL ASSIGNMENT MECHANISMS...........................................................................................37 Introduction................................................................................................................... ..........37 Public School Assignment Problem and the Assignment Mechanisms.................................41 Boston Mechanism..........................................................................................................42 Top-Trading Cycles Mechanism.....................................................................................43 Gale-Shapley Deferred Acceptance Mechanism.............................................................43 Equal Treatment Criterion...................................................................................................... 44 Transition and Consequences.................................................................................................50 Concluding Remarks............................................................................................................. .54 4 THE EFFECTS OF OPEN ENROLLMENT ON SCHOOL CHOICE AND STUDENT OUTCOMES..........................................................................................................................58 Introduction................................................................................................................... ..........58 Policy Background and Data Description...............................................................................64 Policy Background..........................................................................................................64 Data Description..............................................................................................................66 5 PAGE 6 Im pact of Open Enrollment on School Choice.......................................................................68 Incidence of Opting out...................................................................................................68 Composition of the Opt out Students..............................................................................69 Opting Out and Student Test Scores.......................................................................................70 Where do Students Opt out?............................................................................................71 Ordinary Least Squares (OLS) Results...........................................................................73 Instumental Variables (IV) Results.................................................................................74 Disentangling the Reasons Unde rlying the Detrimental Impact.....................................78 Change in school quality..........................................................................................78 Outsider effect..........................................................................................................79 Alternative explanations...........................................................................................80 Impact of Opting out Disadvantaged Students.............................................................81 Falsification Test....................................................................................................................83 Robustness Checks.............................................................................................................. ...84 Concluding Remarks............................................................................................................. .85 5 CONCLUSIONS.................................................................................................................. 107 APPENDIX A NASH EQUILIBRIUM STRATEGIES IN EXAMPLE 2-2................................................109 B DEMONSTRATING THE THREE ASSIGNMENT MECHANISMS USING EXAMPLE 3-1.................................................................................................................... .110 Boston Mechanism...............................................................................................................110 Top-Trading Cycles Mechanism..........................................................................................110 Gale-Shapley Deferred Acceptance Mechanism..................................................................110 C THE TWO ALTERNATIVE MECHANISM S WHEN EACH SCHOOL HAS THE SAME PRIORITY RANKING............................................................................................112 Top-Trading Cycles Mechanism..........................................................................................112 Gale-Shapley Deferred Acceptance Mechanism..................................................................112 LIST OF REFERENCES.............................................................................................................114 BIOGRAPHICAL SKETCH.......................................................................................................116 6 PAGE 7 LIST OF TABLES Table page 2-1 The key equilibrium properties of the two assignment mechanisms.................................34 2-2 Expected payoff matrices of the pref erence revelation game for Example 2-2.................35 2-3 Expected payoff matrix for Example 2-4...........................................................................36 3-1 Characteristics of the assignment mechanisms..................................................................56 3-2 Public school assignments and the assignment probabilities in Example 3-1...................57 4-1 Descriptive statistics..................................................................................................... .....89 4-2 Default school versus target school ch aracteristics: the y ear before opting out................93 4-3 The impact of opting out on test scores: OLS results........................................................94 4-4 Student characteristics and proximity................................................................................95 4-5 The impact of opting out on test scores: IV results...........................................................96 4-6 The impact of opting out on test scores: pre-policy versus post policy IV results..........97 4-7 The impact of opting out on reading sc ores: identifying the impact of changing school quality IV results..................................................................................................98 4-8 The impact of opting out on reading sc ores: by grade of opt out IV results...................99 4-9 The impact of opting out on reading sc ores: elementary versus middle school nontransition graders IV results...........................................................................................100 4-10 The impact of opting out on reading scor es: by years afte r opting out IV results.........101 4-11 Student characteris tics across subgroups.........................................................................102 4-12 Default school versus target school characteristics during the y ear before opting out disadvantaged versus advantaged students...................................................................103 4-13 The impact of opting out on reading sc ores: disadvantaged students IV results...........104 4-14 Falsification test: linear probability model estimates......................................................105 4-15 The impact of opting out on test sc ores: IV results robustness checks.........................106 7 PAGE 8 LIST OF FI GURES Figure page 4-1 Post-policy elementary school attend ance areas in Pinellas County Schools....................87 4-2 Post-policy middle school attendanc e areas in Pinellas County Schools..........................88 4-3 Percentage of opt out students in Pinellas County Schools, 2001 2005..........................90 4-4 Kernel density estimates achievement percentile at the s ending school for nontransition grade opt out students........................................................................................91 4-5 Kernel density estimates achievement percentile at the sending school for transition grade opt out students........................................................................................92 8 PAGE 9 Abstract of Dissertation Pres ented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy THREE ESSAYS ON SCHOOL CHOICE: THE CASE OF OPEN ENROLLMENT PROGRAMS By Umut Ozek August, 2009 Chair: David N Figlio Major: Economics Open enrollment programs such as inter-district and intra-district school choice, which allow parents to send their children to public sc hools outside of the neighborhood they reside, have become increasingly popular in the United States. This study analyzes open enrollment programs along two dimensions. Chapter 2 and Ch apter 3 examine the theoretical aspect by evaluating different assignment mechanisms th at are commonly used to implement these programs. Chapter 4, on the other hand, looks into the empirical aspect of such programs by analyzing their impact on households public school choice behavior and student outcomes. The findings presented in this study mi ght provide useful insights in the design of new public school choice programs. Chapter 2 of this study evaluates the Nash equilibrium characteristics of the Boston mechanism (BM), a commonly employed means of determining public sc hool assignments of students under open enrollment, unde r the informational circumstances that arise in major school districts using this mechanism in the U.S. The results suggest significan t departures from the current body of knowledge in the li terature, implying that the applicability of the earlier findings on the equilibrium properties of the BM in these school districts is rather limited. 9 PAGE 10 10 Chapter 3 shows that two of the strategyproof alternatives pr oposed in the recent literature as replacements for the BM fail to satisfy a standard 14th Amendment equal protection requirement, which is satisfied by the BM. These fi ndings may help to explain the persistence of the BM in many school districts despite its seri ous flaws discussed in the recent economics literature. Chapter 4 analyzes households response to the introduction of in tra-district school choice and examines the impact of exercising this fo rm of school choice on student test scores in Pinellas County Public Schools. The results in dicate that households react strongly to the incentives created by such programs. However, I find that those who opt out of their default public schools often perform significantly worse on standardized tests than similar students who stay behind. The results further s uggest that the impact of opting out varies significantly with respect to grade and socio-econom ic status of the students. PAGE 11 CHAP TER 1 INTRODUCTION Breaking the link between households residen tial choice and school choice has been the driving force behind the school choice reforms duri ng the last two decades in the United States. Branching from this common objective, such reforms employ different instruments to provide alternatives to households ne ighborhood public school. These inte nded alternatives include private schools as in private school voucher programs, new publicly funded schools as in charter and magnet schools or other traditional public schools as in open enrollment programs. Despite the extensive focus on vouchers and charter schools in the current literature1, open enrollment programs such as inter-distr ict and intra-distri ct choice programs2 remain to be the most frequently exercised form of school choice by policy-makers and households. As of 2005, 27 states have passed legisl ations mandating the school distri cts within their boundaries to implement intra-district school choice and 20 states have adopted legislations mandating the school districts to participate in the inter-district choice program of the state (ECS, 2005). There is also an increasing trend in the percentage of households participating in open enrollment programs. Between 1993 and 2003, the percentage of students attending a public school other than their neighborhood schools increased from 11% to 15% in the United States (NCES, 2006). 1 A number of these articles including Fernandez and Rogerson (1996), Nechyba (1999, 2000, 2003) and Ferreyra (2007) branch from the multi-community public good pr ovision framework initialized by Tiebout (1956), whereas others including Epple and Romano (1998), Manski (1992), Hoyt and Lee (1996) and McMillan (2005) use singlecommunity general equilibrium models to simulate the impact of private school voucher programs. Developing a theoretical model with varying assumptions that introduces private alternatives to an existing public school system, this line of research typically employs simulations based on the theoretical model with the exception of Ferreyra(2007) who actually estimates the theoretical model. Even though some of these articles such as Epple and Romano (1998) incorporate open enrollment into their models, the focus has been on the consequences of vouchers. 2 Ideally, intra-district school choice prog rams allow parents to choose any public school within their school district whereas inter-district school choice programs allow parents to send their kids to public schools outside of their school district. 11 PAGE 12 The objective of this study is twofold. First, I exam ine th e theoretical aspect of open enrollment programs by evaluating different assign ment mechanisms that are commonly used to implement such programs. Second, I look into the empirical aspect of open enrollment programs by analyzing their impact on hous eholds public school choice be havior and student outcomes. The findings presented in this study might pr ovide useful insights for policy-makers and researchers in the design of ne w public school choice programs. Chapter 2 evaluates the Nash equilibrium ch aracteristics of the Boston mechanism (BM), a commonly employed means of determining public school assignments of students, under the informational circumstances that arise in major sc hool districts using this mechanism in the U.S. The results suggest significant de partures from the current body of knowledge in the literature. First, in reality, the BM might produce unstable assignments in equilibrium. Second, a transition from the BM to the Gale-Shapley Deferred Ac ceptance mechanism, which recently replaced the BM in Boston Public Schools, might lead to effici ency losses. Third, the pr eference of strategic parents, who might be the important stakeholders in the school dist rict, is not a valid explanation to the persistence of the BM in major school distri cts. An important policy implication is that the applicability of the earlier findi ngs on the equilibrium properties of the BM is rather limited. The equal protection clause of the 14th Amendment applied to school choice mechanisms implies that students with the same expressed public school preferences and in public schools same priority categories (e.g., re siding within a walk zone) must be treated equally. A strict application w ould require that such students ge t the same school assignments. When capacity constraints result in a scarcity of slots at school s, a weaker application would require that if two students who are in the same priority category for a given school rank that school as their first choices with all other choices equal, such st udents have the same probability 12 PAGE 13 13 of attending that public school. Ge neralizing this argument, Chap ter 3 introduces a new criterion to evaluate public school assignment mechanisms and shows that all of the strategy-proof mechanisms proposed in the recent literature as replacements for th e BM fail to satisfy this weak application of equal protection, wh ich is satisfied by the BM. Afte r demonstrating this result, I go on to show that one of these alternatives na mely the Top-trading Cycles Mechanism is worse yet in the following sense. Given two equivalent students, the latter mechanism might assign the lottery loser to the desi red school while assigning the one who wi ns the lottery to a less desired school. The findings in this study might provide an explanation fo r the persistence of the Boston mechanism in most of the school district s using variants of this mechanism. Chapter 4 analyzes households response to the introduction of in tra-district school choice and examines the impact of exercising this fo rm of school choice on student test scores in Pinellas County Public Schools, one of the larg est school districts in the United States. The results indicate that households react strongly to the incentives created by such programs leading to significant changes in the fre quency of exercising a lternative public schoo ling options, as well as changes in the composition of the opt out students. Howeve r, using proximity to public alternatives as an instrument for opting out of the assigned public sc hool, I find no significant benefit of opting out on student achievement and that those who opt out of their default public schools often perform significantly worse on standa rdized tests than sim ilar students who stay behind. The results further suggest that the s hort-run detrimental eff ects of opting out are stronger for students who opt out cl oser to the terminal grade of the school level, yet weaker for disadvantaged students, who typi cally constitute the proposed targ et of school choice reforms. PAGE 14 CHAP TER 2 GAMES OF SCHOOL CHOI CE UNDER UNCERTAINTY Introduction School choice reforms and their effectivene ss in improving the quality of the public school system in the U.S. remain a major topic of debate among policy makers and researchers. The main objective for most of these school choi ce reforms is to provide as equal access to quality education as possible for all students regardless of their soci o-economic status. Along these lines, inter-district and intr a-district school choice programs, which allow parents to choose schools outside of the neighborhood they reside, have become in creasingly popular in the last decade1. Along with increased parental choice has come the need to implement well-behaving public school-student assignment mechanisms, whic h have been evaluated along three desirable properties in the ec onomics literature: 1. STRATEGY-PROOFNESS. A preferred public school assignm ent mechanism avoids creating incentives for parents to play complicated ga mes. Hence, truthful parental ranking of schools should be a dominant strategy. Strategy -proofness of the assignment mechanism is then desirable. 2. STABILITY. An assignment set is defined to be stable if there is no school-student pair ( i ,s ) such that student i prefers school s to her current assignment and either school s prefers student i to at least one of th e students assigned to it or school s has at least one empty seat. Absent stability, there exists justified envy in the assignments providing incentives for parents to seek legal action to overturn school assignments. 3. EFFICIENCY. For the public school assignment problem in the context of this study2, only the welfare of students is considered for Pareto efficiency, since the schools are regarded 1 According to the estimates from the 1 999-2000 school year, 71 percent of the school districts in the West, 63 percent in the Midwest, 44 percent in the South and 19 percent in the Northeast employed these school choice programs in the U.S. (NCES, 2006). 2 For the public school assignment problem discussed in this study, priority categories mandated by the school districts are employed alon g with student preferences to determine the public school assignments. Since these rankings do not necessarily correspond to schools pref erences, only the students preferences are considered for efficiency. On the contrary, there are cases such as th e high school assignments in NYC where schools determine their own priorities. In that case, scho ol preferences as well as student prefer ences might be taken into account for welfare considerations. 14 PAGE 15 as objects to be consum ed by students. Pareto efficient assignments are obviously desirable. One of the most commonly used student assignment mechanisms is the Boston mechanism, so named because of its use until r ecently in Boston. This mechanism is still being used in other major school districts including Cambridge (MA), Charlotte (NC), Denver (CO), Hillsborough (Tampa, FL), Miami-Dade (FL), Minneapolis (MN), Seattle (WA) and Pinellas (St.Petersburg, FL)3. This mechanism has the virtue of removing the ambiguity in assignment decisions by imposing explicit rules4. However, the mechanism has a major weakness: it is not strategy-proof (Abdulkadiroglu and Sonmez, 2003). In other words, the Boston mechanism induces parents to play a difficult preference revelation game in thes e school districts, the details of which are discussed below. The current body of knowledge in the literature pertaining to the characteristics of the Nash equilibrium assignments under the Boston m echanism relies heavily on the assumption that the preference revelation game induced by th is mechanism takes place under a complete informational setting where schools have strict priority rankings over students that are common knowledge5. In fact, in all of the major school di stricts using the Boston mechanism, public school preferences over students are determined by broad priority categories mandated by the school district (e.g. sibling already in the schoo l). This generates the need to break the ties 3 In 2003, over one million students were enrolled in public schools within the boundaries of these school districts. The assignments for a significant number of public schools are determined using the Boston mechanism in these districts. For instance, in Denver, th is mechanism is used for traditional public school assignments (approximately 79% of all public schools), whereas, in Seattle, the assignments for every public school in the district are decided with the use of the Boston mechanism. 4 The absence of such explicit rules creates potential co nflicts between school author ities and parents who question the fairness of school assignments, providing incentives fo r parents to seek legal action to overturn their school assignments. Abdulkadiroglu and Sonm ez (2003) cite cases in Mississippi and Wisconsin where the assignment decisions were overruled due to the ambiguity created by th e school assignment mechanism. 5 The subsequent section provides a detailed discussion of this line of research. 15 PAGE 16 between stu dents in the same priority categorie s before the assignment algorithm can be applied. In all of the aforementioned school districts, the ties between student s are broken in some random fashion and the assignments are determined using the Boston algorithm after all of the student applications are received. Therefore, under the Boston mechanism, games of school choice, in reality, take place in the presence of uncertainty about schools strict priority rankings over students. This study provides a reality-check by ev aluating the Boston mechanism along the aforementioned dimensions of merit under the r eal-world informational setting that arise in major school districts using varian ts of the Boston mechanism. Th e analysis suggests significant deviations from the findings of the previous litera ture. First, the Boston mechanism might lead to unstable assignments and provide parents incentives to seek lega l action to overturn their school assignments in these school distri cts. Furthermore, the results indi cate that the stable assignments under students true preferences are not n ecessarily Nash equilibrium outcomes. The second result of this study indicates that, due to the unc ertainty created by the timing of the tie-breaking, a transition to one of the alternatives called the Gale-Shapley Deferred Acceptance (GS-DA) mechanism, which recently replaced the Boston mechanism in Boston Public Schools, might result in Pareto inferi or assignments. This implies that neither the transition in Boston guaranteed efficiency gains nor would such a tran sition in other major school districts using the Boston mechanism necessa rily be beneficial in terms of efficiency. The third point of this study is to show that the preferences of st rategic (sophisticated) parents, who might be the important stakeholders in the school dist rict, is not a valid explanation to the puzzling persistence of the Boston mechanism in practice. Specifically, the results indicate that, assuming that there is at least one sincere (unsophist icated) student always 16 PAGE 17 revealing her public school pref erences truthfully, strategic st udents m ay end up being assigned to less-preferred schools under the Boston mechan ism than under the GS-DA mechanism, even if they manage to coordinate to achieve the Pareto dominant assignment under the former assignment mechanism. This study demonstrates that the extent to which some of the earlier findings in the literature concerning the equili brium properties of the Boston mechanism can be applied in reality is rather limited. An important policy imp lication is that these findings must be carefully considered by the policy-makers if the Boston me chanism is to be abandoned, since they might provide misleading policy suggestions in these school districts. The analysis proceeds as follows. Section 2 summarizes the current body of knowledge about the Nash equilibrium characteristics of the assignments produced by the Boston mechanism. Section 3 provides a background on th e public school assignment problem, details the Boston and the GS-DA mechanisms as they are applied in major school districts, and shows how the resulting assignments would differ using an example. Section 4 presents the main results and Section 5 concludes. Literature Review While recently abandoned in Boston Public Schools, the Boston mechanism remains to be one of the most commonly used means of determining public school assignments in major school districts. Despite its common use, iron ically, Abdulkadiroglu and Sonmez (2003) show that the Boston mechanism fails to satisfy seve ral desirable properties of a well-behaving assignment mechanism. First, under the Boston m echanism, truthful revelation of public school preferences is not necessarily a dominant strategy for all parent s, inducing parents to play 17 PAGE 18 com plicated preference revelation games.6 Consequently, the Boston mechanism might fail to produce Pareto efficient assignments even though it guarantees Pareto efficiency provided that the true preferences are submitted. Despite these weaknesses of the Boston mech anism, Ergin and Sonmez (2006) show that the Boston mechanism will result in stable assign ments in equilibrium. They further show that the set of stable assignments under students true preferences corresponds to the set of Nash equilibrium assignments under the Boston mechanism. This is perhaps surprising since, in equilibrium, preferences of students are not truthfully revealed in general. Nevertheless, the assignments satisfy the stability property in equilibrium. Another mechanism that has been proposed and recently implemented is the GaleShapley Deferred Acceptance (GS-DA) mechanism7. Compared to the Boston mechanism, the GS-DA mechanism has the desirable feature of being strategy-proof8. In addition, it guarantees stable assignments, though Pareto efficiency is not guaranteed9. 6 More specifically, this mechanism induces some parents to rank their safe schools, for which they have relatively higher probability of admission, higher than their true rankings. It is worth noting that this feature of the Boston mechanism contradicts the main objectiv e of open enrollment programs, since the safest school, for which they have the highest priority of admission, for each parent is the neighborhood public school. 7 After pointing out the weaknesses of the Boston mechanism, Abdulkadiroglu and Sonmez (2003) propose two alternative assignment mechanisms: the GS-DA mechanism and the top-trading cycles (TTC) mechanism. After the publication of this study, Boston Public Schools contacted the authors to design a new public school assignment mechanism. In 2006, the GS-DA mechanism was implemented in Bo ston, replacing the Boston mechanism (Abdulkadiroglu et. al. (2005) and Abdulkadiroglu et. al. (2006)). 8 See Dubins and Freeman (1981) and Ro th (1982). Recently, Abdulkadirog lu et. al. (2007) examine the impact of two different tie-breaking methods on the efficiency and strategy-proofness of the GS-DA mechanism; single tie breaking where each student is given a random number to be used at every school and multiple tie-breaking where each student is assigned a different random number to be used at each school. Their main theoretical result indicates that a GS-DA mechanism that uses sing le tie-breaking is not dominated by any other mechanism that is strategyproof for students. 9 It has been well documented in the previous literature that given strict student preferences and strict school priorities, there exists no other stable assignment that Pareto dominates the assignment produced by the GS-DA mechanism; however overall Pareto effi ciency is not guaranteed. Erdil and Ergin (forthcoming) show that when there are indifferences in school priorities as in the public school assignment problem discussed in this study, there may exist another stable assignment that Pareto dominates the GS-DA outcome. 18 PAGE 19 Ergin and S onmez (2006) also show that ev en though neither the Boston mechanism nor the GS-DA mechanism guarantee Pareto efficienc y, the assignments achieved by the latter will always weakly Pareto dominate the Boston mech anism assignments. Following this result, they state that a transition to the GS -DA mechanism may result in signi ficant efficiency gains in the major school districts using vari ants of the Boston mechanism. Given the numerous aspects along which th e GS-DA mechanism is superior to the Boston mechanism as shown in the previous lite rature, there has been an increasing curiosity among researchers as to why othe r school districts have not yet followed in the footsteps of Boston Public Schools and replaced the Bost on mechanism. Recently, Pathak and Sonmez (forthcoming) suggest the existe nce of important stakeholders who benefit from the Boston mechanism as a possible explanation to this puzz le. Specifically, their main result indicates that the school to which a strategic st udent, who plays a best response, is assigned under the Paretodominant equilibrium of the Boston mechanism is weakly better than her outcome under the GSDA mechanism when there is at least one sincer e student who always reveals her public school preferences truthfully. In other words, in the presence of sincere students, strategic players weakly prefer the Boston mechanism to the GSDA mechanism if they can coordinate to achieve the Pareto-dominant assignment set under the Boston mechanism10. The key assumption behind these results is th at the students have complete information about their relative priority positions at each sc hool when making their school choices. In other words, students are assumed to be acting as if they know the results of the tie-breaking lotteries 10 It is a well-known fact that the games of school choice induced by the Boston mechanism might have multiple equilibria and the Pathak and Sonmez (fortcoming) result regards the equilibrium wherein the strategic players obtain their Pareto-dominant assignments among these Nash equilibria with some sincere players. 19 PAGE 20 before reporting th eir preferences11. However, in reality, students have to take into account the uncertainty created by the lotteries when making their choices. Put differently, in practice, players of this game (strategic students) choos e the strategy (public school ranking) that yields the highest expected utility among different possi ble rankings of public schools, some of which, if submitted, may yield different outcomes (assignments) with known probabilities depending on the lottery result12. The following subsections show that this uncertainty has significant consequences on the equilibrium properties of the Boston mechanism. Public School Assignment Problem and the Two Assignment Mechanisms In a public school assignment problem, there are n students and k public schools each of which has a given number of slots available. Equilibrium assignments depend on students reported preferences, priorities of schools over st udents, and the assignment mechanism. It is assumed that each student has a utility function over the k public schools with strict preferences, which is here assumed to be common knowledge. Students first submit their preferences, i.e., a strict ranking of the k schools, and the assignments use the set of submitted (ordinal) rankings. Schools have priority rankings of students, base d on broad priority categories mandated by the school district (e.g., residing in a wa lk zone), and lotteries are used to break the ties between the students in the same priority categories. St udents know school capacities, the rules of the assignment mechanism, and the pr iority categories of schools when they submit their rankings. I 11 In the last section of their article, Ergin and So nmez (2006) examine a case wher e the complete information assumption is violated. They discuss a scenario where there is uncertainty about the strict preference ordering of a student. Finding the Nash equilibrium given this incomplete information setting, they show that all Nash equilibrium outcomes are not necessarily stable an d a student may be better-off under the Boston mechanism th an under the GSDA mechanism. This study enhances these results in tw o ways. First, identifying the real-world informational structure, the results obtained in this study provide more practical policy implications. Furthermore, this study not only confirms these results under this setting, but also extends them in some important aspects. 12 In the analysis that follows, I preserve the assumption that student preferences are common knowledge as in the related literature. Hence, it is assumed that the game of sc hool choice is not a game of incomplete information where some players (students) have private information. 20 PAGE 21 focus on the fact that the outcom es of the lott eries are not known when students submit their preferences, while I also compare such equilibria to the case in the literature where schools have strict rankings of students as if lotteries were first publicly conducted. How the submitted preferences and school priorities interact to yield assignmen ts depend on the rules of the assignment mechanism, the details of which I discuss next13. Students submit their preferences to maximize their expected util ities in Nash equilibrium. Boston Mechanism Under the Boston mechanism, a student who is not assigned to his first choice is considered for his second choice only after th e students who ranked that students second choice as their first choices. More spec ifically, all of these major sc hool districts employ the following general scheme in their public school assignments14; First step: School districts a nnounce the assignment algorithm, the priority categories, and the way the lottery will be conducted to br eak the ties between students in the same priority category15. The major school districts usin g the Boston mechanism differ considerably in their choices and definitions of priority categori es; however, the most commonly used are sibling and attendance z one priorities. For in stance, in Boston, the following priority categories are currently being used; 1. Students who have siblings cu rrently attending that school and who live in the walk zone of the school. 2. Students who have siblings curre ntly attending that school. 3. Students who live in the w alk zone of the school. 13 School priorities and the assignment mechanism are given so schools are not players in the game. This is in contrast to some two-sided matching problems as discussed in Gale and Shapley (1962). 14 Documentation and more detailed information on the public school assignment procedures in the school districts listed are available upon request. 15 The school districts differ significantly in the ways they use the priority categories along with the lottery outcome to rank the applicants. In Boston, the applicants for a gi ven school are first ranked with respect to the priority categories and then the outcome of the lottery is used to rank those within the same priority category. In MiamiDade, on the other hand, a weighted lottery is conducted where more random numbers are generated for those in higher priority categories. The rankings are then constructed using the best random number for each applicant. 21 PAGE 22 4. Students who do not fall into the three categories above. Second step : Observing these, each student subm its a ranking of her preferred schools. The outcome of the tie breaking is unknown at this time. Third step: Given the applicant pool, the lotte ry is conducted and each applicant is ranked according to the pre-specified priority categ ories and the outcome of the tie-breaking. Fourth step: The assignment of students based on the studen t preferences and the strict school priorities. In the first round, only the first choices of st udents are considered. Based on the schools priority rankings of student s, the seats at each school are assigned one at a time. In the nth round, only the nth choices of the students who coul d not be placed in the (n-1)st round are considered. The proced ure is terminated when th ere are no unassigned students remaining. The crucial point in this public school assignmen t procedure for this analysis is the timing of the lottery to break the tie be tween students in the sa me priority category. In all of the school districts listed, the students ar e required to submit th eir school preference ra nkings before the tiebreaking takes place. Therefore, there is uncer tainty about schools pr iority rankings over students at the time when the school c hoice game among students takes place. To illustrate how the Boston mechanism work s, consider the following example by Roth (1982): Example 2-1 : Assuming complete information16 and that the students truthfully reveal preferences, consider the followi ng preferences and the priority rankings of the three students ( i1, i2, i3) and three schools ( a, b, c ) each of which has only one seat. i1: b a c a: i1 i3 i2 i2: a b c b: i2 i1 i3 i3: a b c c : i2 i1 i3 16 Under complete information, as in the previous literature I assume that the students observe the strict priority rankings of schools and the true preferences of other students. This implies that when making their school choices, students know the applicant pool, each ap plicants true preferences and the ou tcome of the tie-breaking in addition to the assignment procedure. 22 PAGE 23 Applying the Boston m echanism to this example, one obtains the assignments ( i1, b), ( i2, c ) and ( i3, a)17. Note that this example demonstrates how the Boston mechanism might induce students to misrepresent their preferences. If i2 had misrepresented her preferences by listing b as her first choice, the school in which she has the highest priority, she woul d have been assigned to b instead of c and would have been better-off. Hence, the Boston mechanism is not strategyproof. Gale-Shapley Deferred Acceptance Mechanism Unlike the Boston mechanism, none of th e assignments are guaranteed until the assignment algorithm terminates using the GS-DA m echanism. The fourth step of the assignment procedure in the Boston mechanism is modified as follows: Step 1: Each students first choice is consid ered. Each school puts all applicants into a queue unless the number of applicants is higher than its capacity, or rejects the ones ranked lower than its capacity in its priority ranking ot herwise, while placing the rest in its queue. Step k: The rejected applicants next choices are considere d. Comparing the new applicants with the applicants already in th e queue, each school replaces the students on its queue based on its priority rankings. The pro cess terminates when no student is rejected and each student is assigned to the school w hose queue she belongs to when the algorithm terminates. The main advantage of this approach over the Boston mechanism is that it is strategyproof. It also implies stable assignments, but Pareto efficiency is not guaranteed (Roth, 1982). When applied to Example 2-1, the GS-DA mech anism yields the stable assignment set ( i1, a), ( i2, b) and (i3, c )18. However, note that the matching ( i1, b), ( i2, a) and (i3, c ) is Pareto superior to the 17 Only the first choices are considered; given the priorities, i3 is assigned to a and i1 is assigned to b i2 is rejected from a and is assigned to c since the only seat available in school b is occupied by i1. 18 At the end of step 1, i2 gets rejected from a i3 is in the queue of a and i1 is in the queue of b i2 goes to b ; at the end of step-2, i1 gets rejected from b i2 is in the queue of b and i3 is in the queue of a i1 goes to a ; at the end of the step3, i3 gets rejected from a i1 is in the queue of a and i2 is in the queue of b i3 goes to b ; at the end of the step-4, i3 gets rejected from b i2 is in the queue of b and i1 is in the queue of a i3 goes to c ; at the end of the step-5, nobody gets rejected, i1 is assigned to a i2 is assigned to b and i3 is assigned to c and the process terminates. 23 PAGE 24 previous outcom e. The first thr ee columns of Table 2-1 summarize the key properties of the two assignment mechanisms under complete information. Results Stability in Equilibrium Given an arbitrary set of strict true student preferences over schools and pre-lottery school prior ity rankings over students nTTTT ,,,21 kPPPP ,,,21 let denote t set of stable assignm ents and TNresent the Nash equilibrium assignments under the Boston m echanism assuming that all students are strategic. PTS ,he repP Proposition 2-1: Under the aforementioned real-wor ld informational setting, the following conditions are not necessarily true: i. PTNPTS ,,ii. PTSPTN ,, Proof: Consider the following example. Example 2-2 : Assum e that there are three students ( i1, i2, i3) and three schools ( a, b, c ) each of which has only one seat. Assume further that i1 and i2 fall into the same priority category for a, and i2 has higher priority than i1 for b. Also assume that i3 has lower priority than i1 and i2 for both a and b, but has a higher priority than the others for school c Consider the following student preferences and school preferen ces corresponding to these priorities: i1 : a b c a: i2 i1 i3 or i1 i2 i3 i2 : a b c b: i2 i1 i3 i3 : b c a c: i3 i1 i2 The utilities of the students from being assigned to each school are as follows: a b c i1 a1 b1 c1 i2a2 b2 c2 24 PAGE 25 i3a3 b3 c3 where 333 222 111acb cba cba Notice that there are two states of nature; the one where the tie between i1 and i2 for school a is broken in favor of i1, and the one where i1 loses the lottery. There are six possible assignments in this example: i1 i2i3A1 a b c A2 a c b A3b a c A4b c a A5c a b A6c b a Among these assignments, A1 and A3 are stable under students true preferences and the pre-lottery priorities. Hence, 31,, AAPTS Since there are 3 schools to choose from, each student has 6 strategies, i.e. 6 ways to rank the schools. For each state of nature, there are six 6x6 matrices with the corresponding payoffs (utilities) as determined by the assignments of the Boston mechanism. Table 2-2 gives the expected payoff matrices under this scenario where the row player is i1, the column player is i2, the matrix player is i3 and jj j jj j jj jcbf cae bad 2 1 2 1 2 1 j = 1, 2, 3. Consider the first statement of Proposition 1: all Nash equilibrium outcomes are stable. In order for this statement to hold, there should not ex ist a Nash equilibrium st rategy set that results in an unstable assignment. Looki ng at Table 2-2, given students i2 and i3 report school a and 25 PAGE 26 school b as their firs t choices respectively (play axx19 and bxx respectively), the strategy set of student i1 yields the following expected utilities: abc acbbacbcacabcba e1 e1 b1 b1 c1 c1 Therefore, if then student i1 will always report school a as her first choice and play axx given i2 and i3 play axx and bxx respectively. If i1 plays axx and i3 plays bxx the strategy set of i2 results in the followi ng expected utilities: 11be abc acb bac bca cab cba e2 e2 b2 b2 c2 c2 Likewise, if then student i2 will always play axx given i1 and i3 play axx and bxx respectively. Finally, if both i1 and i2 play axx, i3 will always be guaranteed a seat at her most preferred option, school b, if she reports school b as her first choice. Hence, given i1 and i2 play axx, i3 will always play bxx 22be This im plies that, if the following conditions hold: 1112 1 bca (2-1) 2222 1 bca (2-2) then i1: axx, i2: axx and i3: bxx will be a subset of the set of al l Nash equilibrium strategies for the overall game. There are two possible se ts of assignments for this case: 1) If i1 wins the lottery, the Boston mechanism w ill result in the following assignments: ( i1, a), ( i2, c ), ( i3, b). 2) If i2 wins the lottery, the Boston mechanism w ill result in the following assignments: ( i1, c ), ( i2, a), ( i3, b). 19 i2 playing axx means that she either plays abc or acb 26 PAGE 27 Notice th at the Nash equilibrium assignme nts obtained above are not stable under the pre-lottery priorities a nd true preferences: if i1 wins the lottery, i2 prefers b to her assignment (school c ) and school b prefers i2 to i3. If i2 wins the lottery, i1 prefers b to her assignment (school c ) and school b prefers i1 to i3. Therefore, given conditions (2-1 ) and (2-2), this example shows that all Nash equilibrium assignments are not necessarily stable in practice. Now consider the second implication: al l stable assignments under students true preferences are Nash equilibrium outcomes. In this context, this implies that both A1 and A3 are Nash equilibrium assignments. In order to see if this condition is satisfied in this example, one needs to check whether all strategy combinations that result in assignments A1 and A3 are Nash equilibrium strategies. Assume that the conditions (2 -1) and (2-2) are still satisfied. This implies that; 33333 222222 111111aecfb cfbeda cfbeda Finding the Nash equilibria of the overall game, one can observe20 that the strategy set i1: axx, i2: axx and i3: bxx is not only a subset of the set of Nash equilibrium strategies as shown earlier, but this set of strategies corresponds to th e set of all Nash equilibrium strategies of the overall game as indicated by the bold-faced expect ed utilities in Table 2-2. Since this set of strategies yields the assignments A2 or A5 depending on the outcome of the lottery, neither A1 nor A3 can occur as a result of Nash equilibrium strategies. Therefore, the stable assignments under students true preferences are not Nash equilibrium outcomes no matter how the tie is broken between i1 and i2. 20 The derivation is given in the Appendix. 27 PAGE 28 Com bining these two results, this example demonstrates how the uncertainty created by the lottery changes the stability property obta ined under the complete information assumption. The set of Nash equilibrium outcomes, {( i1, a ), ( i2, c ), ( i3, b); ( i1, c ), ( i2, a), ( i3, b )} and the set of stable assignments under stude nts true preferences, {(i1, a), ( i2, b), ( i3, c ); ( i1, b ), ( i2, a ), ( i3, c )} are two distinct sets. Therefore, the Boston mech anism, in practice, might result in unstable assignments providing parents incentives to seek legal action to overturn their assignments. Pareto Efficiency and Transition to the GS-DA Mechanism Proposition 2-2: Under the aforementioned real-world informational setting, a transition from the Boston mechanism to the GS-DA mechanis m might lead to efficiency losses; however, Pareto efficiency is still not gua ranteed under the Boston mechanism. Proof: The following example is sufficient to prove Proposition 2-2. Example 2-3: Assume that there are four students ( i1, i2, i3, i4) and four schools (a, b, c, d) each of which has only one seat. Assume further that i2 and i3 fall into the same priority category for a. Consider the following student preferences and school rankings: i1: b a c d a : i1 i3 i2 i4 or i1 i2 i3 i4 i2: a b c d b : i2 i1 i3 i4 i3: a c b d c : i3 i1 i2 i4 i4: c d b a d : i4 i3 i1 i2 The utilities of the students from being assigned to each school are as follows: a b c d i1 a1 b1 c1 d1 i2a2 b2 c2 d2 i3a3 b3 c3 d3 i4 a4 b4 c4 d4 28 PAGE 29 where 4444 3333 2222 1111abdc dbca dcba dcab Under the Boston mechanism, in the case wher e all players report th eir most preferred schools as their first choices, i3 has a 0.5 chance of being assigned to school a (if she wins the lottery) and 0.5 chance of being assigned to school d (if she loses). Therefore, the expected payoff from playing axxx for i3 in the case where i1, i2 and i4 report their most preferred schools as their first choices is )( 2 133da On the other hand, if she does not reveal truthfully and report school c as her first choice, she will be assigned to school c no matter what the other players do. Given all other players reveal their first choices truthfully, i3s best reply will be not to reveal truthfully and play cxxx if 333)( 2 1 cda 21. Building on this intuition, I now construct a Nash equilibrium. Given that i1 plays bxxx and i3 and i4 play cxxx i2s best reply will be to play axxx since by doing so, she is guaranteed a seat at her favorite school Furthermore, if i2 and i4 report schools a and c as their first choices respectively, and i3 plays cxxx there is no risk for i1 to reveal her most preferred option truthfully and she will play bxxx Finally, given that i1 and i2 reveal truthfully and i3 plays cxxx i4 will always be assigned to school d no matter what she reveals; hence she is indifferent between all of her strategies. Hence, i1: bxxx i2: axxx i3: cxxx and i4: cxxx constitutes a set of Nash e quilibrium strategies given 333)( 2 1 cda which results in the assignments ( i1, b), ( i2, a ), ( i3, c ) and ( i4, d). 21 Notice that given that all other players reveal their first choices truthfully, student i3 will definitely be assigned to school d if she reports school b or school d as her first choice. Therefore, given all other players reveal their first choices truthfully, student i3 will never report school b or school d as her first choice. 29 PAGE 30 W hen applied to this example, the GS-DA mechanism will result in the assignments ( i1, a), ( i2, b), ( i3, c ) and ( i4, d) if the tie is broken in favor of i3 or the assignments ( i1, b), ( i2, a), ( i3, c ) and ( i4, d) otherwise. However, note that the assi gnments achieved by the Boston mechanism weakly Pareto dominate the GS-DA mechanism outcomes if 333)( 2 1 cda On the other hand, each student submitting thei r second-preferred schools, in which they have the highest priority, as their first choices is also a Nash equilibrium strategy set. If these rankings are submitted, notice that both the Bo ston mechanism and the GS-DA mechanism yield the assignment set ( i1, a), ( i2, b ), ( i3, c ) and ( i4, d), which is Pareto dominated by ( i1, b ), ( i2, a ), ( i3, c ) and ( i4, d). Hence, the Boston mechanism does not guarantee Pareto efficiency under the aforementioned informational setting either. This result is particularly important since a transition from the Boston mechanism to the GS-DA mechanism recently took place in Boston Public Schools. Even though the latter mechanism is superior in terms of strategy proofness, the result obtai ned above shows that switching from the Boston mechanism to the GSDA mechanism in practice does not guarantee a Pareto improvement; it may even cause a Paretia n loss. The last three columns of Table 2-1 summarize the first two results obtained in this article by revising the properties of the two assignment mechanisms under the real -world informational setting. Transition in the Presence of Sincere Students In the analysis that precedes, all students we re assumed to be strategic players who play best response while making their school c hoices under the Boston mechanism. However, empirical evidence in the previous literature suggests significantly diverse levels of sophistication among students22. Based on this evidence, suppose that there are two types of 22 See Abdulkadiroglu, Roth, Pathak and Sonmez (2006). 30 PAGE 31 students in the school district: sincere (unsophisticated) students who always reveal truthfully and strategic (sophisticated) stude nts who realize the incentives created by the assignm ent mechanism and play best response. Proposition 2-3: Under the aforementioned real-w orld informational setting, i. Different Nash equilibrium strategy sets under the Boston mechanism might result in different assignments for the sincere students. ii. Strategic students might be assigned to less-preferred schools under the Boston mechanism than the GS-DA mechanism even if they cooperate to achieve the Pareto dominant Nash equilibrium assignm ent set under the Boston mechanism. Proof: Consider the following example: Example 2-4: Assume that there are three students (i1, i2, i3) and three schools ( a, b, c ) each of which has only one seat. Consider the following student preferences and priority rankings: i1: b a c a: i1 i2 i3 or i1 i3 i2 i2: a b c b: i2 i1 i3 i3: a b c c : i2 i1 i3 Suppose that i3 is a sincere player who always reveals her public school preferences truthfully whereas i1 and i2 are strategic players. There are two states of nature; the one where the tie between i2 and i3 is broken in favor of i2, and the one where i3 wins the lottery. Given that i3 always plays abc Table 2-3 provides the e xpected payoff matrix of this game under the Boston mechanism where the row player is i1, the column player is i2 and the payoffs are defined the same way as in Example 2-2. Given that there are two sets of Nash equi librium in this gam e under the Boston mechanism characterized by i1: axx, i2: bxx and i1: bxx, i2: axx The Boston mechanism then results in the following assignments in equilibrium: 22be 31 PAGE 32 i1: axx, i2: bxx i1: bxx, i2: axx If the tie is broken in favor of i2 ( i1, a ), ( i2, b), ( i3, c )( i1, b ), ( i2, a), ( i3, c ) If the tie is broken in favor of i3( i1, a ), ( i2, b), ( i3, c )( i1, b ), ( i2, c ), ( i3, a ) Notice that the former set of Nash equilibrium strategies provides expected payoffs of and whereas the latter produces and for students i1 and i2 respectively. Therefore, given that the Pareto-dominant equilibrium strategy set for the two strate gic students under the B oston mechanism is i1: bxx, i2: axx. On the other hand, when applied to this example, the GSDA mechanism yields the assignments ( i1, b), ( i2, a), ( i3, c ) if the tie is broken in favor of i2 or ( i1, a), ( i2, b), ( i3, c ) if i2 loses the lottery. 1a2b2e1b2e2b Two points are worth noting. First, in the stat e o f nature where the tie is broken in favor of i3, the two Nash equilibria under the Boston mech anism yield different assignments for the sincere student. Hence, multiplicity is an issu e for sincere students under the Boston mechanism in practice. Second, the school s that the strategic student i2 is assigned to under the Paretodominant equilibrium outcome of the Boston mechanism are weakly worse than the GS-DA mechanism assignments. Therefore, even if the tw o strategic students coor dinate to achieve the Pareto-dominant assignment set under the Boston mechanism, i2 will weakly prefer the GS-DA mechanism to the Boston mechanism in this case. Despite the convincing evidence that the Bo ston mechanism is dominated by the GS-DA mechanism along almost all of the desirable prop erties of a well-behaving mechanism, the major school districts using the Boston mech anism have been reluctant to abandon this mechanism in favor of the alternative. The third result in this study suggest s that the preferences of sophisticated parents, who might be the important stakeholders in the school dist rict, is not a valid explanation to this mystery, since the strategic players might weakly prefer the GS-DA mechanism in these school districts. 32 PAGE 33 Conclusions One of the most commonly used student assign ment mechanisms with explicit rules is the Boston mechanism, so named because of its use until recently in Boston Public Schools. Even though this mechanism is superior to some other pre-existing public school assignment mechanisms, it has a major weakness: it is not strategy-proof. In other words, under the Boston mechanism, some students may benefit from misrep resenting their true preferences. As a result, the Boston mechanism induces students to play a complicated preference revelation game where students payoffs depend on others revealed preferences, school priorities and th e rules of the mechanism. This study evaluates the Boston mechanism along the desirable properties of a wellbehaving assignment mechanism under the informationa l setting that arises in most of the major school districts using variants of this mechanism. The results indi cate three significant departures from the findings of the previous literature. First, in practice, the Boston mechanism might result in unstable assignments providing parents incenti ves to seek legal action to overturn their assignments. Second, a transition to one of the proposed alternatives called the Gale-Shapley Deferred Acceptance mechanism may result in efficiency losses under the real-world informational setting. Third, in the presence of sincere students who always reveal truthfully, the school a strategic student receiv es under the Pareto-dominant outcome of the Boston mechanism, in reality, might be weakly worse than the outcome under the student-optimal stable mechanism. An important policy implication of this study is that the curren t body of knowledge related to the equilibrium properties of the Boston mechanism must be carefully considered by the policymakers if it is to be abandoned, since these earlier findings might pr ovide misleading policy suggestions in practice. 33 PAGE 34 34 Table 2-1. The key equilibrium properties of the two assignment mechanisms Under complete information Under uncertainty Strategyproof Guarantees stable assignments Guarantees Pareto efficient assignments Strategyproof Guarantees stable assignments Guarantees Pareto efficient assignments Boston mechanism No Yes No No No No GS-DA mechanism Yes Yes No Yes Yes No Even though neither mechanism guarantees Pareto efficient assignments, the GS-DA mechanism assignments always weakly Pareto dominate the Boston mechanism assignments under complete information. The GS-DA mechan ism assignments do not necessarily weakly Pareto dominate the Boston mechanism assignmen ts under student uncertainty. There are even cases as discussed in this study where the Boston mechanism assignments weakly Pareto dominate the GS-DA mechanism assignments. PAGE 35 Table 2-2. Expected payoff matrices of th e preference revelation game for Example 2-2 abc acb abc acb bac bca cab cba abc acb bac bca cab cba abc (d1,d2,c3) (d1,e2,f3) (a1,b2,c3) (a1,b2,c3) (a1,c2,b3) (a1,c2,b3) (d1,d2,c3) (d1,d2,c3) (a1,b2,c3) (a1,b2,c3) (a1,c2,b3) (a1,c2,b3) acb (e1,d2,f3) (e1,e2,b3) (a1,b2,c3) (a1,b2,c3) (a1,c2,b3) (a1,c2,b3) (d1,d2,c3) (d1,d2,c3) (a1,b2,c3) (a1,b2,c3) (a1,c2,b3) (a1,c2,b3) bac (b1,a2,c3) (b1,a2,c3) (c1,b2,a3) (c1,b2,a3) (b1,c2,a3) (b1,c2,a3) (b1,a2,c3) (b1,a2,c3) (c1,b2,a3) (c1,b2,a3) (b1,c2,a3) (b1,c2,a3) bca (b1,a2,c3) (b1,a2,c3) (c1,b2,a3) (c1,b2,a3) (b1,c2,a3) (b1,c2,a3) (b1,a2,c3) (b1,a2,c3) (c1,b2,a3) (c1,b2,a3) (b1,c2,a3) (b1,c2,a3) cab (c1,a2,b3) (c1,a2,b3) (c1,b2,a3) (c1,b2,a3) (c1,b2,a3) (c1,b2,a3) (c1,a2,b3) (c1,a2,b3) (c1,b2,a3) (c1,b2,a3) (c1,b2,a3) (c1,b2,a3) cba (c1,a2,b3) (c1,a2,b3) (c1,b2,a3) (c1,b2,a3) (c1,b2,a3) (c1,b2,a3) (c1,a2,b3) (c1,a2,b3) (c1,b2,a3) (c1,b2,a3) (c1,b2,a3) (c1,b2,a3) bac bca abc acb bac bca cab cba abc acb bac bca cab cba abc (e1,e2,b3) (e1,e2,b3) (a1,b2,c3) (a1,b2,c3) (a1,c2,b3) (a1,c2,b3) (e1,e2,b3) (e1,e2,b3) (a1,b2,c3) (a1,b2,c3) (a1,c2,b3) (a1,c2,b3) acb (e1,e2,b3) (e1,e2,b3) (a1,b2,c3) (a1,b2,c3) (a1,c2,b3) (a1,c2,b3) (e1,e2,b3) (e1,e2,b3) (a1,b2,c3) (a1,b2,c3) (a1,c2,b3) (a1,c2,b3) bac (b1,a2,c3) (b1,a2,c3) (a1,b2,c3) (a1,b2,c3) (b1,c2,a3) (b1,c2,a3) (b1,a2,c3) (b1,a2,c3) (a1,b2,c3) (a1,b2,c3) (b1,c2,a3) (b1,c2,a3) bca (b1,a2,c3) (b1,a2,c3) (c1,b2,a3) (c1,b2,a3) (b1,c2,a3) (b1,c2,a3) (b1,a2,c3) (b1,a2,c3) (a1,b2,c3) (a1,b2,c3) (b1,c2,a3) (b1,c2,a3) cab (c1,a2,b3) (c1,a2,b3) (c1,b2,a3) (c1,b2,a3) (c1,a2,b3) (c1,a2,b3) (c1,a2,b3) (c1,a2,b3) (c1,b2,a3) (c1,b2,a3) (c1,a2,b3) (c1,a2,b3) cba (c1,a2,b3) (c1,a2,b3) (c1,b2,a3) (c1,b2,a3) (c1,a2,b3) (c1,a2,b3) (c1,a2,b3) (c1,a2,b3) (c1,b2,a3) (c1,b2,a3) (c1,a2,b3) (c1,a2,b3) cab cba abc acb bac bca cab cba abc acb bac bca cab cba abc (d1,d2,c3) (d1,d2,c3) (a1,b2,c3) (a1,b2,c3) (a1,b2,c3) (a1,b2,c3) (d1,d2,c3) (d1,d2,c3) (a1,b2,c3) (a1,b2,c3) (a1,b2,c3) (a1,b2,c3) acb (d1,d2,c3) (d1,d2,c3) (a1,b2,c3) (a1,b2,c3) (a1,b2,c3) (a1,b2,c3) (d1,d2,c3) (d1,d2,c3) (a1,b2,c3) (a1,b2,c3) (a1,b2,c3) (a1,b2,c3) bac (b1,a2,c3) (b1,a2,c3) (a1,b2,c3) (a1,b2,c3) (b1,a2,c3) (b1,a2,c3) (b1,a2,c3) (b1,a2,c3) (a1,b2,c3) (a1,b2,c3) (b1,a2,c3) (b1,a2,c3) bca (b1,a2,c3) (b1,a2,c3) (a1,b2,c3) (a1,b2,c3) (b1,a2,c3) (b1,a2,c3) (b1,a2,c3) (b1,a2,c3) (a1,b2,c3) (a1,b2,c3) (b1,a2,c3) (b1,a2,c3) cab (b1,a2,c3) (b1,a2,c3) (a1,b2,c3) (a1,b2,c3) (d1,d2,c3)1 (a1,b2,c3) (b1,a2,c3) (b1,a2,c3) (a1,b2,c3) (a1,b2,c3) (d1,d2,c3)1 (a1,b2,c3) cba (b1,a2,c3) (b1,a2,c3) (a1,b2,c3) (a1,b2,c3) (b1,a2,c3) (a1,b2,c3) (b1,a2,c3) (b1,a2,c3) (a1,b2,c3) (a1,b2,c3) (b1,a2,c3) (a1,b2,c3) 35 Between i1 and i2, whoever wins the lottery is assigned to school a whereas the loser is assigned to school b under these strategies: i1: abc i2: abc i3: abc ; i1: axx i2: axx i3: acb ; i1: axx i2: axx i3: cxx Under the strategy set i1: acb i2: abc i3: abc if i1 wins the lottery, assignments are ( i1, a ), ( i1, b ) and ( i3, c ). If i2 wins the lottery, assignments are ( i1, c ), ( i2, a ) and (i3, b ). Under the strategy set i1: abc i2: acb i3: abc if i1 wins the lottery, assignments are ( i1, a ), ( i2, c ) and (i3, b ). If i2 wins the lottery, assignments are (i1, b ), ( i2, a ) and ( i3, c ). Under the strategy sets i1: acb i2: acb, i3: abc and i1: axx i2: axx i3: bxx if i1 wins the lottery, assignments are ( i1, a ), ( i2, c ) and (i3, b ). If i2 wins the lottery, assignments are ( i1, c ), ( i2, a ) and (i3, b ). PAGE 36 Table 2-3. E xpected payoff matrix for Example 2-4 abc acb bac bca cab cba abc (a1,b2,c3) (a1,c2,b3) (a1,b2,c3) (a1,b2,c3) (a1,c2,b3) (a1,c2,b3) acb (a1,b2,c3) (a1,c2,b3) (a1,b2,c3) (a1,b2,c3) (a1,c2,b3) (a1,c2,b3) bac (b1,e2,e3)1 (b1,e2,e3)1 (c1,b2,a3) (c1,b2,a3) (b1,c2,a3) (b1,c2,a3) bca (b1,e2,e3)1 (b1,e2,e3)1 (c1,b2,a3) (c1,b2,a3) (b1,c2,a3) (b1,c2,a3) cab (c1,d2,d3)2 (c1,d2,d3)2 (c1,b2,a3) (c1,b2,a3) (b1,c2,a3) (b1,c2,a3) cba (c1,d2,d3)2 (c1,d2,d3)2 (c1,b2,a3) (c1,b2,a3) (b1,c2,a3) (b1,c2,a3) Under the strategy set i1: bxx, i2: axx, between i1 and i2, whoever wins the lottery is assigned to school a whereas the loser is assigned to c Under the strategy set i1: cxx i2: axx, between i1 and i2, whoever wins the lottery is assigned to school a whereas the loser is assigned to b. 36 PAGE 37 CHAP TER 3 EQUAL TREATMENT AS A MEANS OF EVALUATING PUBLIC SCHOOL ASSIGNMENT MECHANISMS Introduction Open enrollment programs such as inter-distr ict and intra-district school choice, which allow parents to send their children to public schools outside of the neighborhood they reside, have become increasingly popular in the United St ates during the last tw o decades. As of 2005, 27 states had passed legislation mandating school districts to implement intra-district school choice, and 20 states had mandated the school districts within thei r boundaries to participate in the inter-district choice program of the state (ECS, 2005). There is also an increasing trend in the percentage of households part icipating in open enrollment programs. Between 1993 and 2003, the percentage of students atte nding a public school other than their assigned neighborhood schools increased from 11% to 15.4% in the United States (NCES, 2006). In the ideal setting, absent frictions, open enrollment programs allow parents to send their children to any public school with in the boundaries of a region that contains, but is not limited to, the households neighborhood. In this scenario, public school assignments are trivial; each student is assigned to the public school of her choice within these boundaries. However, in practice, parents are typically limited in their public school choices by non-boundary constraints, especially public school capacities. The presence of such constrai nts necessitates other parents public school preferences to be taken into account in order to determine the public school assignment of a given student, which turns the public school assignment into a complicated problem and obligates the school districts to employ centralized assignment mechanisms. These assignment mechanisms have so far been evaluate d in the economics literature along three major dimensions: 37 PAGE 38 1. STRATEGY-PROOFNESS. A preferred public school assignm ent mechanism avoids creating incentives for parents to play complicated game s. Hence, truthful parental ranking of schools should be a dominant strategy. Strategy-proofness of the assignment mechanism is then desirable. 2. STABILITY. An assignment set is defined to be stable if there is no school-student pair ( i ,s ) such that student i prefers school s to her current assignment and either school s prefers student i to at least one of th e students assigned to it or school s has at least one empty seat. Absent stability, there exists justified envy in the assignments, providing incentives for parents to seek legal action to overturn assignment decisions. 3. EFFICIENCY. For the public school assignment problem in the context of this study, only the welfare of students is consider ed for Pareto efficiency, since the schools are regarded as objects to be consumed by students.1 Pareto efficient assignments are obviously desirable. One of the most commonly used student assignment mechanisms is the Boston mechanism, so named because of its use until rece ntly in Boston. This mechanism is still being used in other major school districts including Cambridge (MA), Charlotte (NC), Denver (CO), Hillsborough (Tampa, FL), Miami-Dade (FL), Minneapolis (MN), Seattle (WA) and Pinellas (St.Petersburg, FL).2 Despite its common use, ironically, th e previous literatu re has shown that the Boston mechanism fails to satisfy any of th e aforementioned properties of a well-behaving assignment mechanism in practice.3 Given these results, two altern atives have been proposed to replace the Boston mechanism: the Gale-Shapl ey Deferred Acceptance (GS-DA) mechanism, which, in fact, replaced the Boston mechanism in Boston Public Schools in 2006, and the Toptrading Cycles (TTC) mechanism. Both of these alternatives have been shown to dominate the 1 For the public school assignment problem discussed in this study, priority categories mandated by the school districts are employed alon g with student preferences to determine the public school assignments. Since these rankings do not necessarily correspond to schools pref erences, only the students preferences are considered for efficiency. On the contrary, there are cases such as th e high school assignments in NYC where schools determine their own priority rankings. In that case, school preferen ces as well as student preferences might be taken into account for welfare considerations. 2 In 2003, over one million students were enrolled in public schools within the boundaries of these school districts. The assignments for a significant number of public schools are determined using the Boston mechanism in these districts. For instance, in Denver, th is mechanism is used for traditional public school assignments (approximately 79% of all public schools), whereas, in Seattle, the assignments for every public school in the district are decided with the use of the Boston mechanism. 3 See Abdulkadiroglu and Sonmez (2003) and (Ozek, 2009). 38 PAGE 39 Boston m echanism on important dimensions such as strategy-proofness.4 Therefore, based on the established criteria to evaluat e public school assignment mechanisms society might significantly gain by transitioning from the Boston mechanism to either one of the alternatives. Given these findings, there has been an incr easing curiosity among re searchers in recent literature as to why the Boston me chanism is still being used in almost all of the aforementioned major school districts. Pathak and Sonmez (forthcoming) conclude by stating that: The reason why the mechanism (the Boston mechanism) continues to be used is a puzzle, which might be related to the anal ysis here. Chubb and Moe (1999) argue that important stakeholders often c ontrol the mechanisms of reform in education policy. In the context of student assignment mechanisms, th e important stakeholders may be strategic parents who have invested en ergy in learning about the mechanism, and the choice of the Boston mechanism may reflect their preferences. This study sheds light on this mystery by introducing a new dimension of merit and evaluates the Boston mechanism and the alternatives proposed in the recent literature along this dimension, namely equal treatment. A direct im plication of the Equal Protection Clause of the 14th Amendment in public school assignment context would seem to require that students with the same public school preferences and in public schools same priority categories must be treated equally.5 In the presence of binding public sc hool capacity constraints, a weaker application would require that if two students w ho are in the same priority category for a given school truthfully reveal the same public school preferences with that school as their first choices, the assignment mechanism imply an equal probability of assignment to the school of interest. 4 While both alternatives achieve stra tegy-proofness, there is a trade-off be tween stability and Pareto efficiency when different schools have different priority rankings of students. The GS-DA mechanism attempts to assign each student to her highest possible public school choice for which she has high enough priority to be assigned. Thus, this mechanism produces stable assignments, though Pareto ef ficiency is not guaranteed. On the other hand, the TTC mechanism trades priorities of students among themselves starting with the students with highest priorities, producing Pareto efficient, but not necessarily stab le assignments. See Abdulkadiroglu and Sonmez (2003), Abdulkadiroglu et. al. (2005), Abdulkadiroglu et. al. (2006). 5 The legal precedent is further discussed below. 39 PAGE 40 Generalizing this weak applicati on of equal protection, this study presents a new criterion, which I refer to as equal treatm ent of the equal to evaluate public school assignment mechanisms. Evaluating the Boston mechanism and the two proposed alternative mechanisms along this new dimension, the main finding of this study shows that none of these public school assignment mechanisms satisfy the equal treatme nt criterion except the Boston mechanism. One of these alternatives, namely th e TTC mechanism, is worse yet in the following sense. Assume that two students ( i1 and i2) who are in the same priority category for a given school s reveal the same public school preferences with school s being their first choices. The latter mechanism might result in assignments where even though i1 wins the lottery over i2 for school s i2 gets assigned to school s while i1 is assigned to a less desired school! The literature has demonstrated that the Bo ston mechanism is dominated in important respects by clever alternatives. It is not the intention of this study to show incompatibility between these traditional norms of merit and the equal treatment criterion. In fact, the findings presented in the subsequent sections of this study indicate that, under certain procedural constraints, it is possible for school districts to benefit from the desirable properties of the aforementioned alternatives w ithout suffering the legal conse quences. An important policy implication of this study, to the contrary, is that careful consideration must be given by the school districts if the Boston mechanism is to be replaced by one of th ese alternatives, since serious and costly legal consequences can be anticipated from such a transition under the assignment procedures commonly employed in reality.6 6 In a recent case, in 20 00, parents filed a lawsuit against Seattle Pub lic Schools for using r ace as a tie-breaker to determine the public school assignments, claiming that the tie-breaker violated the Equal Protection Clause of the 14th Amendment (PICS v. Seattle School District No. 1 (05908)). In 2007, the U.S. S upreme Court ruled that race cannot be a factor in assignment of children to public schools in America. The school district hired outside lawyers to work with a team of in-house attorneys for the lawsuit, the legal costs of which totaled approximately $434,000, excluding staff time, over the seven years. The total cost of this case for Seattle Public Schools can potentially exceed $1,000,000 fo llowing a new lawsuit by the parents to recove r their legal costs (Blanchard, Jessica. 2007. 40 PAGE 41 Public School Assignment Proble m and the Assignment Mechanisms In a public school assignment problem, there are n students ( i1, i2,, in) and k public schools ( s1, s2,, sk) each of which has a certain number of seats available (c1, c2,, ck). Public school assignments depend on the students reported preferences the schools priorities over students, and the assignment mechanism. It is assumed that each student has a utility function over the k public schools with strict pr eferences. Students first subm it their preferences, i.e., a strict ranking of the public schools. Public school assignments are then determined based on the set of submitted (ordinal) ranking s. Schools have priority rankings of students, based on broad priority categories mandated by the school district (e.g., residing in a walk zone). A single random lottery is conducted to break the ties between the students in the same priority categories after the students s ubmit their preferences.7 How the submitted preferences and school priorities interact to yield assignments depends on the assi gnment mechanism, which is applied after the random lottery is conducted. I illust rate three of these mechanisms using the following example: Example 3-1: Let n = k = 3 and ( c1, c2, c3) = (1, 1, 1). In other words, suppose that there are three students (i1, i2, i3) and three schools ( s1, s2, s3) each of which has only one seat available. Public school preferences of students and pre-lottery priority rankings of students at each school are given as: i1: s1 > s2 > s3 s1: i3 > i1 = i2 i2: s1 > s2 > s3 s2: i2 > i3 > i1 i3: s2 > s1 > s3 s3: i1 > i2 > i3 Schools legal fight could get more costly. http://seattlepi.nwsource.co m /local/321970_ race30.html accessed 04/ 23/2008). 7 The school districts differ in the ways they use the priority categories along with the lottery outcome to rank the applicants. In Boston, the app licants for a given school are first ranked w ith respect to the priority categories and then the outcome of the lottery is used to rank those within the same priority category. In Miami-Dade, on the other hand, a weighted lottery is conducted where more random numbers are generated for those in higher priority categories. The rankings are then constructed using the best random number for each applicant. In this study, we focus on the former noting that the results also apply to the latter. 41 PAGE 42 where > indicates strict pr eference f or students and higher priority category for schools whereas = indicates that the tw o students are in the same prio rity category for the given school. Boston Mechanism Under the Boston mechanism, a student who is not assigned to his first choice is considered for his second choice only after th e students who ranked that students second choice as their first choices. Formally, the algorithm is as follows: In the first step, only the first choices of students are considered. Based on the schools post-lottery priority rankings of students, the seats at each sch ool are assigned one at a time until either there are no seats left or there is no student left who has listed it as her first choice. In the nth step, only the nth choices of the students who c ould not be placed in the (n-1)st round are considered. Based on the schools postlottery priority ranki ngs of students, the seats at each remaining school are assigned one at a time until either there are no seats left or there is no student left who has listed it as her nth choice. Applying this algorithm to Example 3-1, gi ven that each student reveals their public school preferences truthfully to illustrate, th e Boston mechanism results in the assignments ( i1, s1), ( i2, s3) and (i3, s2) in the state of nature where the tie between i1 and i2 for school s1 is broken in favor of i1 or the assignments ( i1, s3), ( i2, s1) and ( i3, s2) otherwise.8 A detailed description of the application of this mechanism to Example 3-1 and to those mechanisms that follow is provided in Appendix-A. Recent literature has shown that truthful revelation of public school preferences is not necessarily the weakly dominant strategy for eac h parent under the Boston mechanism; strategy8 Notice that truthful revelation for each student might not be a Nash equilibrium strategy set under the Boston mechanism in Example 3-1. For instance, under the assumption that students have perfect knowledge of other students true preferences and the lottery outcome before they submit their school choices, if the tie between i1 and i2 is broken in favor of i1, truthful revelation of all students is not a Na sh equilibrium whereas in the other state of nature, truthful revelation is one of the multiple Nash equilibria. Under a more realistic informational setting where students make their school choices under uncertainty due to the timing of the random tie-breaking in practice, all three students might reveal truthfully depending on the relative like/dis like of each student for each school. Furthermore, the existence of sincere students in reality, wh o always reveal their preferences truthfully, increases the likelihood of truthful revelation (Abdulkadiroglu, Roth, Pathak and Sonmez (2006); Pathak and Sonmez (forthcoming)). 42 PAGE 43 proofness fails. The Boston m echanism then induces parents to play a complicated preference revelation game where the payoffs are determin ed by the preferences of the students over schools, the priorities of schools over students, and the rules of the mechanism. Furthermore, the Nash equilibrium assignments of this game guarantee neither Pareto efficiency nor stability.9 Top-Trading Cycles Mechanism Abdulkadiroglu and Sonmez (2003) show that the TTC mechanism is strategy-proof and Pareto efficient; however, stability is not guaranteed. The formal algorithm works as follows: Step 1: Each student points to her favorit e school and each school points to the highest priority-ranked student. Assign all students in a cycle to th e schools they point to and remove them from the cycle.10 Also remove a school from th e available schools list if its capacity becomes full. Step k: Apply the same algorithm to the remaining students and schools. The process terminates when there are no remaining cycles. When applied to Example 3-1, for the true pref erences which are expressed in equilibrium, the TTC mechanism results in the assignments ( i1, s3), ( i2, s1) and (i3, s2) for both outcomes of the lottery. Gale-Shapley Deferred Acceptance Mechanism Unlike the previous two mechanisms, none of the assignments are guaranteed until the assignment algorithm terminates under this m echanism. The algorithm works as follows: Step 1: Each students first choice is considered. Each school puts al l applicants into its queue unless the number of applicants is highe r than the number of seats available at the school. Otherwise, each school rejects the applicants ranke d lower than its number of empty seats using its post-lottery priority ranking, while placing the rest of the applicants in its queue. 9 See Abdulkadiroglu and Sonmez (2003). While the findings of Ergin and Sonmez (2006) suggest that the Boston mechanism produces stable assignments in equilibrium, Ozek (2009) shows that this result relies on a complete information assumption, which is not satisfied in practice due to the timing of lotteries. 10 A cycle is an ordering of distinct students and schools ( s1, i1, s2, ... sk, ik) where s1 points to i1, i1 points to s2, sk points to ik, ik points to s1. In a public school assignment problem, we know that there is at least one cycle, since the number of students and schools are finite. 43 PAGE 44 Step k: The rejected applicants next choices are considered Comparing the new applicants with the applicants already in th e queue, each school replaces the students on its queue based on its priority rankings. The process terminates when no student is rejected and each student is assigned to the school w hose queue she belongs to when the algorithm terminates. Applying to Example 3-1, again using the tr ue preferences, the GS-DA mechanism yields the assignments ( i1, s3), ( i2, s2) and ( i3, s1) if the tie is broken in favor of i1 or produces ( i1, s3), ( i2, s1) and ( i3, s2) otherwise. Even though it preserves strategy-proofne ss and guarantees stable assignments, the GS-DA mechanism does not necessarily result in Pareto efficient assignments11. Notice that in the state of nature where i1 wins the lottery, the resulting assignment is Pareto dominated by ( i1, s3), ( i2, s1) and ( i3, s2). Table 3-1 summarizes the characteristics of the three assignment mechanisms. The findings in the table lead one to wonder why the ot her major school districts using variants of the Boston mechanism have not yet followed in the footsteps of the Boston Public Schools, which abandoned the Boston mechanism in favor of th e GS-DA mechanism in 2006. The next section presents the main findings and provides a pot ential explanation to this puzzle. Equal Treatment Criterion Absent frictions, public school assignments under an open enrollment regime are trivial; each student is assigned to the public school of he r choice within the bounda ries of a region that contains, but is not limited to, the neighbor hood where the student resides. When capacity constraints result in a scarcity of slots at public schools, school districts need to implement centralized assignment mechanisms as well as criteria to classify the students in order to 11See Dubins and Freeman (1981) and Ro th (1982). Recently, Abdulkadirog lu et. al. (2007) examine the impact of two different tie-breaking methods on the efficiency and strategy-proofness of the GS-DA mechanism; single tie breaking where each student is given a random number to be used at every school and multiple tie-breaking where each student is assigned a different random number to be used at each school. Their main theoretical result indicates that a GS-DA mechanism that uses sing le tie-breaking is not dominated by any other mechanism that is strategyproof for students. In this study, random lottery is single tie-breaking. 44 PAGE 45 determ ine the assignments in public schools where the number of applicants is greater than the number of seats available. For this purpose, broad priority categories (e.g. sibling currently attending the school) are commonly used and the ties between students in the same priority categories are broken using an equal-proba bility random lottery, which preserves the ex ante equivalence of such students, before the as signment algorithm can be applied.12 Based on these priority categories, the true and the submitted preferences of students, I introduce a criterion called equal treatment to evaluate public school assignment mechanisms. For an arbitrary set of submitted st rict student preferences over schools and pre-lottery school priority rankings over students let nSSSS ,,,21 kPPPP ,,,21 prSIt em, denote the set of students who are in a given priority category p for school sm and submit the same public school rankings truthfully SSt e with sm being their rth choices.13 Under these student preferences, pre-lottery school priority rankings, and a given assignment mechanism, let represent the conditional probability of being assigned to her jth choice for an arbitrary student PS ,|jtPr prSIit emt,, .14 12 For instance, in Boston, the following priority categories are used: (1) Students who have siblings currently attending that school and who live in the walk zone of the school; (2) Students who have siblings currently attending that school; (3) Students who live in the walk zone of the school; (4) Students who do not fall into the three categories above. Furthermore, each applicant is assigned a random numbe r, which is used to break the ties between students in the same priority categories when necessary. 13 In practice, the cardinality of some of the sets prSIt em,, is likely to be large for several reasons. First, the number of choices students are allowed to make, as determined by the school districts, is typically small (up to three or five choices). Second, as mentioned earlier, broad priority categories are commonly used in major school districts to determine the public school assignments in over-demanded sc hools. Furthermore, the realistic possibility that many students rank schools based on a common observed school quality hierarchy and the existence of sincere students (see Abdulkadiroglu, Roth, Pathak and Sonmez, 2006; Pathak and Sonmez, forthcoming), who always reveal their preferences truthfully, increases the likelihood that multiple students in the same priority category for a given school truthfully reveal the same public school preferences. 14 Notice that given student preferen ces, school priorities and the assignment mechanism, the public school assignments depend on the outcome of the random lottery. 45 PAGE 46 Definition: For all values of r and p the m embers of the set t eS prSIt em, are identical for school sm and should be treated equally for an assignment in that school. For any arbitrary pair of students prSIiit emut,,, a public school assignment mechanism violates the equal treatment of the equal if and only if PSrPSru,|Pr,|tPr given that for all PSjt,|Pr PSju,|Pr r j .15 First consider the simple case demonstrated by Example 3-1 where students i1 and i2 are in the same priority category for s1 and have the same true pr eferences over schools with s1 as their first choices. Provided that they reveal trut hfully, these two students are equivalent from the point of view of the school di strict for a seat in school s1.16 The equal treatment criterion guarantees that i1 and i2 are provided equal chance of assignment to school s1 for which they are equivalent, regardless of the submitted preferences of all other students. Absent equal treatment, the assignment mechanism discriminates against one of the two equivalent students, making her assignment to school s1 less likely. On the other hand, when r > 1, PSrPSru t,|Pr,|Pr does not necessarily imply that the equal treatment criterion is violated; th e lower probability of assignment to school sm might be the result of that student being assigne d to a public school that she prefers to sm for all 15 It is worth mentioning that this definition of equal treatment introduces an important practical limitation when evaluating assignment mechanisms under which truthful revelation of preferences is not a dominant strategy: a dimension of merit based on ex-ante equivalence of students with identical preferences and identical actions is counterintuitive if the actions of students might change depending on the outcome of the lottery. For instance, in Example-1, assuming that the students know the outcome of the lottery (along with the true preferences of other students) before they submit their preferences, if the tie between i1 and i2 is broken in favor of i1, i1 submitting s1s2s3, i2 submitting s2s1s3 and i3 submitting s1s2s3 is a Nash equilibrium strategy set, whereas in the other state of nature, i1, i2 and i3 might submit (s1s2s3), (s1s2s3) and (s2s1s3) respectively in equilibrium under the Boston mechanism. A stricter definition of equal treatment that re quires identical students to reveal the same preferences truthfully for all states of nature would solve this issue. Notice that the main results extend to this strict definition of equal treatment. Furthermore, random tie-breaking commonly takes place after students submit their preferences in major school districts, eliminating the aforementioned possibility in reality. 16 See footnote 8 for a detailed discussion of truthful revelation in Example 3-1. 46 PAGE 47 outcom es of the lottery (i.e. 1,|Pr PSju for some r j which implies that ). The latter condition in the equal treatm ent criterion ru les out this possibility by imposing that all members of the set 0,|Pr PSru p rSIt em, have equal probability of assignment to each public school they rank higher than sm given S and P Hence, 1,|Pr PSju if and only if for some 1,|Pr PSjt r j which in turn implies that Pr,|0,| Pr PSr PSru t. Proposition 3-1: Among the three aforementione d public school assignment mechanisms, the Boston mechanism is the onl y one that satisfies the equal treatment requirement.17 Proof: Example-1 is sufficient to show that neither of the two strategy-proof alternatives mentioned earlier satisfies th e equal treatment criterion.18 In this case, in order to comply with this requirement, an assignment mechanism needs to provide students i1 and i2 equal probability of assignment to school s1, since i1 and i2 report s1, for which they are equivalent, as their first choices. Table 3-2 presents these probabilities as well as the public school assignments for the two possible outcomes of the lott ery under the two mechanisms. Under both of thes e alternatives, i1 has no chance of being assigned to s1 whereas the equivalent student i2 is guaranteed a seat in s1 under the TTC mechanism and has 0.5 chance of being assigned to s1 under the GS-DA mechanism. Furthermore, the TTC mechanism is worse 17 This statement extends to a weaker definition of equal treatment under which identical students for a given school are defined as those in the same priority category and subm it the same preferences with that school as one of their choices. On the other hand, if identical students were defi ned solely based on priorities and true preferences, Boston mechanism would fail to satisfy the equal treatment criteri on as well as the two alternatives (formal proof is available upon request). However, it is worth noting that these alternative definitions become questionable if the assignment mechanism induces gaming: under the former alternative, identical students might have different true preferences even though they submit the same school pr eferences whereas under the latter definition, identical students might have different actions. 18 Example 3-1 illustrates a case where th e number of public schools equals th e number of choices each parent can make, which is typically pre-determined by the school distri ct. It is worth noting that th e analysis extends to cases where the number of public schools exceeds the allowed number of choices. 47 PAGE 48 yet in the following sense. The latter m ech anism results in assignments where even though i1 wins the lottery over i2 for s1, i2 is assigned to s1. The GS-DA mechanism, on the other hand, avoids such cases by assuring stable assignments.19 I now show that the Boston mechan ism satisfies equal treatment. Let denote the step of the B oston mechanism in which student ta prSIt em, is assigned to a public school. Under the Boston mechanism, since only the nth choices of the remaining stude nts are considered in the nth step, we know that PSr ,| aPSrtt tPr,|Pr for all possible values of r When r > 1, suppressing S and P, this can be written as: 1 Pr*1|Pr Pr rarararatt ttt tt (3-1) If for some 1Pr jt r j then the second probability on the right-hand side of Equation 3-1 equals zero. This implies that 0,|Pr,|Pr PSrPSru t for an arbitrary pair of students prt e,, SIiimut, since juPr jtPr for all r j as required by the equal t condition. reatment On the other hand, if 1,|Pr PSjt for any r j the second probability on the righthand side of Equation 3-1 is non-zero, and Equation 3-1 can be written as: 1 21Pr1*1|Pr1*1|Pr Prr j tt ttt ttt tta jaja rarara (3-2) Lemma: 1|Pr1|Pr jaja jajau uu ttt for all r j Proof: Consider 1|Pr jajattt when j = 2: 19 Since the GS-DA mechanism guarantees stable assignments with respect to post-lotte ry priority rankings, there can not exist a school-student pair such as ( i1, s1) where i1 strictly prefers s1 to her current assignment and i1 has higher priority than at l east one student assigned to s1 ( i2) with respect to post-lottery priority rankings. 48 PAGE 49 1|2Pr 1Pr1 2Pr 1Pr1 2Pr 1|2Pr u uu uu uu tt tt tttaa a a a a aa since for all jju tPrPr r j Likewise, when j = 3, 2|3Pr 1Pr11|2Pr1 3Pr 2|3Pr u uu tt ttt tt tttaa a aa a aa since for all jju tPrPr r j and 1|2Pr1|2Pr u uu tttaa aa. By iteration, one can eas ily show that 1|Pr1 | Pr j att ajtjajau uu for all r j Therefore, following Equation 3-2, in order to prove that the Boston mechanism satisfies the equal treatment condition, we only need to show that 1|Pr1|Pr rara rarau uu ttt Provided that and 1 rat1 rau ti there are two types of students who will be considered for a seat in school sm prior to and under the Boston m echanism: ui 1. The students who rank sm higher than r (if r > 1). 2. The students who are in a hi gher priority category than p for sm and rank school sm as their rth choices. Given S and P let denote the number of remaining seats at school sm in the rth step of the Boston mechanism after the two types of students satisfying the latter criteria are assigned, and rcm prSInt em, denote the number of members of the set prSIt em, ,. There are three cases to consider for equal treatment as determined by the relative values of rcm and prSInt em, ,. In the extreme cases where or rcm0 prSInrct em m,, 1 we know that 1 Pr |Pr | ru auu at rarattr under the Boston mechanism, since neither of the tw o students will be assigned to sm if the former condition is satisfied 49 PAGE 50 01|Pr1|Pr rara rarau uu ttt 11|Pr1|Pr whereas both will be guaranteed a seat in sm rara rarau uu ttt under the latte r condition. On the other hand, if prSInrct em m,, 0 the outcome of the random lottery will determine which member(s) of prSIt em,, will be assigned to sm. The equal-probability random lottery implies that there are prSInt em, ,! equally-likely ways to break the ties between the members of prSIt em, ,. For a given outcome of the lottery, let prSIiRt emt,, | represent the post-lottery ranking of student prSIit emt,, among the members of prSIt em,, Under the Boston mechanism, will be assigned to sm only if ti rcprm ,, SIt em| iRt Therefore, prSIn rc rcprSIiRraram t emt t tt,,|Pr1|Pr PSrPSru t,|Pr,|Pr t em m,, is independent of which shows that for an arbitrary pair of students ti prIiimut, St e, ; and com pletes the proof. Transition and Consequences Recent literature has indicated several aspects along which the Boston mechanism is dominated by the proposed alternatives as summ arized in Table 3-1. Following such findings, Boston Public Schools (BPS) abandoned the Bo ston mechanism in favor of the GS-DA mechanism in 2006. Among the desirable features of the GS-DA mechanism, strategy-proofness played a key role in the decision of BPS as suggested by the following memorandum of Superintendent Thomas W. Payzant on May 25, 2005:20 The most compelling argument for moving to a new algorithm is to enable families to list their true choices of sc hools without jeopardizing their chances of being assigned to any school by doing so. We know that the current algorithm (Boston mechanism), because 20 The memorandum, in its entirety, is available upon request. 50 PAGE 51 it pr ioritizes families first choices, requires fa milies to be strategic in their selection of school preferences The findings presented in the previous secti on, on the other hand, suggest that such desirable properties come with a burden. A trans ition from the Boston mechanism to either of the alternatives proposed in the re cent literature might create an arbitrary dis tinction between equivalent students, some of whom might be harmed as a result of this policy change. For instance, in Example 3-1, provide d that they reveal their prefer ences truthfully, as a result of such a transition, student i1 would be obligated to sacrif ice her right to attend school s1 and be worse-off in order for the school district to benefi t from the desirable features of the alternative, whereas i2 would be rewarded with hi gher probability of assignme nt to her first choice. The use of broad priority cat egories along with random lottery results to rank students is the main reason behind this artificial classification between identical students. Therefore, there are two paths school districts can pursue in order to benefit from the desirable features of the alternative mechanisms while a voiding such classifications. In one extreme, school districts can choose cr iteria that strictly rank students at each school and thus eliminate the random lotteries.21 Then, trivially, equal treatment is satisfied since no two students can be ranke d the same by any school. A common example of this practice is the use of proximity-based measures such as d istance from the applicants primary address to the school, which reduces the lik elihood of two students falling in to the same priority category and thus minimizes the need to implement random lotteries, as evidence d in Seattle Public Schools (WA) and Pinellas County (FL).22 If ranking students on a m easured ability scale is 21 We know that the desirable properties of the alternativ e mechanisms are not reliant on the choice of priority categories. Hence, if school districts us e priority categories that lead to stri ct rankings of students, the strategyproofness, stability and/or efficiency proper ties of these mechanisms will still prevail. 22 Pinellas County School Board uses the shortest driv ing distance from the applican ts primary address to the public school computed to the nearest hundredth of a mile whereas Seattle Public Schools employ the straight-line 51 PAGE 52 socially acceptable, with the sam e ranking at each school, another exam ple of this approach arises. At the other extreme, school districts can avoid the aforementi oned classification by completely eliminating priority categories a nd conducting a single random lottery, which breaks the ties between all students in the same way for each school, to determine the public school assignments. Proposition 3-2: Given that each school has the same priority ranking of students as determined by a single random lottery, the two alternative mechanisms satisfy the equal treatment condition. Proof: In the case where each school has the same priority ranking, the two alternative mechanisms produce the same assignments as the random serial dictato rship (RSD) mechanism which works as follows: order all students with a random lottery and assign the first student to her first choice, the next stude nt to her top choice among the remaining slots, and so on. 23 Therefore, it suffices to show that the random se rial dictatorship mechan ism satisfies the equal treatment requirement. In the absence of priority categories, rSIprSIt em t em,,, represents the set of students who reveal the same public sc hool preferences truthfully ( Se) with school sm as their rth choices. For a given outcome of the lo ttery, an arbitrary student, rSIit emt, will be assigned to her rth choice if and only if the number of students who ha ve higher priority rank ings and are assigned distance from the primary address to the public school as a criterion to determine public school assignments. In both cases, the student living closer to the public school is given higher priority. 23 A detailed explanation is provided in Appendix-B. We know that the RSD mechanism is strategy-proof and produces Pareto efficient assignments (Abdulkadiroglu an d Sonmez, 2003). Furthermore, the resulting assignments will be stable with respect to reveal ed student preferences and the post-l ottery priority ranking. Therefore, employing a single random lottery along with the alte rnative mechanisms to determine the public school assignments preserves the appeali ng features of the alternatives. 52 PAGE 53 to sm is less than the number of seats available at sm ( cm). However, notice that this is a symmetric problem for all members of rSIt em, since, ex ante, all students have the same priority at each school and all possible priority ra nkings of students are equally-li kely. Thus, the probability of assignment to school sm for it depends on r the number of members of the set rSIt em, rSInt em, the number of seats available at school sm ( cm), the number of students ( n ), and the revealed public school pr eferences of students ( S ), none of which is stude nt-specific given that the student is a member of rSt em, I Therefore, each member of rSIt em, has equal probability of assignment to school sm under the RSD mechanism. While not problem free, these assignment pr ocedures suggest that school districts might benefit from the appealin g features of the alternative assignment mechanisms without violating equal treatment. The equilibrium implications di ffer markedly across the alternatives however. Using proximity as a criterion to determine public school assignments counteracts the main objective of open enrollment programs by implicit ly reducing the number of feasible school choices available to parents. C onsider the extreme case where pa rents share a common perceived quality hierarchy of schools. Then, with the pr oximity criterion, housing prices would ultimately conform to the hierarchy, households would sort by income and preference for school quality, and a neighborhood schooling system would effectively emerge. If schools instead rank students by a standardized ability measure, again assuming a common ranking of school qualities, then ability stratification across the hierarchy can be predicted. In the alternative with school rankings based solely on a lottery, perhaps the purest form of school choice, one can predict a representative cross section of students across a given school qua lity hierarchy. Hence, the choice of the assignment procedure has prof ound implications for access to schools along the quality hierarchy so that the pr eferred approach requires an e xpression of social preferences. The 53 PAGE 54 results, on the other h and, suggest that a transition from the Boston mechanism to one of the aforementioned alternatives in a hybrid assignmen t process where broad priority categories along with random lottery results are used, such as the recent transition in Boston Public Schools, can be expected to induce legal challenges. Concluding Remarks One of the most commonly exercised forms of school choice is the open enrollment program, which allows parents to send thei r children to public schools outside of the neighborhoods they reside. When cap acity constraints result in a scarcity of seats at public schools, such programs necessitate the implemen tation of centralized public school assignment mechanisms, among which the Boston mechan ism is commonly implemented by the school districts. Despite its co mmon use, previous literature has shown that the Boston mechanism is dominated by clever alternatives along almost a ll of the desirable featur es of a well-behaving assignment mechanism, leading to increased curi osity among researchers in the recent literature as to why none of the school di stricts using variants of the Bo ston mechanism have yet followed in the footsteps of Boston Public Schools, wh ich abandoned the Boston mechanism in favor of one the alternatives in 2006. This study adds to this debate by introducing a new criterion, which I refer to as the equal treatment requirement, to evaluate public school assignment mechanisms. The main finding provides a possible explanation to the persiste nce of the Boston mechanism in major school districts, showing that none of the assignment mechanisms proposed in the recent literature as replacements for the Boston mechanism satisfy a direct implication of the Equal Protection Clause of the 14th Amendment, which is satisfied by the Boston mechanism. The first section of the 14th Amendment, which is co mmonly known as the Equal Protection Clause, reads in part as follows: 54 PAGE 55 No State shall deny to any person within its jurisdiction the equal protection of the law s. Over the recent decades, the Supreme Cour t has developed a three-tiered-scrutiny approach to analysis under th e Equal Protection Clause. Even though education is typically not considered as a fundamental right at the federal-level as ev idenced in San Antonio School District v. Rodriguez, there have been numer ous state-level cases over the last four decades where the Court ruled that educ ation is a fundamental right an d classifications burdening ones right of education falls under th e strict scrutiny category, which requires the government (i.e. the school district in this case) to show that the challenged classification serves a compelling state interest and that the classification is necessary to serve that interest. 24 However, from the discussion in the previous secti on, we already know that the ar tificial classification between identical students, which would arise as a resu lt of a transition from the Boston mechanism to one of the alternatives, is not necessary to achieve the desirable f eatures of the alternative assignment mechanisms. Previous literature has convincingly indicated that, given the established criteria to evaluate public school assignment mechanisms in the economics literature, it is socially beneficial to abandon the Boston mechanism in fa vor of one of the propos ed alternatives in major school districts. An important policy impli cation of the findings pres ented in this study, on the contrary, is that such a transition might initiate serious legal consequences. 24 San Antonio School District v. Rodriguez, 411 U.S. 1 (1973). The state-level cases include Alabama Coalition for Equity v. Hunt (Ala. Circ. Ct. 1993); Serrano v. Priest, 5 Cal. 3d 584, 487 P.2d 1241 (1971); Horton v. Meskill, 172 Conn. 615, 376 A.2d 359 (1977); Milliken v. Green, 389 Mich. 1, 203 N.W.2d 457 (1972); Richland County v. Campell, 294 S.C. 346, 364 S.E.2d 470 (1988); Pauley v. Kelly, 162 W.Va. 672, 255 S.E.2d 859 (1979); Kukor v. Grover, 148 Wis.2d 469, 436 N.W.2d 568 (1989); Washakie County School District No.1 v. Herschler, 606 P.2d 310 (Wyo. 1980). In perhaps the most prominent of such state-level cases, commonly referred to as Serrano I the California Supreme Court found that property-wealth based funding of education violates both state and federal equal protection, both because educatio n is fundamental and because wealth is a suspect cat egory (Enrich, 1995). 55 PAGE 56 Table 3-1. C haracteristics of the assignment mechanisms Boston TTC GS-DA Strategy-proof Stable Pareto Efficient Ozek (2009) shows that the Boston mechanis m will fail to produce stable assignments, since the complete information assumption is violated in practice. 56 PAGE 57 57 Table 3-2. Public school assignments and the assignment probabilities in Example 3-1 Lottery winner Assignment probability to s1 i1 i2 i1 i2 TTC (i1, s3), ( i2, s1), ( i3, s2) (i1, s3), ( i2, s1), ( i3, s2) 0 1 GS-DA (i1, s3), ( i2, s2), ( i3, s1) (i1, s3), ( i2, s1), ( i3, s2) 0 0.5 PAGE 58 CHAP TER 4 THE EFFECTS OF OPEN ENROLLMENT ON SCHOOL CHOICE AND STUDENT OUTCOMES Introduction Improving the quality of elementary and sec ondary education remains atop the political agenda in the United States, which annually sp ends roughly 1.5 times more money per pupil on primary and secondary education than the aver age member of the Organization for Economic Cooperation and Development (OECD) 1. Yet, the additional resources allocated to education do not fully translate into higher student achieveme nt: the U.S. students perform worse than the OECD averages on international tests in math, reading and science2. Increasing parental choice has been one of th e leading themes of the educational policy implemented to enhance academic achievement in the U.S. during the last two decades. The main objective of such policies is to level th e playing field in term s of access to quality education for disadvantaged st udents who cannot otherwise affo rd the higher-quality schooling options. Along these lines, open enrollment programs such as inter-district and intra-district school choice, which allow parents to send thei r children to public schools outside of the neighborhoods in which they reside, have beco me increasingly popular. As of 2005, 27 states had passed legislation mandating sc hool districts to implement in tra-district school choice, and 20 states had adopted legislation mandating that school di stricts participate in the inte r-district choice program of their state (ECS 2005). There is also an increasing trend in the percentage of households participating in open enrollment pr ograms. Between 1993 and 2003, the percentage 1 OECD (2008). In 2004, the per-pupil spending on primary (secondary) education in the U.S. was $9,156 ($10,390) compared to the OECD average of $6,252 ($7,804). 2 OECD (2003). In 2003, as part of the Programme for International Student Assessment (PISA), OECD tested the 8th graders in member countries on subj ects including math, reading and science. The average test scores for U.S. students were 483, 491 and 495 in math, science and reading respectively compared to OECD averages of 500, 499 and 494. 58 PAGE 59 of students attending a public school other than their neighborhood schools increased f rom 11% to 15.4% in the United States (NCES, 2006). This study analyzes households response to th e introduction of public school choice in the form of open enrollment in Pinellas County Schools (PCS), one of the la rgest school districts in the U.S., and examines the impact of exerci sing this form of school choice on student test scores. Having abandoned the zoning regime with court-ordered busing, which had been used to prevent racial segregation for more than three decades, and im plemented intra-district school choice in 2003, PCS provides an appealing case to analyze the im pact of increased educational opportunities on households school choice behavior3. Using the entire elementary and middl e school student popul ation attending 4th through 8th grades between 2001 and 2005 in PCS, the results indicate that households reacted strongly to the incentives created by the open enrollment program, leading to significant increases in the rate of students who opt out of their default schools. Among th e transition-grade students (6th graders who transitioned from elementary school to middl e school at the beginning of the school year), the implementation of open enrollment increased th e percentage of students who opt out of their default middle school from 8% to 33%, whereas for non-transition grade students, the opt out rate increased from 7% to 16% in the year fo llowing policy adoption. The findings also reveal significant changes in the com position of opt out students fo llowing the polic y change. The policy change, by reducing the implicit cost of opting out for students, smoothed-out the prior achievement levels of the traveling students, attracting more mediocre students to opt out. 3 In the U.S. context, the focus of the previous literature has been mainly on the impact of increased public school choice on student outcomes and households school choice in a regime where school choice has already been introduced See Cullen, Jacob and Levitt (2005, 2006); Has tings, Kane and Staiger (2008). There are several exceptions in the international context though. An important example is Fiske and Ladd (2000), who examine the impact of a dramatic school choice reform on households behavior in New Zealand. 59 PAGE 60 Having established that households responded to the incentives created by the open enrollm ent program, I then examine the impact of exercising this form of school choice on test scores. By expanding the set of feasible public schools available to e ach household, open enrollment programs might enha nce student achievement in tw o ways. First, students, who cannot otherwise afford higher qua lity schooling options, might be able to attend higher quality public schools or schools that better match their interests a nd needs under the open enrollment regime. Furthermore, if the increasing compe tition among public schools improves the efficiency of the public provision of education, ope n enrollment programs will enhance student achievement by increasing the overall quality of public education. The extent to which open enrollment improves student achievement relies on households willingness and ability to send their children to higher quality public schools in the presence of open enrollment4. However, testing these predictions has been proven difficult due to the highly selective nature of opting out. In other words, if those who opt out of their default public schools differ from their peers who stay behind along unobservable characteristics such as intrinsic motivation to excel, traditional ordinary least-squares approach fails to provide unbiased estimates of the causal relationship between opting out and st udent achievement. A recent body of research makes use of randomized lotteries, which are commonly employed by school districts and schools to determine the assignments in oversubsc ribed public schools, to deal with this issue5. 4 In the ideal setting, absent frictions, open enrollment pr ograms allow parents to send their children to any public school within the boundaries of a region that contains, but is not limited to, the households neighborhood. However, in practice, parents are typically limited in their public sc hool choices by non-boundary constraints, especially public school capacities, restricting households ability to send th eir children to higher quality public schools. Furthermore, households might place more we ight on non-academic characteristics of pub lic schools such as proximity, limiting the competitive pressure public schools face under ope n enrollment (Hastings, Kane and Staiger, 2008). 5 Some examples in the open enrollment context are Cullen, Jacob and Levitt (2006); Cullen and Jacob (forthcoming); Hastings, Kane and Staiger (2008). Hastings and Weinstein (forthcoming), on the other hand, use natural and field experiments in which some parents are ra ndomly provided information about school quality. They find that those who receive th e information are more likely to send their children to higher quality schools and those who attend higher quality schools perform better on standardized tests. 60 PAGE 61 Com paring the student outcomes between the lott ery-winners and lotterylosers, these studies typically find no significant be nefit of attending selective public schools on stude nt test scores6. However, these estimates will not necessarily re flect the true impact of exercising the school choice provided by open enrollment on student outc omes for the entire student body if those who participate in lotteries differ fr om the entire student population7. Cullen, Jacob and Levitt (2005), on the ot her hand, employ instrumental variables approach to estimate the causal relationship betw een opting out of the assigned public school and student outcomes. Using proximity to the closest public alternative as an instrument for opting out, their results reveal that, other than for st udents who opt out to high school career academies, there is no significant impact of opting out of th e assigned high school at the end of 8th grade on the probability of dropping-out during the high school years. Using proximity to the relevant public alte rnatives as an instrument for opting out of the default public school, I estimate the impact of opting out on stude nt test scores for elementary and middle school stud ents between grades 4 and 8. Despite the similar use of proximity for identification, this study extends Cullen, Jaco b and Levitt (2005) along two important dimensions. First, I am able to use te st scores as the outcome of interest, since I eliminate the selection problem caused by dr op-outs by excluding high school students. Moreover, the findings presented in this study provide a more comp lete picture about the impact of exercising this form of sc hool choice on student outcomes, si nce the dataset I employ enables 6 Even though no significant effect of winning the lottery on the average lottery participant is the main conclusion of all these studies, some studies find significant benefits of opting out for certain subgroups. For instance, Hastings, Kane and Staiger (2008) find that children of parents with strong preference for academic quality experience significant gains in test scores as a result of attending their chosen school, while children whose parents weighted academic characteristics less heav ily experience academic losses. 7 Randomized lotteries become necessary when there are mo re applicants than the numb er of seats available at a given school. If the demand for a public school is correlated with the schools quality, then lotteries will take place more frequently at higher quality public schools. Therefore, it is quite likely th at the lottery participants have higher tastes for quality education than non-participants. 61 PAGE 62 m e to analyze the impact of non-transition gr ade opting out as well as transition grade opting out on test scores. The findings reveal no significant benefit of opting out on student test scores and that the students who opt out of their de fault schools often perform significantly worse in reading than similar students who stay: the av erage traveling student scores r oughly one-quarter of a standard deviation lower in reading. Give n the substantially different nature of opting out for transition grade students and non-transition grade students, I further disaggregate the analysis into these two groups. The IV analysis on th e two sub-samples indicate that the detrimental impact of opting out on reading scores for the entire sample is mainly driven by the non-transition graders. The transition-grade students, on the other hand, neither bear a ny significant costs nor benefit from opting out of thei r assigned middle schools. There are several competing mechanisms th rough which opting out might affect student achievement in a negative way. One explanation is that frictions such as binding public school capacity constraints limit the ability of those w ho opt out to exercise higher quality schooling options. Comparisons between the default and targ et schools during the sc hool year before the opt-out reveal that the traveli ng students did not expe rience significant change s in school quality compared to their peers who stayed behind in our sample.8 Moreover, opting out might have deteriorating effects on traveling students achievement levels if being an outsider at the new school leads to a decline in the intr insic motivation of the students. A direct implication of the outsider effect is that thos e who opt out closer to the terminal grade of the school level will experi ence higher achievement losses, since the lack of time and incentives to become an insider might tr anslate into more severe declines in intrinsic 8 For instance, the average gain in t eacher experience for the traveling studen ts is 0.1 years, whereas the average teacher experience in the entire sample is 13.2 years. 62 PAGE 63 motivation. Sim ilarly, keeping the proximity to the terminal grade of the school level constant, elementary school students are expected to suffer more from non-transition grade opting out, since their new peers at the target school spe nd more time together, making it harder for the traveling students to become insiders. Finally, if getting used to the new school environment is positively correlated with students intrinsic motivation, one would expect to see an improvement in student achievement at th e end of the second y ear after opting out. The results provide evidence s upporting the first two implicati ons of the outsider effect: those who opt out two years before the terminal grade of the school leve l benefit significantly (one-third of the standard deviat ion) in terms of math scores whereas opting out one year before the terminal grade is associated with significant declines in bot h reading and math test scores. Furthermore, comparing the impact of non-tran sition grade opting out between elementary and middle school students, the result s indicate that the former gr oup suffers significantly from nontransition grade opting out in terms of both ma th and reading scores, whereas there is no statistically significant impact for the latter group. However, I find no improvement in the achievement levels of the trave ling students over time: the short-run detrimental effects of opting out on student achievement persist at the end of the second year after opting out. Finally, I estimate the impact of opting out on the disadvantaged students, the target student group of most school choice reforms, as determined by the poverty level and the performance of their default public schools. The results indicate that opting out of a highperforming or a low-poverty default school lead s to a significant decline in reading scores whereas the disadvantaged students neither suffer nor benefit from opting out. Furthermore, the results reveal that this differe nce can not be explained by the di fferential gains/losses in school quality experienced by these two groups. 63 PAGE 64 The analysis proceeds as follo ws. The subsequen t section de tails the recent school choice policy change in Pinellas C ounty Schools and introduces the data. Section 3 examines households response to the adoption of open enro llment policy in PCS. S ection 4 analyzes the impact of exercising this form of school choice on st udent test scores and provides a falsification exercise and robustness checks. Section 5 pres ents the concluding re marks and the policy implications of the findings. Policy Background and Data Description Policy Background In order to examine the impacts of incr easing public school choice, I use the recent school-choice policy change in one of the largest school district s in the U.S., Pinellas County Schools (PCS), which adopted its intra-distri ct choice program in 2003. Prior to open enrollment, for over three decades, public school assignments in the district were determined using a zoning regime with f orced busing, under which house holds residential choices had direct implications on the public school thei r children will attend; however, a minority of students was forced to attend other public school s to avoid racial segr egation. Students could also voluntarily opt out of their default schools using Special Attendance Permits (SAP)9. During the pre-policy period, the majority of the students w ho attended a public school other than their zoned schools were in the latter category: during the 1999-2000 school year, 6,048 (5.3%) out of the 114,500 enrolled students in PCS were able to attend a different public school than their zoned schools using SAPs10. 9 Special Attendance Permit (SAP) grants students the privilege of attending a school in another attendance zone. Students are granted SAPs under extenuating circumstances including, but not limited to, child care needs, a family hardship or the medical condition of the child. Other factors including the racial diversity and the capacity of the target school are also consid ered in processing SAP requests. 10 In the sample, prior to open enrollment, the rate of students who attended public schools other than their zoned schools is roughly 6%, which implies that only 600 students were forced to opt-out in that school year. Given this 64 PAGE 65 Under the new school-choice regime, the school di strict is divided into four attendance areas for elem entary schools and three attendance areas for middle schools as shown in Figures 4-1 and 4-2. The attendance areas at each grade level were determined based on factors including population density, public school capacities and educational offe rings. Non-traditional public schools including countywide fundamental schools, magnet programs, charter schools and high school academies, each of which has a separate application procedure and timeline, were excluded from this choice plan11. During the first year of the program, each student was required to submit a list of her preferred schools, which could include any tradi tional public school within the boundaries of her attendance area, whereas in the subsequent years, only the transition-grade students were required to submit their preferences.12 The non-transition graders were automatically assigned to their current schools unless they subm itted a list of their preferences. Given the submitted student preferences, if the number of applicants exceeded the number of seats available at a given public sc hool, assignments were determined using the following priority categories and the assignment mechanism commonly referred to as the Boston mechanism13: evidence, I assume that all of the pr e-policy opt-outs are voluntary throughout the remainder of the study, since the dataset I employ does not allow me to identify th e bused students during the pre-policy period. 11 The main difference between countywide fundamental schools and traditional public schools is that the former type admits any student in the district regardless of th e residential location, depending on capacity constraints. 12 Prior to the introduction of open enrollment in PCS, Family Education and Information Centers (FEIC) were established to provide all parents information on school choice description and opportunities, available schools by attendance area, choice applications, transportation and school programs in order to assist them in choosing the appropriate schools for their children. As part of the par ent outreach program, FEIC staff was also required to visit libraries, day-care centers and community centers, and to speak to parent groups about the registration process and the academic programs. 13 Besides PCS, the Boston mechanism is also being used in some major school districts such as Cambridge, Charlotte, Denver, Hillsborough County, Miami-Dade County, Minneapolis and Seattle. Under the Boston mechanism, a student who is not assigned to his first ch oice is considered for his second choice only after the students who ranked that students second choice as their first choices. Thus, a student might lose her priority at a 65 PAGE 66 1. Grandfathering and Extended Grandfathering Priority a. CONTINUATION (GRANDFATHERING) PRIORITY. Allows students to remain at the school of attendance until promotion to the next grade level or the student otherwise leaves the school. b. EXTENDED GRANDFATHERING PRIORITY. Allows students to remain at the school of attendance and progress through each school level previously assigned to the parent/guardians address until the student graduates from high school or the family moves out of the residence used to determine the progression of schools14. 2. FAMILY PRIORITY. Used to assign family members to the same school where family is defined as those who reside together as a family at the same address. 3. PROXIMITY PRIORITY. Provides increased likelihood that a family living closest to a school will be selected to attend the school if that is the familys first choice. In the first step of the school choice pl an called controlled ch oice (2003-2007), racial diversity, which employs minimum and maximum racial percentages to ensure diversity, was also used as an additional criterion for student assignments15. Data Description The data includes a panel of the entire PCS elementary and middle school students attending 4th through 8th grades between 2001 and 2005. I exclude three t ypes of public school students from the analysis: high school students due to the samp le selection issue created by students who drop-out; students at tending non-traditional public schools such as charter schools, magnet programs and countywide fundamental school s, since these schools are not included in the PCSs choice program; and stude nts attending kindergarten through 4th grade, since the public school unless she lists that school as her first choi ce. One major issue with this assignment mechanism is that truthful revelation of public school preferences is not necessarily a weakly dominant strategy for households: it is not strategy-proof (Abdulkadiroglu and Sonmez, 2003). 14 In other words, this preference allows students to stay at their neighborhood schools to which they were assigned before the open enrollment program based on their residences. 15 For the years between 2003 and 2007, there were court-ordered ratios in place to help the district make the transition from the 1971 court order for desegregation to a unitary school system. During these four years, the maximum percentage of black students for any school was 42 percent. The minimum percentage of black students for a school was determined by the percentage of black st udents residing within each attendance area. Since the 2007-2008 school year, racial diversity has no longer been used to determine public school assignments. 66 PAGE 67 standard ized testing in PCS begins in the third grade and I use previous years test score as a proxy for students intrinsic abili ty. These restrictions result in 105,791 remaining observations. The primary outcome of interest is student test scores, which are derived from the Stanford-9 and Stanford-10 Achievement Test s (SAT-9 and SAT-10) and are given in the national percentile ranking (NPR) format. In addition to test scores, the dataset includes individual student characteristic s such as race, gender, free-lunc h status and, more importantly for the analysis, residential loca tion and school attended. I define opt out students as those who opted out of their default public schools and attended another traditi onal public school at the beginning of the school year. For each student, th e default school is defined as follows. For students who did not move to a different attend ance zone during the summer before the academic year, the default public school is either the public school attended duri ng the prior school year if the student was in a non-transition grade (3rd, 4th, 6th and 7th grades) or the attendance zone middle school if the student wa s in the transition grade (5th grade) during the previous school year. If the student moved to a different attenda nce zone during the summer before the academic year, the default public school is the attendance zone elementary or middle school at the new residence. There are two residential identifiers in the da taset: the physical resi dential address of the student and the transportation grid in which the student resides16. Using these two variables, I identify the mover students, who changed their residences du ring the summer before the school year, as well as the attendance zone in which the students residence is located17. Furthermore, the physical address of the student enables me to calculate the dr iving distances to alternative 16 Pinellas County Schools is divided into approximately 900 transportation grids. For each student residing within the boundaries of a given grid, transportation is provided to only one public school at each school level. 17 Using the transportation grids, I find the attendance zone for a given public school at each school year by aggregating the grids in which the majority of studen ts attend that public school during that year. 67 PAGE 68 public schools at the students school level, which I use to in strum ent for opting out in the regression analysis. Table 4-1 provides the descriptive statistics for the entire sample as well as sub-samples based on grade level and opt-out status. The average PCS student scores slightly above the national median in both reading and math. Approximately 11% of a ll students opted out of their default schools and the opt-out rate is significantly higher for transition-grade students. The racial distribution in the sample is very similar to the racial distributi on of the general population in the U.S. with the exception of Hispanics, who are underrepresented in the sample. There are substantial differences between th e students who opt out of their default schools (opt out stude nts) and those who stay (non-opt out students) in terms of their observed characteristics. Opt out student s perform significantly worse on standardized tests during the year prior to opting out, are mo re likely to be free-lunch stud ents, African-American and are more likely to have changed resi dences during the summer before opting out. It is also worth noting the differences between th e students who opt out after the transition grade and those who opt out after a non-transition grad e. Transition-grade opt-out students have significantly higher prior achievement levels than non-transition grade opt-out students and are more similar to nonopt out students at the same grade level. The average non-transition grade opt out student is ranked roughly 4 percentiles lowe r than the average transitiongrade opt-out student in both reading and math tests du ring the previous year. Impact of Open Enrollment on School Choice Incidence of Opting out Reducing the costs associated with opti ng out of the default public school, open enrollment programs allow students, who could not otherwise afford to exercise other traditional public schooling options, to opt out Therefore, one would expect an increase in the rate of 68 PAGE 69 students who opt out of their de fault schools with the introduction of this policy. Figure 4-3 presents the opt-out rates in PCS between 2001 and 2005 for the entire sample as well as for transition grade and non-tra n sition grade students. The implementation of open enrollment at the end of 2002-2003 school year in PCS had a significant impact on the opt-out rate for the entire sample in the years following the policy adoption. The opt-out rate more than doubled in the first year after the poli cy change, from 7% to 18%, and then declined slightly in the following year. Compari ng the two sub-gr oups, the results indicate that the tran sition graders reacted mo re to the increasing school choice. With the enactment of the choice program, the rate of opting out among transiti on graders quadrupled from 8% to 33% in the first year and furthe r increased to 38% in the second year. The nontransition grade opt-out rate, on th e other hand, increased from 7% to 16% during the first year and then declined to 12% in the following year.18 Composition of the Opt out Students During both pre-policy and postpolicy periods, each student will opt out of her default public school if the discontent or the anticipated disple asure with the default school overwhelms the cost of opting out. Therefore, by lowering the cost, open enrollment programs will induce less-discontented or less-motivated to opt-out students to opt-out. If those who opted-out before the enactment of open enrollment were mainly the bad apples with the lowest achievement levels in their original schools, then open enrollment will induce students with relatively higher achievement levels to exercise other public alternatives. On the other hand, if those who opted out pre-policy were mainly the high-achievers in their sending schools, who 18 There are two possible explanations to this decline. Firs t, the significant increase in the opt-out rates after the 5th grade during the first year after the policy change might have resulted in a decline in the rate of students who opt out at the end of the 6th grade during the second year. Furthermore, sinc e non-transition grade students were not required to submit their public school preferen ces after the first year of policy adoption, the increasing cost of opting out might have altered the school choice behavior of the marginally-displeased parent. 69 PAGE 70 w ere dissatisfied with the quality of their defa ult public school, open enrollment will result in relatively low achievers to opt out. Figures 4-4 and 4-5 present the Kernel de nsity estimates for the prior achievement percentiles of the opt-out stude nts in the sending school compared to their peers at the same grade level. During the pre-policy period, for th e non-transition graders, the opt-out students were mainly the lowest achievers in their sending schools: approximately 25% of the nontransition grade opt-out st udents were in the lowest two deci les of the grade-level achievement distribution at their send ing schools. As predicted, increasing choice attracted relatively higherachievers to opt out and this rate declined to 15% with the enactment of open enrollment. On the contrary, those who opted-out of thei r default middle schools at the end of the 5th grade during the pre-policy period were mainly th e highest achievers at their sending schools: approximately 31% of the transiti on grade opt out students were in the highest two deciles of the grade-level achievement distribut ion at their sending schools in both reading and math. The open enrollment program induced relatively lower achie vers to opt-out for this subgroup: only 18% of the post-policy opt-out students were in the highest two deciles in the post-policy period19. In each case, the Wilcoxon test for equality of pre-po licy and post-policy achievement distributions provides further evidence that the policy change altered the co mposition of opting out students significantly. Opting out and Student Test Scores The extent to which exercisi ng the school choice provided by open enrollment translates into higher student achievement depends on hous eholds primary motives behind opting out and households ability to exercise higher-quality schooling optio ns. If households are more 19 I repeat the same analysis using the previous year test scores (absolute achievement at the sending school rather than relative) for the opt-out students; however, the conclusion remains unchanged. 70 PAGE 71 achievem ent-oriented in their public school choice s, then open enrollment will result in more students attending higher-quality sc hools or schools that are be tter matches to their needs, leading to improvements in the achievement levels of the opt-out student s due to the increased school input and possibly increased motivation. Moreover, the increased competition for students among public schools might lead to an improvement in the overall quality of public education, increasing the school input for a ll students. However, these pred icted effects of open enrollment will be limited if frictions such as public school ca pacity constraints restri ct students ability to exercise higher quality public schooling options On the other hand, if households make their public school choices primarily based on non-ac ademic characteristics of schools such as proximity, then students should experience no in crease in the school inpu t and consequently no benefit from opting out. Where do Students Opt out? In order to identify the mechanisms thru wh ich exercising this form of school choice impacts student test scores, it is essential to ex amine the extent to which students were able to exercise higher-quality public schooling options. For this purpose, I compare the default school and target school of the opt-out student s along three major dimensions: non-academic characteristics, direct measur es of school quality and indir ect measures of school quality. Driving distance to the student s residence is the main component of the non-academic characteristics. For direct academic measur es, I use the Florida A+ program school grade20, the average math and reading scores, average teacher experience, % teachers w ith advanced degrees, which serve as a proxy for the inst ructional quality, % free-lunch students and % gifted students, 20 Since 1999, as part of Floridas A+ plan, public schools have been annually evaluated based on their students performance in the statewide Florida Comprehensive Assessment Test (FCA T). The grade of each school, which may range between A and F, depends on (1) overall performance of their students on FCAT; (2) the percentage of eligible students who take the test and (3) whether or not students have made annual learning gains in reading and math, with particular attention to the reading and math scores of the lowest 25% of students in the school. 71 PAGE 72 which m easure the peer quality, in the school year prior to th e opt-out. Indirect academic measures include the crime rates, % in-school suspensions and % out-sch ool suspensions during the school year prior to the opt-out. The first column of Table 4-2 presents the pre-policy mean of the difference between the default school and target school characteristics (target minus defa ult) for the opt-out students. The results suggest that the aver age student opts out to a publi c school 0.5 miles farther away from her residence. Prior to open enrollment, th e average opt-out student attended a school with slightly higher average test scores (0.4 percen tiles in both reading and math), % of advanced degree teachers (0.32%), higher average teacher experience (0.18 years), higher % of gifted students (0.87%) during the year preceding to the opt-out. However, dur ing this period, the average opt-out student did not experience any statistically si gnificant changes in terms of indirect academic measures. On the other hand, during the post-policy period, the gains experien ced by the opt-out students are only statistically different than zero for two of the seven direct academic measures. In contrary to the pre-policy period, the averag e opt-out student in th e post-policy period opted out to safer public schools: on average, the target school had 0.3 less crimes per 100 students, 0.6% less in-school susp ensions and 0.1% less out-school su spensions. However, the t-test results for the equality of pr e-policy means and post-policy m eans, along with the Wilcoxon test results presented in the third and fourth columns of Table 4-2 respectively, indicate that the gains experienced by the opt-out student s are not statistically differe nt between pre-policy and postpolicy periods for the majority of the school characteristics. Overall, despite the fact that some of the di fferences in school quality between the target and default schools are statistically different from zero at conventional levels, none is 72 PAGE 73 econom ically significant. For instance, the average opt-out student in the sample attends a school with only 0.07 years of higher average teacher experience, whereas the average teacher experience for the public schools in the sample is 13 years. Ther efore, although students travel more in order to opt out of th eir default public schools, their ta rget schools are very similar to their default schools along observed characteristics. Ordinary Least Squares (OLS) Results In order to quantify the relationship between opting out and student achievement, I first estimate the following equation using OLS: itatsgGt it ti it itXXyO y 5 41,3 10 (1) where represents th e end-of-year test score of student i during year t standardized to mean zero and unit variance; is an indicator for students who opted out of their default public schools at the beginning of the school year; represents the previous year test score of student i in the sam e subject to control for the intrinsic ability of the student; denotes the vector of stu dents characteristics such as ra ce, gender, an indicator for whether the student changed residences during the summer before the school year and free-lunc h status, which serves as a proxy for the socio-economic status of the student; ityitO1, tiyitXg is a grade fixed-effect to control for the test score differences between grades; and ts is a default school-year fixed-effect to control for the time-varying school input at the default school21. In order to control for the time-invariant and time-varying neighborhood inputs, I use attendance zone fixed-effects (a ) and the average student characteristics at the trans portation grid level for each year ( GtX) respectively. 21 Using default school-year fixed effects, I intend to compare the opt-out students to their similar peers who stayed behind. If a student opts out to a higher quality school, the impact of the relative increase in her school input compared to her peers at the default school will show in the coefficient of the opt-out' variable. 73 PAGE 74 Table 4-3 provides the OLS estim ates of 1 the parameter of interest, for the entire sample as well as the transition graders and non-tr ansition graders. The resu lts suggest that there is no statistically significant imp act of opting out on test scores for the entire sample. The results further suggest that the impact of opting out is quite differe nt for the two subgr oups of interest. Transition-grade students who opt out of their default schools perform slightly better than similar students who stay behind in r eading whereas the opposite is tr ue for non-transition graders: opting-out, on average, is associated with declin es of 2 and 3% of the standard deviation in reading and math respectively for non-transition graders. Instrumental Variables (IV) Results The major problem with the OLS analysis in this context is the inability to control for all differences between those who opt -out and those who stay behind including intrinsic motivation to excel, which is positively correlated with st udent achievement. If those who travel are more academically-motivated than similar students who stay behind, OLS results will overestimate the true impact of opting out on st udent test scores. Furthermore, the OLS results will provide overestimates/underestimates of the true impact of opting out if those who stay behind suffer/benefit from the departure of their peers who opted out. However, this source of bias should be rather limited in the sample due to the relatively low opt-out rates (0.11 for the entire sample). In order to deal with this selection issue, I instrument for opting out using the sum of the reciprocal driving distances to the relevant public alternatives. Specifically, the instrument is defined as follows: 74 PAGE 75 2003 1 2003 1 Proximityittif srd tif srdtij area tijSs jit Ss jit where d ( rit, sj) is the driving distance betw een the residence of student i at time t ( rit) and school sj, denotes the set of public schools at the school level of student i at time t other than the default public school within the attendan ce area of student i at time t and denotes the set of public s chools at the school level of student i at time t other than the default public school in the entire districtarea itSitS22. Compared to the previously used measures of proximity, this instrument captures the students access to pub lic alternatives better, since it does not confin e choice to the closest alternative, while realizing the negative relationship between the distance to the public alternative and the relevance of that alternative for the household. Proximity has been shown to be a signifi cant determinant of hous eholds public school choice23. This is especially true in PCS where p roximity to the public school is used as a priority category to determine the public school assignments after th e enactment of open enrollment. In the sample, for 77% of the students who stayed, the default public school is one of the three closest public schools whereas this number is 59% among the traveling students. The validity of the instrument relies on the condition th at households, which are similar in observable characteristics and reside within the attendance z one of a given public school, are not stratified along any unobserve d dimension such as the tast e for education, which would 22 In order to construct the instrument, I first find the driving distances from the residential address of each student to each public school at the students school level in the di strict using Mapquest. This requires the calculation of driving distances between roughly 32,000 residential addresses for elementary school students and 74 elementary public schools (2,350,000 distances), and 40,000 residential addresses for middle school students and 21 middle schools (840,000 distances). By identifying and excluding the default school for each student, I then calculate sum of the reciprocal distances to the rele vant public alternatives for each student. 23 See Hastings, Kane and Staiger (2008). 75 PAGE 76 sim ultaneously impact the probability of opting out and student achievement, with respect to the proximity to the relevant public alternatives24. Naturally, households re sidential choices are not random; most households make their residential choices with school characteristics as one of the determinants. However, provided that a household chooses to reside within the boundaries of an attendance zone, it is unlikely that the proximity to the relevant alternatives will play a significant role in the households residential choice within that zone25. Table 4-4 provides further evidence on the vali dity of the instrument. Each row in the table presents the estimated impact of the pr oximity measure on various uncontrolled student characteristics in equation (1), controlling for the same covariates in the original model with the addition of the distance to the default public scho ol. The F-stats presented in the third column suggest that the instrument has no statistically signif icant impact on any of these uncontrolled characteristics conditional on the covariates listed in equation (1). One must realize the two im portant limitations of this study while interp reting the IV results. First, this study ignores the possibility that, in the absence of choice, students could opt out of their assigned public schools by relocating or exercise pr ivate or non-traditional public schooling options such as charter sc hools and countywide fundamental schools26. This limitation makes it hard to compare the well-being of the opt-out students in the presence and the absence of open enrollment. Furthermore, if the increas ing competition between public schools, which is 24 This follows, since I control for residential location via attendance-zone fixed effects in the model. 25 It is worth emphasizing the significance of residential lo cation controls for the analysis The instrument is clearly correlated with the school density in the area where the household resides. If those who reside in urban areas with relatively high student population and school density differ on unobservable achievement-related characteristics than those who chose to reside in relatively rural areas, then the instrument will impact student achievement in ways other than its impact on opting out. However, restricting the variation in the instrument to within attendance zones, which have an average area of three square miles, I overcome this issue. 26 Roughly 15% of the entire K-12 student body in Pinellas County attend private schools, whereas the percentage attending non-traditional public options is significantly lower (0.5% in charter schools and 4% in countywide fundamental public schools). 76 PAGE 77 expected to im pact the low-performing public schools disproportionately, leads to an improvement in the overall quality of public education, the IV re sults will provide underestimates of the true impact of opting out on test scores.27 The second row of Table 4-5 pres ents the first-stage results of the IV regression for the entire sample, non-transition grade students and the transition grad e students. In addition to the covariates defined earlier in equation (1), I also control for the dr iving distance to the default school, which is expected to have a positive imp act on opting out. The firs t-stage results indicate exceptionally strong correlation between proximity to alternative schools and the probability of opting out of the default public sc hool. The students with more nearby public school alternatives, as determined by the proximity measure, are mo re likely to exercise other public schooling options and the relationship is ex tremely statistically significant fo r the entire sample as well as the two subgroups, as indicated by the F-tests of significance of the excluded instrument for each regression. The second-stage results, which are reported in the first row of Table 4-5, confirm the earlier prediction on the di rection of the bias in the OLS estimates. For the entire sample and the non-transition graders, IV results suggest a significantly stro nger negative impact of opting out on reading test scores than the OLS estimates The average opt-out st udent is ranked roughly one-fourth of the standard deviation lower in read ing test scores than a similar student who stays behind. This detrimental impact of traveling on reading scores is sl ightly higher for nontransition graders. On the other hand, there is neither any statistically significant benefit/loss associated with opting out on math scores nor for those who opted out of their default middle schools at the end of the 5th grade. 27 This statement is true assuming that those who opt out of their default public schools attend higher-quality public alternatives. However, the earlier results indicate that this has not been the case in Pinellas County. 77 PAGE 78 Table 4-6 compares the pre-policy and post-pol icy im pacts of opting out on student test scores. The post-policy opt-out is associated with a significantly higher reduction in reading test scores compared to pre-policy: the pre-policy av erage opt-out student is ranked roughly one-fifth of the standard deviation lower in reading compared to the one-third of the sta ndard deviation reduction after open enrollment. One possible explanat ion to this puzzling result is the change in the composition of the opt-out students with th e enactment of open enrollment. The new optout students are more mediocre and possibly less-motivated to excel than their pre-policy counterparts. Another plausible e xplanation is that it takes time for parents to comprehend the new system and make good choice s for their children. This is es pecially true in PCS, where a relatively complicated mechanism, commonly know n as the Boston mechanism, is used to determine the public school assignments. No t being strategy-proof, the Boston mechanism makes it even harder for parents to submit the optimal list of public school preferences by providing some parents incentives to misreport their preferences28. Disentangling the Reasons Underlying the Detrimental Impact Change in school quality There are several competing mechanisms th rough which opting opt might impact student test scores. If those who opt -out of their default public schools ar e able to exercise higher-quality schooling options, all else constant, opting out is expected to lead to an increase in the traveling students test scores. In order to quantify the impact of changing school quality for the traveling students on test scores, Table 4-7 presents the estimated im pact of opting out on reading scores with and without controls for the change in the school qua lity experienced by those who opt out. The first 28 See Abdulkadiroglu and Sonmez (2003) for more detailed information about the Boston mechanism. 78 PAGE 79 colum n replicates the first co lumn of Table 4-5, whereas th e second, third and the fourth columns introduce attended school characteristics29, attended school fixed-effects and attended school-year fixed effects to the model respective ly. Since our baseline model includes default school-year fixed-effects, the difference in the es timated coefficients of the opt-out variable between the first specif ication and the others should provi de the impact of changing school quality caused by the opt-out on reading scores. The estimated impact of opting out remains relatively stable across specifications confirming th e earlier finding that students, on average, opt out to similar schools and he nce do not experience significan t improvements in their school inputs. Outsider effect The second mechanism through which opting out might affect stude nt achievement is changing intrinsic motivation of th e traveling students. If being an outsider at the new school leads to a decline in the intrinsic motivation of the student, then opting out might have a detrimental impact on test scores. If the outsider e ffect is valid, then students who opt out of their default schools closer to the terminal grade of the school leve l are expected to suffer more from doing so, since they will have less time and incentives to be come acquainted with the new environment. Furthermore, keeping the proximity to the terminal grade of the school level constant, elementary school stude nts are expected to suffer mo re from non-transition grade opting out, since their new peers at the target scho ol are likely to have spent more time together, which makes it harder for the traveling student to become an insider. Finally, if the intrinsic 29 Attended school characteristics include the school grade in the previous year, average test scores in the previous year, average teacher experience, % t eachers with advanced degrees, % gift ed students, % free-lunch eligible students, crime rates and suspension rates. 79 PAGE 80 m otivation of the traveling students increases as they become more familiar to the new school, one might expect the negative impact of opting out to vanish in the long run. The IV results presented in Table 4-8 support the first predic tion: those who opt out of their default public schools one year before th e terminal grade of the school level experience significant declines in bo th reading (roughly one-fourth of the st andard deviation, yet marginally significant at conventional levels) and math (half of the standard deviation) whereas opting-out of the default public school two years before the terminal grade leads to a significant improvement (one-third of the sta ndard deviation) in the math sc ores of the traveling students. Comparing the impact of non-transition grade opting out between elementary school students and middle school stude nts, the findings presented in Table 4-9 provide evidence supporting the second prediction. Non-transition grade opting out during elementary school years is associated with significant declines in both re ading (one third of the standard deviation) and math (one fourth of the standard deviation), wher eas there is no statistica lly significant impact of non-transition grade opting out on te st scores during middle school. Table 4-10 examines the long-term impacts of opting out on student test scores. In addition to the covariates used previously, the re gressions in the second co lumn also include the student characteristics two y ears after the initial opt out30. Contrary to the pr edictions, those who opt out perform significantly worse compared to si milar students who stayed behind at the end of the second year after opting out. Alternative explanations One must be cautious in interpreting the resu lts presented in the previous subsection. Since the instrumental variables approach in this context deals only with the within subgroup 30 These characteristics include indicators for whether the student opted out or changed residences at the end of the first year after the initial opt-out. 80 PAGE 81 selection into opting out the differences between the estim ated impacts of opting out across subgroups might be driven by the differences be tween the traveling students in different subgroups. While by no means conclusive, the comp arisons between the opt-out students in different subgroups presented in Table 4-11 provi de evidence against the latter possibility. The equality of mean characteristics between those who opt out two years before the terminal grade and one year before the terminal grade is reje cted at 5% significance level for only 2 of the 8 characteristics discussed. Comparisons between the elementary school no n-transition grade optout students and their counterparts in middle school reach the same conclusion: the equality of means is rejected at 5% significance level for only 3 out of the 8 characteristics. Therefore, given the substantial heterogeneity in the impact of opting out across subgroups, the findings presented in Tables 4-9 and 4-10 can be regarded as evid ence supporting the outsider effect discussed in the previous subsection. Impact of Opting out Disadvantaged Students One of the main objectives of school choice reforms such as open enrollment is to enable the disadvantaged students such as those from low-SES families, who cannot otherwise afford better schooling options, to attend higher-quality public sc hools. If the disadvantaged students are more likely to opt out to higher-quality public schools compared to th eir default public schools, they are expected to benefit more or suffer less from opting out, since they will experience higher gains in school quality relative to the advantaged students. I define disadvantaged st udents in two ways: with re spect to the poverty and the performance levels of their default public school s. High poverty schools are defined as schools in which the majority of the students are free-lunch eligible in at least three of the five years between 2001 and 2005, whereas the opposite indicates low pove rty. High performing schools 81 PAGE 82 are defined as having received a grade of A in at le ast three years during this time period, whereas the opposite indicates low performance. Table 4-12 presents comparisons between the gains in school quality experienced by the two subgroups. The results indica te that the disadvantaged st udents experience significantly higher gains in school quality compared to thei r advantaged counterpart s regardless of the definition of disadvantaged.31 The IV estimates presented in the first set of rows of Table 4-13 partially verify the expectations: opting out is associ ated with a significant decline in reading test scores for the advantaged students under both definitions. Th ose who opt-out of th eir low poverty default schools are ranked one-third of the standard devia tion lower in reading th an similar students who stayed behind, whereas opting-out of a high pe rforming default school is associated with a decline of approximately one half of the standard deviation in reading. On the other hand, the disadvantaged students neith er suffer nor benefit from opting out of their default schools in terms of test scores. How much of this difference can be explained by the differen ce in gains in school quality experienced by the disadvantaged opt-outs and the advant aged opt-outs? The second set of rows in Table 4-13 introduces attended school characteristics to the earlier specifi cation to control for the gains in school quality. The estimated impact of opting out does not change significantly across specifications for each subgroup and subjec t indicating that the aforementioned difference in the estimated impact is not driven by the differential gains in school quality between the two groups. 31 For instance, those who opt-out of their low performing default schools attend public schools with 4% less freelunch eligible students, whereas those who opt-out of their high performing default schools experience an increase of 5% in the percentage of free-lunch eligible students. 82 PAGE 83 Another possible, yet not testab le, ex planation to this diffe rence between advantaged and disadvantaged students is the possibility of a pri son break effect: regardless of where they optout, disadvantaged students might be experiencing improvements in their intrinsic motivations just because they were able to get away from the undesirable environment at the default school. This positive effect might be offsetting the outsider effect mentione d earlier, re ducing the detrimental impact of opting out.32 Falsification Test So far, I have attributed households changing school choice behavior to the introduction of open enrollment in Pinellas County Schools. However, it is quite possible that these behavioral changes had taken place due to so me idiosyncratic factors or other concurring educational reforms other than open enrollment. In contrary to the previous literature, the existence of pre-policy data enables me to test this possibility. For this purpose, I propose the following falsification exercise. Consider an elementary school student residing in attendance-area A. Prior to the adoption of open enrollment, the cost of opting out to a public school in attendance area-A for the student should be similar to opting out to an y public school in another attendance area given that the two schools are equidistant to the students residence. Ther efore, pre-policy, proximity to the alternative public schools in attendance area-A (policy alternat ives) as well as the proximity to the alternative public schools in other post-policy attenda nce areas (non-policy alternatives) should have explanatory power on the students opt-out probability. 32 One must realize the obvious possibility that this diffe rence might be caused by the differences between the disadvantaged and advantaged opt-outs along observable and unobservable characteristics. Advantaged opt-outs have significantly higher prior achievement levels, less likely to be free-lunch eligible and less likely to be black compared to disadvantaged opt-outs. If the impact of opting out is heterogeneous with respect to these characteristics, the difference can also be explained by this scenario. 83 PAGE 84 On the other hand, by allowing the stude nt only to choose among public schools in attend ance area-A, the new choice plan in PCS effectively decreased the relative cost of attending area-A elementary schools for the student This implies that, post-policy, the relevant alternatives are only the ones w ithin the attendance area of th e student. Therefore, after the policy change, only the proximity to th e policy alternatives should have explanatory power on the students opt-out probability. Table 4-14 presents the linear probability model estimates where the outcome of interest is the likelihood of opting-out. In addition to th e two proximity measures, the model includes the covariates described in Table 4-5. As predic ted, during the pre-policy period, both proximity measures have statistically significant impacts on the likelihood of opting out. However, after the policy change, only the proximity to policy alternatives has significant impact on households public school choice. These results provide evidence that the households changing school choice behavior can be regarded as a r eaction to the adoption of open enrollment. Robustness Checks In order to check the robust ness of the results, I employ two alternative proximity measures to instrument for opting out. The first a lternative is to use the m ean of the distances to all relevant public alternatives in the IV regression. The first two columns of Table 4-15 present the IV estimates using th e first alternative proximity measure. The first-stage results suggest strong correlation between the instrument and the prob ability of opting out and the instrument coefficient has the expected negative sign. More importantly, the second stage results indicate that the estimated impact of opting out does not change w ith this alternative instrument. The second alternative instrume nt is the distance to th e closest relevant public alternative. However, notice that this measure has the disadvantage of assuming that students 84 PAGE 85 choice is lim ited to the closest relevant alternatives. Nevertheless, the previous conclusions remain unchanged. Concluding Remarks One of the most commonly exercised forms of school choice is the open enrollment program, which allows parents to send thei r children to public schools outside of the neighborhoods in which they reside. By expa nding the set of feasible public schools for households, such programs are predicted to impact households public school choice as well as student achievement. Using the recent school-choice po licy change in Pinellas C ounty Schools, I first examine the impact of open enrollment on households pub lic school choice behavior. I find significant changes in the frequency of opting out of the default school and attending another traditional public school, as well as the composition of those who exercise this choice with the adoption of the open enrollment policy. I then attempt to quantify th e causal relationship between ex ercising this form of school choice and student test scor es. Using proximity to relevant traditional public alternatives as an instrument for opting out, the re sults indicate no signi ficant benefit of opti ng out and that those who opt out often perform signifi cantly worse on standardized tests than similar students who stay behind. Furthermore, I find that the impact of opti ng out is significantly heterogeneous with respect to the grade of opting out. The findings s uggest that those who opt out during elementary school years suffer significantly bot h in terms of reading and math scores, whereas there is no statistically significant impact on middle school students. Furthe rmore, those who opt out two years before the terminal grade of the school level significantly benefit in terms of their math scores whereas opting out one year prior to the terminal grade is associated with a significant 85 PAGE 86 declin e in both reading and math test scores. Fina lly, the results indicate that the negative effect of opting out is non-existent for disadvantaged students, who typically constitute the proposed target of school choice reforms. Such detrimenta l effects seem to persist at the end of the second year after the initial opt out. An important policy implication of the findi ngs presented in this study is that open enrollment programs fail to improve the achievement levels of those who exercise this form of choice. One reason underlying this conclusion is the highly c onstrained choice environment provided by such programs, which does not enable students to ex ercise higher quality schooling options. The results indicate that those who opt out of their default public schools, on average, attend similar schools along various measures of school quality. Along with the negative effect of opting out on the intrinsic mo tivation of the students due to being an outsider at the new school, lack of gains in school quality might explain the detrimental impact of opting out. 86 PAGE 87 Figure 4-1. Post-policy elem entary school attendance areas in Pinellas County Schools 87 PAGE 88 88 Figure 4-2. Post-policy middle school attendance area s in Pinellas County Schools PAGE 89 Table 4-1. Descriptive statistics Non-Transition Grade Students Transition Grade Students All Students Non-Opt Out Students Opt Out Students All Students Opt Out Students All Students Opt Out Students Reading NPR-This Year 57.740 58.729 49.616 58.515 48.541 54.189 52.560 (27.798) (27.566) (28.356) (27.559) (27.66) (28.601) (29.986) Math NPR-This Year 61.707 62.729 53.321 61.972 52.024 60.495 56.871 (27.696) (27.388) (28.771) (27. 609) (28.133) (28.062) (30.171) Opted Out Last Year 0.109 0.097 0.162 (0.311) (0.296) (0.369) Moved Last Year 0.175 0.129 0.549 0.164 0.599 0.227 0.413 (0.38) (0.336) (0.498) (0.37) (0.49) (0.419) (0.493) Reading NPR-Last Year 56.627 57.562 48.947 57.286 47.827 53.607 52.011 (28.007) (27.82) (28.352) ( 28.068) (27.984) (27.522) (29.122) Math NPR-Last Year 60.336 61.334 52.141 60.749 51.004 58.444 55.253 (27.907) (27.657) (28.596) (27. 933) (28.025) (27.712) (29.888) Free Lunch 0.415 0.390 0.624 0.413 0.646 0.427 0.562 (0.493) (0.488) (0.485) (0.492) (0.478) (0.495) (0.496) White 0.742 0.759 0.601 0.743 0.586 0.740 0.641 (0.437) (0.427) (0.49) (0.437) (0.493) (0.439) (0.48) Black 0.152 0.138 0.264 0.151 0.281 0.156 0.216 (0.359) (0.345) (0.441) (0.358) (0.45) (0.363) (0.412) Hispanic 0.053 0.051 0.073 0.053 0.071 0.054 0.079 (0.224) (0.219) (0.26) (0.223) (0.257) (0.226) (0.27) Female 0.494 0.496 0.481 0.495 0.485 0.490 0.472 (0.5) (0.5) (0.5) (0.5) (0.5) (0.5) (0.499) N 105,791 94,303 11,488 86,827 8,415 18,964 3,073 89 PAGE 90 0% 5% 10% 15% 20% 25% 30% 35% 40% 20012002200320042005 All Students Transition Grade Students Non-Transition Grade Students Figure 4-3. Percentage of opt out students in Pinellas County Schools, 2001 2005 90 PAGE 91 A B Figure 4-4. Kernel density es timates achievement percentile at the sending school for nontransition grade opt out students. A) Reading test scores B) Math test scores. Both densities were estimated with an Epanec hnikov kernel function and halfwidth of 5 percentiles. 91 PAGE 92 A B Figure 4-5. Kernel density es timates achievement percentile at the sending school for transition grade opt out students. A) Reading test scores B) Math test scores. Both densities were estimated with an Epanec hnikov kernel function and halfwidth of 5 percentiles. 92 PAGE 93 93 Table 4-2. Default school versus target school characteristics: the year before opting out Pre-Policy Post-Policy pre = post Wilcoxon p-value (Driving Distance) 0.481*** 0.544*** 0.427 0.052 (School Grade) 0.022 -0.026** 0.009 0.011 (Mean Reading Score) 0.366*** 0.094 0.093 0.039 (Mean Math Score) 0.393*** 0.127 0.116 0.035 (% Free-Lunch) 0.163 -0.372* 0.106 0.289 (% Gifted) 0.867*** 0.056 0.000 0.000 (% Teachers with Advanced Degree) 0.379** 0.047 0.118 0.138 (Average Teacher Experience) 0.178*** -0.003 0.003 0.003 (Crime Rate) -0.025 -0.327*** 0.000 0.000 (% In-School Suspensions) 0.037 -0.588*** 0.001 0.090 (% Out-School Suspensions) 0.052 0.116** 0.458 0.838 N(all opt out) 3977 6464 The first two columns present the mean diffe rence between the defa ult school and target school characteristics (target minus default) for the opt out students in the subgroups indicated. The third column provides the p-values of the t-te sts for equality of the mean difference between the subgroups and the last colu mn presents the p-values for the Wilcoxon rank-sum test for the distributional equality of the two subgroups. *, ** and *** indi cate that the null hypothesis for equality to zero is rejected at the sign ificance levels of 10, 5 and 1% respectively. PAGE 94 Table 4-3. The impact of opting out on test scores: OLS results All Students Non-Transition Grade Students Transition Grade Students Reading Math Reading Math Reading Math Opt out -0.005 -0.017 -0.022** -0.026** 0.039** 0.021 (0.010) (0.012) (0.009) (0.012) (0.018) (0.019) Adjusted-R2 0.66 0.68 0.66 0.68 0.67 0.67 N 104,830 104,830 85,893 85,893 18,937 18,937 For each regression, test scores are standardized to mean zero and unit variance within the subgroup. All regressions include i ndividual student characteristics (previous year test score in the same subject, free-lunch status, race, gender), indicator for whether the student changed her residence during the summer before the opt out, grade fixed-effect s, default school fixed-effects, year fixed-effects, defau lt school-year fixedeffects, attendance zone fixed-effects a nd transportation grid characteristics. Robus t standard errors, clustered at the school level, are given in the parentheses. *, ** and *** represent statistical significance at 10, 5 and 1% respectively. 94 PAGE 95 Table 4-4. S tudent characteristics and proximity Dependent Variable Ln(Proximity) N F-Stat Reading score two years earlier -0.510 73,248 2.30 (0.332) Math score two years earlier -0.179 73,248 0.19 (0.411) Repeated a grade this year 0.001 102,902 0.42 (0.002) Repeated a grade last year 0.002 46,414 0.59 (0.002) Changed residences two years earlier 0.014 25,058 1.36 (0.012) Reading score three years earlier -0.517 45,055 1.90 (0.375) Math score three years earlier 0.523 45,055 0.92 (0.544) All regressions include the covariates described in Table 4-3. Each row presents the results for the regressions where the variable indicated is the dependen t variable. Robust standard errors, clustered at the school level, are given in the parentheses. *, ** a nd *** represent statistical significance at 10, 5 and 1% respectively. 95 PAGE 96 Table 4-5. The impact of opting out on test scores: IV results All Students Non-Transition Grade Students Transition Grade Students Reading Math Reading Math Reading Math Opt out -0.236*** -0.062 -0.250** -0.048 -0.056 -0.103 (0.089) (0.100) (0.115) (0.113) (0.129) (0.149) First Stage Results Ln(Proximity) 0.086*** 0.086*** 0.078*** 0.078*** 0.132*** 0.132*** (0.006) (0.006) (0.006) (0.006) (0.015) (0.015) F-Stat (Excluded Instr.) 238.39 238.39 184.68 184.68 76.03 76.03 N 104,830 104,830 85,893 85,893 18,937 18,937 For each regression, test scores are standardized to mean zero an d unit variance within the subgroup. In addition to the covariates described in Table 4-3, all regressions include the driving distance to the default public school. Robust standard errors, clustered al th e school level, are given in the parentheses. *, ** and *** represent statistical significance at 10, 5 and 1% respectively. 96 PAGE 97 Table 4-6. T he impact of opting out on test scor es: pre-policy versus pos t policy IV results Pre-Policy Post-Policy Reading Math Reading Math Opt out -0.197* -0.032 -0.312** -0.116 (0.106) (0.164) (0.149) (0.172) First Stage Results Ln(Proximity) 0.082*** 0.082*** 0.076*** 0.076*** (0.008) (0.008) (0.008) (0.008) F-Stat (Excluded Instr.) 115.56 115.56 103.43 103.43 N 63,275 63,275 41,555 41,555 For each regression, test scores are standardized to mean zero and unit variance within the subgroup. All regressions include the covariates describe d in Table 4-5. Robust standard errors, clustered at the school level, are given in the parentheses. *, ** and *** represent statistical significance at 10, 5 and 1% respectively. 97 PAGE 98 Table 4-7. T he impact of opting out on reading scores: identifying the im pact of changing school quality IV results I II III IV Opt out -0.236*** -0.210*** -0.263*** -0.257*** (0.089) (0.086) (0.094) (0.093) Default school-year FE YES YES YES YES Attended school character. NO YES NO NO Attended school FE NO NO YES NO Attended school-year FE NO NO NO YES First Stage Results Ln(Proximity) 0.086*** 0.083*** 0.074*** 0.074*** (0.006) (0.005) (0.005) (0.005) F-Stat (Excluded Instr.) 238.39 239.01 191.27 198.53 N 104,830 103,636 104,141 104,141 For each regression, test scores are standardized to mean zero and unit variance within the subgroup. All regressions include the covariates described in Table 4-5. Attended school characteristics include the school grade in the previous year, averag e test scores in the previ ous year, average teacher experience, % teachers with advanced degrees, % gift ed students, % free-lunch eligible students, crime rate and suspension rates. Robust standard errors, clustered at the school level, are given in the parentheses. *, ** and *** represent statistical significance at 10, 5 and 1% respectively. 98 PAGE 99 Table 4-8. T he impact of opting out on readi ng scores: by grade of opt out IV results Two Years Before Terminal Grade One Year Before Terminal Grade Reading Math Reading Math Opt out -0.157 0.346*** -0.230* -0.535*** (0.161) (0.127) (0.134) (0.225) First Stage Results Ln(Proximity) 0.086*** 0.086*** 0.071*** 0.071*** (0.008) (0.008) (0.008) (0.008) F-Stat (Excluded Instr.) 110.04 110.04 80.10 80.10 N 44,028 44,028 41,685 41,685 For each regression, test scores are standardized to mean zero and unit variance within the subgroup. All regressions include the covariates describe d in Table 4-5. Robust standard errors, clustered at the school level, are given in the parentheses. *, ** and *** represent statistical significance at 10, 5 and 1% respectively. 99 PAGE 100 Table 4-9. T he impact of opting out on read ing scores: elementary versus middle school nontransition graders IV results Elementary School Middle School Reading Math Reading Math Opt out -0.323*** -0.241** -0.112 0.247 (0.135) (0.121) (0.194) (0.172) First Stage Results Ln(Proximity) 0.124*** 0.124*** 0.052*** 0.052*** (0.010) (0.010) (0.007) (0.007) F-Stat (Excluded Instr.) 144 144 55.20 55.20 N 39,693 39,693 46,200 46,200 For each regression, test scores are standardized to mean zero and unit variance within the subgroup. All regressions include the covariates describe d in Table 4-5. Robust standard errors, clustered at the school level, are given in the parentheses. *, ** and *** represent statistical significance at 10, 5 and 1% respectively. 100 PAGE 101 Table 4-10. The im pact of opti ng out on reading scores: by years after opting out IV results One Year After Opting Out Two Years After Opting Out Reading Math Reading Math Opt out -0.236*** -0.062 -0.289** -0.037 (0.089) (0.100) (0.127) (0.143) F-Stat (Excluded Instr.) 244.29 244.29 139.94 140.19 N 104,830 104,830 54,939 54,939 For each regression, test scores are standardized to mean zero and unit variance within the subgroup. The regressions reported in the first two columns include the same covariates in Table 4-5. In addition, the last two regressions include indicators for whether the student opted out or moved during the summer prior to the second year. Robust standard e rrors are given in the parentheses. *, ** and *** represent statistical significance at 10, 5 and 1% respectively. 101 PAGE 102 Table 4-11. Student charact eristics across subg roups Opted out one year before the terminal grade Opted out two years before the terminal grade Elementary school opt-out Middle school opt-out Reading NPR-Last Year 47.721 48.124 48.114 47.665 (0.445) (0.431) (0.397) (0.493) Math NPR-Last Year 50.747 51.456 50.562* 51.891 (0.457) (0.422) (0.397) (0.496) Free Lunch 0.622*** 0.670 0.691*** 0.583 (0.008) (0.007) (0.007) (0.008) White 0.595 0.581 0.576** 0.604 (0.008) (0.008) (0.007) (0.008) Black 0.279 0.278 0.287* 0.266 (0.007) (0.007) (0.007) (0.008) Hispanic 0.067 0.076 0.070 0.074 (0.004) (0.004) (0.004) (0.005) Female 0.487 0.481 0.471** 0.502 (0.008) (0.008) (0.007) (0.009) Moved Last Year 0.632*** 0.574 0.597 0.609 (0.008) (0.008) (0.007) (0.008) N 3,971 4,201 4,792 3,380 *, ** and *** indicate that the null hypothesis for equality of means across subgroups is rejected at the significance levels of 10, 5 and 1% respectively. 102 PAGE 103 Table 4-12. Default school versus tar get school characteristics during the year before opting out disadvantaged versus advantaged students Low Poverty High Poverty lp = hp Wilcoxon p-value (School Grade) -0.166*** 0.156*** 0.000 0.000 (Mean Reading Score) -1.364*** 1.819*** 0.000 0.000 (Mean Math Score) -1.172*** 1.682*** 0.000 0.000 (% Free-Lunch) 4.492*** -5.005*** 0.000 0.000 (% Gifted) 0.520*** 0.204*** 0.017 0.000 (% Teachers with Advanced Degree) -1.241*** 1.642*** 0.000 0.000 (Average Teacher Experience) -0.208*** 0.350*** 0.000 0.000 N 5317 5124 High Performing Low Performing hp = lp Wilcoxon p-value (School Grade) -0.269*** 0.213*** 0.000 0.000 (Mean Reading Score) -2.440*** 2.439*** 0.000 0.000 (Mean Math Score) -2.333*** 2.405*** 0.000 0.000 (% Free-Lunch) 4.638*** -4.252*** 0.000 0.000 (% Gifted) -0.679*** 1.252*** 0.000 0.000 (% Teachers with Advanced Degree) -1.304*** 1.428*** 0.000 0.000 (Average Teacher Experience) -0.676*** 0.696*** 0.000 0.000 N 4796 5645 The first two columns present the mean differen ce between the default school and target school characteristics (target minus default) for the opt out students in the subgroups indicated. The third column provides the p-values of the t-tests for equality of th e mean difference between the subgroups and the last column presents the p-values for the Wilcoxon rank-sum test for the distributional equality of the two subgroups. *, ** and *** indicate that the null hypothesis for equality to zero is rejected at the significance levels of 10, 5 and 1% respectively. 103 PAGE 104 Table 4-13. The im pact of opti ng out on reading scores: disadvant aged students IV results I. Without Attended Sc hool Characteristics Low Poverty Default School High Poverty Default School Reading Math Reading Math Opt out -0.338* -0.025 -0.111 -0.070 (0.182) (0.157) (0.104) (0.155) F-Stat (Excluded Instr.) 88.74 88.74 136.89 136.42 N 65,498 65,498 38,643 38,643 High Performing Default School Low Performing Default School Reading Math Reading Math Opt out -0.452** 0.052 -0.048 -0.123 (0.194) (0.165) (0.115) (0.149) F-Stat (Excluded Instr.) 77.67 77.79 143.04 142.80 N 60,050 60,050 53,403 53,403 II. With Attended School Characteristics Low Poverty Default School High Poverty Default School Reading Math Reading Math Opt out -0.353* -0.011 -0.097 -0.059 (0.195) (0.176) (0.101) (0.148) F-Stat (Excluded Instr.) 78.32 78.32 144 143.52 N 65,386 65,386 38,250 38,250 High Performing Default School Low Performing Default School Reading Math Reading Math Opt out -0.477** 0.076 -0.044 -0.127 (0.217) (0.182) (0.120) (0.157) F-Stat (Excluded Instr.) 75.52 75.34 131.33 131.1 N 59,826 59,826 52,966 52,966 For each regression, test scores are standardized to mean zero and unit variance within the subgroup. All regressions include the covariates described in Table 4-5. High poverty school is defined as having the majority of the students free-lunch eligible in at least three of the five years between 2001 and 2005, whereas the opposite indicates low poverty. High performance school is defined as receiving a grade of A in at least three of the years between 2001 and 2005, whereas the opposite indicates low performing. Robust standard errors, clustered at the sc hool level, are given in the parentheses. *, ** and *** represent statistical significance at 10, 5 and 1% respectively. 104 PAGE 105 Table 4-14. Falsif ication test: linear probability model estimates Dependent Variable: Opt Out Pre-Policy Post-Policy Ln(Proximity to Policy Schools) 0.045*** 0.076*** (0.010) (0.015) Ln(Proximity to Non-policy Schools) 0.054*** 0.004 (0.022) (0.035) F-Stat (Policy Schools) 19.44 25.15 F-Stat (Non-Policy Schools) 6.04 0.01 N 63,275 41,555 For each regression, test scores are standardized to mean zero and unit variance within the subgroup. Both regressions include the covariates described in Table 4-5. Robust standard errors,clustered at the school level, are given in the parentheses. *, ** and *** represent statistical significance at 10, 5 and 1% respectively. 105 PAGE 106 106 Table 4-15. The impact of opting out on test scores: IV results robustness checks I II Reading Math Reading Math Opt out -0.223** -0.101 -0.225** -0.089 (0.099) (0.100) (0.105) (0.121) First Stage Results Ln(Mean Distance) -0.125*** -0.125*** (0.009) (0.009) Ln(Distance-Closest Alt.) -0.030 -0.030 (0.002) (0.002) F-Stat (Excluded Instr.) 188.79 188.79 177.69 177.69 N 104,830 104,830 104,830 104,830 For each regression, test scores are standardized to mean zero and unit variance within the subgroup. All regressions include the covariates describe d in Table 4-5. Robust standard errors, clustered at the school level, are given in the parentheses. *, ** and *** represent statistical significance at 10, 5 and 1% respectively. PAGE 107 CHAP TER 5 CONCLUSIONS Open enrollment programs have become the most frequently exercised form of school choice among households and policy-makers in th e United States during the last two decades. This study examines the implementation of open en rollment programs open as well as the impact of increasing parental choice in the form of open enroll ment on student outcomes in major school districts. Three findings ar e worth highlighting. First, the extent to which some of the earlier findings in the literature concerning th e equilibrium properties of the Boston mechanism, a commonly employed means of determining public school assignments of students in major school districts in the U.S., is ra ther limited. Second, careful consideration must be given by the school districts if the Boston mechanism is to be replaced by one of the al ternatives proposed in the recent economics literature, since serious and costly legal consequenc es can be anticipated from such a transition under th e assignment procedures commonl y employed in reality. Third, despite households strong reaction to increasing parental choice, open enrollment programs fail to improve the achievement levels of those w ho exercise this form of choice. The findings presented in this study might prov ide useful insights for policy-makers and researchers in the design of new public school choice programs. An important policy implication of the findings presented in this study is that open enrollment programs fail to improve the achievement levels of those who exercise this form of choice. One reason underlying this conclusion is the highly c onstrained choice environment provided by such programs, which does not enable students to ex ercise higher quality schooling options. The results indicate that those who opt out of their default public schools, on average, attend similar schools along various measures of school quality. Along with the negative effect 107 PAGE 108 108 of opting out on the intrinsic mo tivation of the students due to being an outsider at the new school, lack of gains in school quality might explain the detrimental impact of opting out. PAGE 109 APPENDIX A NASH EQUILIBRIUM STRATEGI ES IN EXAMPLE 2-2 Proposition A-1: Given the conditions (1) and (2), i1: axx, i2: axx and i3: bxx represents the set of all (pure-strategy) Nash equilibrium strategies of the game defined in Example-2. Proof: Looking at Table-2, given th e conditions (1) and (2), c onsider the following set of strategies that can no t be Nash equilibrium: 1. Given i3 plays axx, i1 never plays bxx or cxx and i2 never plays bxx or cxx. 2. Given i3 plays bxx, i1 never plays cxx i2 never plays bxx or cxx i1 never plays bxx or cxx 3. Given i3 plays cxx i2 never plays bxx i1 never plays bxx i2 never plays bxx or cba i1 never plays bxx or cxx i2 never plays bxx or cxx 4. Given i1 and i2 never play bxx or cxx i3 never plays cxx 5. Given i1 plays acb and i3 plays, i2 never plays acb Given i1 and i2 never play bxx or cxx, i3 never plays axx or cxx Therefore, the only possible set of Nash equilibrium strategies is i1: axx, i2: axx and i3: bxx. We know from the prev ious discussion that i1: axx, i2: axx and i3: bxx is a set of Nash equilibrium strategies. Therefore, i1: axx, i2: axx and i3: bxx represents the set of all (purestrategy) Nash equilibriu m strategies of the game defined in Example 2-2. 109 PAGE 110 APPENDIX B DEMONS TRATING THE THREE ASSIGNMENT MECHANISMS USING EXAMPLE 3-1 Boston Mechanism First step: Only the first choices are considered. Given the priorities, i3 is assigned to s2. If the tie between i1 and i2 for school s1 is broken in favor of i1, i1 is assigned to s1. Otherwise; i2 is assigned to school s1. Second step: Depending on the outcome of the ra ndom lottery, the student who gets rejected from s1 is assigned to s3, since the only available seat in s2 is occupied by i3. The algorithm terminates. Top-Trading Cycles Mechanism First step: There is only one cycle: i2 s1 i3 s2 i2 is assigned to s1 and i3 is assigned to s2. Students i2 and i3 are removed from the algorithm as well as schools s1 and s2, which became full. Second step: The only cycle is between i1 and the only remaining school, s3. i1 is assigned to school s3 and the algorithm terminates. Gale-Shapley Deferred Acceptance Mechanism a. If the tie is broken in favor of i1: 1. First step : i1 is in the queue of s1, i3 is in the queue of s2, and i2 gets rejected from school s1. 2. Second step : i2 proposes to school s2 and i3 gets rejected from school s2. 3. Third step : i3 proposes to school s1 and i1 gets rejected from school s1. 4. Fourth step : i1 proposes to school s2 and gets rejected from school s2. 5. Fifth step : i1 proposes to school s3 and the algorithm terminates. Final assignments: ( i1, s3), ( i2, s2), ( i3, s1). 110 PAGE 111 111 b. If the tie is broken in favor of i2: 1. First step : i2 is in the queue of s1, i3 is in the queue of s2, and i1 gets rejected from school s1. 2. Second step : i1 proposes to school s2 and gets rejected from school s2. 3. Third step : i1 proposes to school s3 and the algorithm terminates. Final assignments: ( i1, s3), ( i2, s1), ( i3, s2). PAGE 112 APPENDIX C THE T WO ALTERNATIVE ME CHANISMS WHEN EACH SCHOOL HAS THE SAME PRIORITY RANKING Suppose that there are n students ( i1, i2,, in) and k public schools ( s1, s2,, sk) each of which has at least one seat ava ilable. Assume further that th e students are ranked the following way at each school: i1 i2 i3 in Top-Trading Cycles Mechanism In the first step of the algorithm, all schools point to student i1, and student i1 points to her first choice. Therefore, the only cycle is between student i1 and her first choice; i1 is assigned to her first choice. In the second step, all remaining schools point to student i2 and i2 points to her top choice among the remaining schools. The only cycle is between i2 and her top choice among the remaining schools; i2 is assigned to her top choice among the remaining schools and so on. Notice that this is exactly the same as the random serial dictatorship mechanism. Gale-Shapley Deferred Acceptance Mechanism In the first step of the algorithm, each student proposes to her first choice. Lets examine the assignments of each student with respect to their priority rankings: i. Since each school has at l east one seat available, we know that student i1 will be in the queue of her first choice at the end of the first step. Furthermore, since she has the highest priority ranking at each school, i1 will not be rejected from her first choice at any step of the algorithm, and thus will be assigned to her first choice wh en the algorithm terminates. ii. There are two cases to consider for the assignment of student i2. First, if both i1 and i2 rank the same school, which has only one seat available, as their first choices, i2 will be rejected from her first choice at the end of the first step of the algorithm. If this is the case, in the second step, she will be placed in the queue of her second choice. Otherwise, she will be placed in the queue of her first ch oice at the end of the first st ep. In both cases, she will not be rejected in the following step s of the algorithm, and thus will be assigned to either her first choice or second choice when the algorith m terminates. Therefore, under the GS-DA mechanism, i2 will be assigned to her top choi ce among the remaining schools after i1 is assigned to her first choice. 112 PAGE 113 113 Using the same analogy, one can show that the students ( i3, i4,, in) will be assigned to their top choice among the remaining schools. Th erefore, the GS-DA mechanism produces the same assignments as the random serial dictatorship mechanism. PAGE 114 LIST OF REFERE NCES Abdulkadiroglu, A., Sonmez, T., 2003. School choice: a mechanism design approach. American Economic Review 93, 729-747. Abdulkadiroglu, A., Pathak, P. A., Roth, A. E., Snmez, T., 2005. The Boston Public School Match. American Econom ic Review, 95 (2), 368-371. Abdulkadiroglu, A., Pathak, P. A., Ro th, A. E., Snmez, T., 2006. Changing the Boston Mechanism: Strategyproofness as Equal Access. Unpublished mimeo, Harvard University. Abdulkadiroglu, A., Pathak, P. A., Roth, A. E., 2007. Strategy-proofness versus Efficiency in Matching with Indifferences: Redesigning the NYC High School Match. Working Paper, Harvard University. Chubb, J. E., Moe, T. M., 1990. Politics, Markets and Americas School s. Washington D.C.: Brookings Institution Press. Cullen, J. B. Jacob, B. A., Levitt, S. D., 2005. The impact of school choice on student outcomes: an analysis of the Chicago Public Schools. Journal of Public Economics, 89: 729 760. Cullen, J. B. Jacob, B. A., Levitt, S. D., 2006. The effect of school choice on participants: evidence from randomized lotteries. Econometrica, 74 (5): 1191 1230. Cullen, J. B., Jacob B. A., forthcomi ng. Is Gaining Access to a Selective Elementary School Gaining Ground? Evidence from Randomized Lotteries. In Jonathan Gruber, ed. An Economics Perspective on the Problems of Disadvantaged Youth. Chicago, IL: University of Chicago Press. Dubins, L.E., Freedman, D.A., 1981. Machiave lli and the Gale-Shapley algorithm. American Mathematical Monthly 88, 485. Education Commission of the States, 2005. State Notes: Open Enrollment: 50-State Report Enrich, P., 1995. Leaving In equality Behind: New Direc tions in School Finance Reform. Vanderbilt Law Review, 101 (48), 101-194. Erdil, A., Ergin H., forthcoming. Whats the Matter with TieBreaking? Improving Efficiency in School Choice. American Economic Review. Ergin, H., Sonmez, T., 2006. Games of school choice under the Boston Mechanism. Journal of Public Economics 90, 215-237. 114 PAGE 115 115 Fiske, E. B., Ladd, H., 2000. When Sc hools Compete: A Cautionary Tale. Washington DC: Brookings Institution Press. Gale, D., Shapley, L., 1962. College admissions and the stability of marriage. American Mathematical Monthly 69, 9-15. Hastings, J. N., Weinstein, J. M., fort hcoming. Information, School Choice and Academic Achievement: Evidence from Two Experiments. Quarterly Journal of Economics. Hastings, J. N., Kane T., Stai ger, D., 2008. Heterogeneous Pref erences and the Efficacy of Public School Choice. Nati onal Bureau of Economic Research Paper Working Papers No. 12145 and No. 11805 combined. National Center for Education Statistics, 2006. Trends in the Use of School Choice: 1993 to 2003. National Center for Education Statistics, 2007. Status of Education in Rural America. Organization for Economic Cooperation and De velopment, 2008. Education at a Glance 2008. Organization for Economic Cooperation a nd Development, 2003. Programme for International Student Assessment 2003. Ozek, U., 2009. Games of School Choice under Uncertainty. University of Florida. Working Paper. Pathak, P., Sonmez T., forthcoming. Leveling the Playing Field: Si ncere and Strategic Players in the Boston Mechanis m. American Economic Review. Roth, A. E., 1982. The economics of matching: Stability and Incentives. Mathematics of Operations Research, 7(4), 617-628. PAGE 116 BI OGRAPHICAL SKETCH Umut Ozek received his Bachelor of Science degree in electrical and electronics engineering from the Middle East Technical University in Ankara, Turkey, his Master of Arts degree in economics from the University of Colorado at Denver, and his Ph.D. in economics from the University of Florida. His research interests include the economics of education, public economics and health economics. He has accepted a Research Associate position at the Urban Institute, where he will begin working in September, 2009. 116 |