UFDC Home  Search all Groups  UF Institutional Repository  UF Institutional Repository  UF Theses & Dissertations  Vendor Digitized Files   Help 
Material Information
Record Information

Full Text 
INCOMPLETELY SPECIFIED COMBINATORIAL AUCTION: AN ALTERNATIVE ALLOCATION MECHANISM FOR BUSINESSTOBUSINESS NEGOTIATIONS By JONI LYNNE JONES A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2000 Copyright 2000 by Joni Lynne Jones ACKNOWLEDGMENTS This dissertation, the culmination of my graduate education, could not have been completed without the support, patience and guidance of a multitude of people. One individual is overwhelmingly responsible for my success, Dr. Gary Koehler, my dissertation chair. His willingness to patiently share his expertise coupled with a gentle guiding hand and high expectations inspired me to seek excellence. I would also like to express my appreciation to the members of my committee for freely sharing their time and for their insightful comments. I am especially grateful to Dr. Selcuk Erenguc, my department chair, for recognizing my potential, giving me the opportunity and providing the resources necessary to achieve my goals. I'd like to extend my thanks to Dr. Jacquelyn Rees for providing a welllit path to follow on the quest for my Ph.D. Let me also acknowledge my colleagues in the Ph.D program for their unwavering moral support and friendship throughout my years in the program, especially Dr. Lawrence Nicholson, Ms. Pauline Chin, Ms. Cheryl Aasheim and Mr. Kutsal Dogan. Thanks go to my friends who have been patient, understanding and always encouraging during the many triumphs and failures along the way. I especially appreciate my dear friend, Mr. Thomas Standridge, for his tireless support. Finally, I am forever indebted to my family, particularly my mother, Rose Patton, for always believing in me. TABLE OF CONTENTS page A CKN OW LED GM ENTS ................................................................................................. iii ABSTRACT..................................................................................................................... viii CHAPTERS 1 IN TRODU CTION ......................................................................................................... 1 1.1 Introduction ......................................................................................................... 1 1.2 The Research Problem ......................................................................................... 3 1.3 M otivation ........................................................................................................... 4 1.4 Expected Contributions of this Research ............................................................ 7 1.5 Sum m ary ............................................................................................................. 7 2 A U CTION TH EORY .................................................................................................... 9 2.1 Introduction ......................................................................................................... 9 2.2 Overview ............................................................................................................. 9 2.3 Auction Types ................................................................................................... 12 2.3.1 Single Item A uctions.................................................................................. 12 2.3.2 M multiple Item Auctions.............................................................................. 14 2.4 Fram ew ork: The Benchm ark M odel ................................................................ 19 2.5 Modifying The Benchmark Model Assumptions.............................................. 21 2.6 Electronic Auctions........................................................................................... 27 2.7 Sum m ary ........................................................................................................... 30 3 APPLICATION ENVIRONMENT AND MODEL .................................................... 31 3.1 N etw ork Television Practices............................................................................ 31 3.1.1 Environm ental Constraints......................................................................... 34 3.1.2 N negotiation Strategies................................................................................ 35 3.2 A auction D description .......................................................................................... 37 3.2.1 N otation...................................................................................................... 38 3.2.2 Objective Function..................................................................................... 39 3.2.3 Constraints ................................................................................................. 39 3.2.3.1 Reservation require ent.................................................................... 40 3.2.3.2 M axim um seller coverage.................................................................. 40 3.2.3.3 M axim um spot availability ................................................................ 41 3.2.3.4 Buyer selection indicator ................................................................... 41 3.2.3.5 Cam paign com m ercial length constraint ........................................... 42 3.2.3.6 AntiClutter Control........................................................................... 43 3.2.3.7 Frequency: M ax com m ercials per show ........................................... 43 3.2.3.8 Dem graphic gross im pression guarantee ......................................... 44 3.2.3.9 Show place ent require ent............................................................. 45 3.2.3.10 Pod protection constraints.................................................................. 46 3.2.3.11 Bid specification and ordering........................................................... 48 3.2.3.12 Bidder reservation price..................................................................... 48 3.3 Sum m ary ........................................................................................................... 49 4 CON STRAINT PROGRAM M ING ............................................................................. 50 4.1 Introduction....................................................................................................... 50 4.2 Constraint Satisfaction Problem s ...................................................................... 51 4.3 Arc Consistency ................................................................................................ 53 4.4 System atic Search A lgorithm s .......................................................................... 55 4.4.1 Look Back A lgorithm s............................................................................... 56 4.4.1.1 Chronological Backtracking (BT)...................................................... 56 4.4.1.2 Backjum ping (BJ).............................................................................. 57 4.4.1.3 ConflictDirected Backjum ping (CBJ).............................................. 58 4.4.2 Look Ahead A lgorithm s ............................................................................ 58 4.4.2.1 Forward Checking (FC) ..................................................................... 59 4.4.2.2 M maintaining ArcConsistency (M AC) ............................................... 60 4.4.3 Hybrid Backtracking/Forward Checking Algorithms................................ 60 4.4.4 Im proving Perform ance............................................................................. 61 4.5 ArcConsistency Algorithm s............................................................................. 62 4.6 Stochastic Search Algorithm s ........................................................................... 63 4.7 Constraint Program m ing................................................................................... 64 4.8 Advertising Sales Application........................................................................... 65 4.9 Sum m ary ........................................................................................................... 66 5 HUERISTIC DEVELOPM EN T .................................................................................. 67 5.1 Introduction ....................................................................................................... 67 5.2 Overview ........................................................................................................... 68 5.3 Aggregate SubProblem s................................................................................... 70 5.4 Domain Management Constraint Programming............................................. 73 5.5 M aster Problem ................................................................................................. 76 5.5.1 Sorting Criteria 1........................................................................................ 78 5.5.2 Sorting Criteria 2........................................................................................ 79 5.5.3 Sorting Criteria 3........................................................................................ 79 5.6 Branch and Bound ............................................................................................. 80 5.6.1 Breadth First Search (BFS)........................................................................ 80 5.6.2 Depth First Search (DFS)........................................................................... 81 5.6.3 Fathom ing Criteria..................................................................................... 81 5.7 Determ ining an Upper Bound to PI: ................................................................. 82 5.8 Sum m ary ........................................................................................................... 82 6 SIMULATED BIDDING AGENT DEVELOPMENT................................................ 84 6.1 Introduction ....................................................................................................... 84 6.2 D ata Analysis .................................................................................................... 86 6.3 N um ber of Buyers ............................................................................................. 87 6.4 Demographic Category and Total Gross Rating Points .................................... 87 6.5 Bidder Reservation Price................................................................................... 89 6.6 Com m ercial Lengths and Frequency................................................................ 91 6.7 Lim its on N um ber of Com m ercials................................................................... 92 6.8 Selection of Product Type ................................................................................. 93 6.9 Selection of D esired Show s............................................................................... 93 6.10 Bidding Strategy................................................................................................ 96 6.11 Sum m ary ......................................................................................................... 101 7 EXPERIM ENTAL DESIGN ..................................................................................... 103 7.1 Introduction .................................................................................................... 103 7.2 Perform ance M measures ................................................................................... 103 7.2.1 Efficiency................................................................................................. 104 7.2.2 Optim ality ................................................................................................ 107 7.2.3 A auction Length ........................................................................................ 108 7.2.4 Solution M ethodology Perform ance........................................................ 109 7.3 The Experim ents.............................................................................................. 109 7.3.1 M odification of Bidder Types.................................................................. 10 7.3.2 M odification of Bid Increm ent ................................................................ 111 7.3.3 M odification of Stopping Rule................................................................ 111 7.3.4 M odification of M axim um Round Tim e.................................................. 112 7.3.5 Heuristic Methods versus Integer Programming ..................................... 113 7.4 Sum m ary ......................................................................................................... 114 8 EXPERIM ENTAL RESU LTS................................................................................... 115 8.1 Introduction ..................................................................................................... 115 8.2 A gent Sum m ary Statistics ............................................................................... 115 8.3 Bidder Type Im pact......................................................................................... 118 8.3.1 Sum m ary of Im pact of V various Bidder Types......................................... 118 8.3.2 Analysis of Bidder Type Results ............................................................. 120 8.4 Bid Increm ent Im pact...................................................................................... 121 8.4.1 Analysis of Bid Increm ent Results .......................................................... 122 8.4.2 Sum m ary of Bid Increm ent Findings....................................................... 126 8.5 Analysis of Stopping Rule Effect.................................................................... 127 8.5.1 Trickle Effect ........................................................................................... 128 8.5.2 Sum m ary of Stopping Rule Influence...................................................... 129 8.6 Influence of Seller Reservation Price.............................................................. 129 8.7 Heuristic Performance: Changing Computing Time...................................... 132 8.7.1 Standard H euristic Perform ance .............................................................. 132 8.7.2 FastM ode H euristic Perform ance............................................................. 134 8.8 Heuristic Perform ance Versus Integer Program .............................................. 135 8.9 Sum m ary .........................................................................................................137 9 CO N CLU SION .......................................................................................................... 139 9.1 Introduction ..................................................................................................... 139 9.2 Project O verview ............................................................................................. 139 9.3 Conclusions ..................................................................................................... 140 9.4 Lim itations....................................................................................................... 141 9.5 D direction for Future Research ......................................................................... 142 9.5.1 Expansion of the M echanism ................................................................... 142 9.5.2 Em pirical Investigations .......................................................................... 144 9.5.3 G am e Theory Studies............................................................................... 145 9.6 Sum m ary ......................................................................................................... 145 APPENDICES A DISTRIBUTION OF DEMOGRAPHIC REQUIREMENTS BY CATEGORY........ 146 B REGRESSION OF PRICE AND DEMOGRAPHICS .............................................. 149 C EX PERIM EN TA L RESU LTS................................................................................... 155 LIST O F REFEREN CES................................................................................................. 178 BIO GRA PH ICA L SKETCH ........................................................................................... 188 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Doctor of Philosophy INCOMPLETELY SPECIFIED COMBINATORIAL AUCTION: AN ALTERNATIVE ALLOCATION MECHANISM FOR BUSINESSTOBUSINESS NEGOTIATIONS By Joni Lynne Jones August 20000 Chairman: Gary J. Koehler Major Department: Decision and Information Sciences The popularity of auctions has increased dramatically with their introduction to the Internet. The migration has provided a unique opportunity to harness the power of computing to create new auction forms that were previously unworkable. This research presents a new auction mechanism designed to accommodate the potentially large and often complex problems that are commonly reflected in negotiated sales. I describe a new and innovative way of using an auction mechanism by modifying a combinatorial auction to accept inexact multicriteria package bids. The bids are incompletely specified yet provide enough of a framework to guide, rather than dictate, the choice of goods to satisfy stated needs. The ability of the bidder to prescribe various aspects of the sale, beyond her willingness to purchase goods at a particular price, makes it possible to use this type mechanism to replace or enhance negotiated sales. The allocation of goods requires solving a complex combinatorial problem in real time. In the past this has been considered completely impractical in a conventional auction setting. Utilizing computing resources an online auction of this nature is not only feasible but may provide a way to optimally allocate a given set of bids while satisfying bidder preferences. This auction is applicable to any collection of goods but is most appropriate for complementary goods. As expected, the proposed model becomes computationally intractable as the number of bidders increase, I therefore present simplifying heuristics to make large problems manageable. Television advertising sales provides an interesting arena in which to investigate the use of the incompletely specified auction. The auction will be employed to accommodate sales where approximately 300 to 350 buyers compete for a finite amount of commercial airtime in the upcoming season. Several constraints unique to media buying are included in the model increasing its complexity significantly which include separation of competing ads, meeting bidder's demographic group exposure requirements while ensuring the seller receives her reservation prices, allowing for a variety of commercial lengths, and accommodating specific show placement requests. CHAPTER 1 INTRODUCTION 1.1 Introduction Electronic or Internet based auctions have garnered a great deal of interest in recent years. The renewed popularity of auctions stems from various characteristics unique to this form of commerce. Web based auctions enjoy a much broader audience due to their accessibility to anyone with an Internet connection, thereby growing the buyer pool, increasing competition and thus enhancing profits. The expense and logistics of gathering at one location, a major deterrent to conventional auctions, has been replaced with inexpensive websites. Barriers to entering the electronic auction market have been lowered for all participants. Finally, electronic auctions provide us an opportunity, through harnessing the power of computing, to establish more complex trading rules and handle more complex goods. It is this opportunity upon which we attempt to capitalize in this research. Recent progress in utilizing computing and networking power includes a variety of new auction designs to facilitate the simultaneous sale of multiple items. Various mechanisms have emerged such as the popular Internet based Yankee Auction (Vakrat & Seidman, 1998) and the Groves/Vickery design (Bapna, Goes & Goupta, 1998) in which a specified number of identical items are offered for sale simultaneously with the items going to the top bidders whose aggregate demand equals the number of items for sale. Alternatively, the FCC Spectrum license sales of the early 1990s showed that by executing single items auctions simultaneously, buyers could aggregate a desired collection of goods (Crampton, 1995; McMillan, 1994). None of the above designs allow the bidder to submit a single bid for a combination of heterogeneous items, although it has been shown that in the nonauction environment the value of a bundle of positively correlated goods can be greater than the sum of the individual item values (Bakos & Brynjolfsson, 1998). Allowing the bidder to create a unique bundle of desired goods for which a single bid is submitted, referred to as a package bid, would seem like a logical way to improve the efficiency of auctioning complementary goods by capturing the synergies between product offerings. The most recent advance in auction design, the combinatorial auction, acknowledges this need and profits from the use of computing power. In the past bidding on a group of objects has been considered completely impractical in a conventional setting due to the complexity of winner determination, however utilizing computing resources an online auction of this nature is not only feasible but has proven superior to other multiobject sales mechanisms (DeMartini, Kwasnica, Ledyard & Porter, 1999). All of the auction forms mentioned so far suffer from the fact that the buyer supplies a bid restricted to item and price information. For example, most combinatorial auctions assume the bidders desire multiple goods and have a reservation price for each, thus their package bid consists of the desired quantity of each item and a per unit price, or a single price for a designated collection of items. In either case a priceitem vector is the sole basis for allocating winning bids. Constraints limiting the allocation, within the current combinatorial formats, are generally restricted to meeting reservation prices and product availability. In many cases this does not adequately reflect the needs of either the buyer or seller. Negotiated sales is a prime example of a business process that could benefit from the use of auctions as they provide an effective means of price discovery, especially for products hard to price a priori or when information asymmetries are present (EnglbrechtWiggans, 1980; Milgrom, 1989; Choi & Whinston, 1998). However, incorporating the negotiation process into an auction mechanism requires the bid to contain extended specifications. The complexities generated by the required modifications to existing auctions have discouraged widespread use of electronic negotiation models (Choi, Stahl & Whinston, 1997). This research attempts to develop a new auction mechanism that captures the intricacies necessary to enhance or replace a negotiated environment. In this chapter we provide an overview of our thesis, including a brief description of the mechanism and application environment, our motivation and the research contributions. The research problem is presented in Section 1.2, the motivation for developing the particular model is discussed in Section 1.3 and the anticipated impact of this study is provided in Section 1.4. 1.2 The Research Problem This research attempts to develop an auction mechanism as an alternative to sales usually accomplished through negotiation. The mechanism will need to provide the buyers with an opportunity to constrain the allocation of goods through a multicriteria package bid. To impart flexibility, the mechanism must allow bids that are incompletely specified yet provide enough of a framework to guide the allocation. The ability of the bidder to dictate various aspects of the sale, beyond her willingness to purchase units at a particular price, makes it possible to use this type of mechanism to replace or enhance a negotiated environment. Our research problem is to design an auction mechanism to accommodate the potentially large and often complex problems that are commonly reflected in the negotiated environment. We address this problem by modifying a combinatorial auction to accept inexact multicriteria bids. The bids must extend beyond the current priceitem vector and allow the bidder to guide, rather than dictate, the choice of goods to satisfy stated needs. We address this problem by carefully constructing a mechanism to accommodate the requirements of an industry whose sales are currently conducted exclusively through negotiation. The selection of winning bids in a combinatorial auction is an extremely complex problem, in fact it has been shown to be NPcomplete (Rothkopf, Pekec and Harstad, 1998). Therefore, another question addressed in this study is to ascertain if a heuristic allocation engine can be developed to determine a satisficing solution in real time and if so how effective is it? 1.3 Motivation The primary motivation for studying this problem was an acknowledgement of the need for an auction mechanism that more accurately reflects the demands of the market. Rarely are purchase decisions predicated solely on the price of an item, yet current auction mechanisms make allocation decisions based on this limited criterion. Secondly, the ability to harness computing power has made possible more complex auction mechanisms opening an exciting area of research and we want to expand the models available through this endeavor. Finally, we have identified an industry that may benefit from changing their current business process. We design our auction mechanism to sell commercial airtime for the Network Television Industry. Television advertising airtime is a commodity product that is currently sold through negotiations that are "frequently based on longterm relationships and editorial and demographic synergies, not just getting the lowest price (Weaver, pp.1, 1999)." Units are typically allocated on a first come first serve basis as opposed to being dictated by competitive forces that could enhance the network's ability to achieve an equilibrium based distribution of goods. The complexity of determining an allocation that simultaneously satisfies the market participants' demands restricts the seller's ability to promote competitive bargaining. This complexity is evidenced by advertising agencies widespread use of optimizerr," decision support software to assist in planning and buying media. Optimizers are purported to give knowledge and realtime information to the buyers to help them determine the best mix of media, i.e. network (including various daypart decisions), cable, syndication, billboard and print (Ross, 1998). Some suggest that if networks do not embrace technology they will get left behind (Stewart, 2000). The major networks have already seen an erosion of their market share with cable and syndication among the beneficiaries (Ross, 1998). Further evidence of the need for change in this industry is the appearance of alternative selling mechanisms. A number of webbased auctions have appeared selling excess last minute advertising inventory. These sites, such as AdOutlet.com and Adauction.com, are simplistic in nature selling single units of time that are considered "fire sale" spots or unsold airtime within close proximity to airdate. Airtime is similar to airline seats, at the end of each day unsold commercial airtime can never be recovered. These sites have been criticized for their limited offerings and their focus on "distressed" inventory (Stewart, 2000; Coleman, 1999). Proponents point out the advantage of 24 hour access and price discovery afforded by the sites (Kuchinskas, 1999). Current online advertising auctions are designed for scatter and opportunistic sales that handle shortterm campaigns and or supplemental purchases throughout the year. There is no mechanism designed to assist in "upfront sales," the large onetime sale of spots encompassing annual campaigns. This market provides an interesting arena in which to extend the current design of the combinatorial auction. The combinatorial auction is appropriate for this environment to accommodate synergies between products and consumers' desires for a collection or bundle of goods to meet their annual campaign exposure requirements. In this environment there are also ample substitutes so buyers are not restricted to obtaining a specific item but may be satisfied with any number of the substitutes available. The current versions of combinatorial auctions do not allow for substitutes or bids that do not precisely specify the desired objects. Therefore, a new mechanism needs to be developed to accommodate these market characteristics. Chapter 3 explores this industry in greater detail, including a discussion of constraints imposed on the placement of ads that must be conveyed through the mechanism. 1.4 Expected Contributions of this Research Two major contributions are expected from this research. First, we hope to develop a new businesstobusiness auction mechanism, the generalized model will be germane to environments were the purchase decision criteria extends beyond the price of the item or items desired. This multidimension combinatorial auction will accommodate business models that currently use negotiations as their primary sales mechanism and sales for products and or services that require multicriteria decisions. The other major result from this study is the development of a heuristic designed to determine auction winners and efficiently allocate inventory. We hope the heuristic will provide an optimal or near optimal allocation and thus provide a methodology applicable to other combinatorial optimization problems. 1.5 Summary This chapter has briefly introduced the research project and discussed the merits of the study as well as its anticipated contribution. The remainder of this dissertation is organized as follows. Chapter 2 will present the background literature for auctions, including an overview of classical theory as well as recent discoveries propagated by the changes wrought by the migration of auctions to the Internet. Chapter 3 will introduce the application environment and define the problem using an integer programming model. We review constraint programming, a methodology effective in solving the type of problem represented by our auction in Chapter 4. The fifth chapter describes the heuristic development in detail. We intend to test our mechanism's performance through simulation and in Chapter 6 we present the characteristics of the artificial agents that will represent bidders in our experiments. The agents are modeled to be representative of the industry participants, their characteristics reflect patterns discovered from data provided by a major corporation in the television industry. In Chapter 7 we outline the experiments that will be conducted to judge the performance of the mechanism. Details of experimental results as well as an interpretation of the findings are given in Chapter 8. Finally, conclusions from the research and identification of future directions for this study encompass the remainder if this dissertation. CHAPTER 2 AUCTION THEORY 2.1 Introduction This chapter presents a general overview of Auctions. The goal of this chapter is to review classic auction theory and various extensions that have originated from it. Section 2.2 will provide a general overview of auction theory, specifically defining an auction, its players, various auction categories and the settings that favor their use. Section 2.3 describes various auction types, both single item and multiple object formats. Section 2.4 will present the well accepted classical or "benchmark" auction model. Section 2.5 looks at modeling issues that must be considered and broadens our perspective by reviewing several auction situations that either extend or restrict the benchmark model. Section 2.6 reviews the impact of the Internet on auctions. 2.2 Overview "An auction is a market institution with an explicit set of rules determining resource allocation and prices on the basis of bids from the market participants" (McAfee & McMillan, 1987, pp.701). It attempts to match a buyer with a seller to achieve a market clearing equilibrium price (EngelbrechtWiggans, 1980). Formalized trading procedures govern the players' interaction based on specific rules for competitive bidding and trade execution (Klein & O'Keefe, 1998). Choi and Whinston (1998) describe an 9 10 auction as simply posted prices where the price movements are more rapid and the number of participants greater. Auctions are the simplest and most familiar means of price discovery (EngelbrechtWiggans, 1980). Thus, they are effective when items are hard to price (Beam & Segev, 1998), for instance when demand for an item is difficult to determine a priori or when a product's value varies greatly (as with time sensitive products or services, which tend to lose their value rapidly) ( Chio & Whinston, 1998; Bapna et al.,1998; Milgrom, 1989). Auctions are commonly classified by the goods traded or the rules used that determine the final price (Lengwiler, 1999). Some auctions are performed in real time while others accept bids over time to be matched at a specified later date (Klein & O'Keefe, 1998). A typical auction consists of four major components: players, objects, payoff functions and strategies (EngelbrechtWiggins, 1980). Players include the bidders, the seller and the auctioneer. Most literature views auctions from the perspective of the seller who owns the item(s) for sale and is attempting to maximize his profit. Conversely, the bidder attempts to minimize the price, thereby maximizing her utility. The seller, as the Stackelberg leader or first mover, normally precommits to a set of policies, choosing the auction form and rules (McAfee and McMillan, 1987). A Stackelberg leader is implicitly assumed to commit first to his chosen action and not change that action after receiving additional information, even though changing might be profitable ex post (Rasmusen, 1989). This seeming advantage is tempered by information asymmetries. The seller and buyers do not know the other player's valuation of the object to be auctioned (Choi & 11 Whinston, 1998). The auctioneer, in the traditional setting, is simply a facilitator bringing together the buyer and seller. The number of items offered, value information, physical characteristics and type characterize objects (EngelbrechtWiggins, 1980). Single objects are the common focus of classical auction theory, however objects may be single divisible or indivisible items, a package of nonidentical items, or multiple homogeneous items. Value information broadly defines who knows what. There are two commonly assumed models, the Independent Private Value model (IPV) and the Common Value model. If bidders know with certainty the value they individually place on an item the auction uses the Independent Private Values model. Individual valuations have a common distribution but are statistically independent of the other customers' valuations (Beam, Segev & Shanthikumar, 1996). A Common Value object, on the other hand, is assumed to have a single value but information regarding this value is varied among bidders. The bidders' valuation is dependent on at least one common objective variable, possibly resale value or future royalties (Kagel, 1995; Das & Sundram, 1997; Milgrom, 1989). Due to the statistical dependence inherent in the Common Value model, bidders tend to infer information from others' bids (Beam, et al., 1996). Milgrom and Weber (1982) refer to the positive correlation between bidder valuations in the Common Value model as affiliation. The payoff function involves decisions surrounding the financial transfer of a product such as the award mechanism or the rule used to determine the winning bid, final price and recipient, the presence or absence of a reservation price, and other participation costs (EngelbrechtWiggins, 1980). Occasionally included in the calculation of the 12 payoff function are charges for preparing and submitting bids and participation fees. The price paid by the winner for the object can depend solely on the final bids or on something correlated with the item's values such as royalties (McAfee & McMillan, 1987). In single object auctions, the payoff function routinely awards the object to the highest bidder if that bid exceeds the seller's reservation price. The seller has the right to retain the item should bidding fall below a predetermined minimum amount, referred to as his reservation price (Milgrom, 1989). Finally, strategies refer to how a bidder executes her bid. Milgrom (1989) defines a "pure" bidding strategy as one that is based on a function of the information the bidder knows. If all bidders accurately predict the other participants' bidding strategies and then use that information to select their own strategy, Nash Equilibrium is achieved (Nash, 1950). Nash Equilibrium (or rationalexpectations) strategies maximize the expected utility of the outcome and is used routinely as the strategy of choice in classical auction theory (EngelbrechtWiggins, 1980). 2.3 Auction Types 2.3.1 Single Item Auctions There are four basic types of auctions used when a unique item is bought and sold. They can be classified by the rules that govern the exchange of goods. The rules affect bidding strategies and incentives and thus transaction efficiency. English Auction: By far the most common type of auction, the English auction is an oral, outcry, ascending auction where progressively higher bids are solicited until only one bidder remains. The object is awarded to the remaining bidder at the price equal to 13 her highest bid. At equilibrium the bidder who values the item the most will retain the object at a price equal to their second highest valuation, therefore the English auction is efficient (Milgrom, 1989). Due to the open nature of the auction, bidders can observe the behavior of other bidders, process this information and dynamically modify their reservation prices (under the Common Value model) (Beam, Segev and Shanthikumar, 1996). The dominant bidding strategy used is to bid until the price exceeds the buyer's willingness to pay, normally a small increment below the bidder's true valuation (Beam, et al, 1996). Examples include art, antique and livestock auctions. Dutch Auction: Similar to the English auction, the Dutch auction is an open outcry, oral auction. It differs only in the direction of the bid progression. In a Dutch auction the auctioneer calls out an initial high price and then successively lowers the price until a bidder claims the object, normally by shouting "mine!"(Choi & Whinston, 1998). Unlike the English auction, since the auction concludes with the first bid, bidders cannot gain signal information from the behavior of other bidders (Beam et al., 1996). The dominant bidding strategy used is to claim the object when the bid equals a small increment below the bidder's true valuation (Beam et al., 1996). The increment between the bidder's true valuation and the actual bid is the buyer's surplus. The cut flower market is an example of this type of auction. First Price Auction: This mechanism is also known as the First Price Sealed Bid Auction. Potential buyers submit a single sealed written bid. Bids are opened simultaneously and the item is awarded to the buyer who submitted the highest bid at a price equal to her bid (Milgrom, 1989). As in Dutch auctions, information cannot be gleaned from observing other bidders' behavior, therefore bidding strategies are based on 14 the participant's individual value of the object and the expected behavior of the other customers (Beam, et al., 1996). This auction may be inefficient since the bidding strategy employed is affected by information asymmetries and thus may result in not awarding the item to the highest value bidder (Milgrom, 1989). Government procurement contracts commonly use this auction type. Second Price Auction: This is also known as the Vickrey or sealedbid second price auction. As with the first price sealed bid auction, buyers submit sealed bids with the highest bidder claiming the object. However, as the name implies, the price paid for the item is equal to the second highest bid rather than the winning bid (Vickrey, 1961). Bidding one's true valuation is the dominant strategy for this auction since the object will be awarded at some increment below the winning bid, ensuring consumer surplus (EngelbrechtWiggans, 1980). The Vickrey auction duplicates the principle characteristics of the English auction and, due to the similar theoretical properties discussed in depth in section 2.4, as well as notational efficiency, English auctions are customarily modeled as Second Price auctions (Milgrom, 1989). Variations of the four general categories have spawned a multitude of auctions. For example, fees can be charged for participation or the seller may impose a reservation price below which he will not sell (McAfee & McMillan, 1987). 2.3.2 Multiple Item Auctions Vickrey (1961) first proposed the "multiple auction" where several identical units are sold to bidders who desire but one item. Two variations of the progressive auction, simultaneous and sequential, are most commonly used to facilitate multiunit sales. How 15 the auction(s) are conducted within the multiunit environment (i.e., sequentially or simultaneously) propagate a variety of alternative designs. Additionally, recent progress has been made in facilitating package bidding within a multiobject variation of the traditional English auction. Sequential auctions utilize one of the standard or modified auction forms but execute them serially until all goods have been exhausted (Vickery, 1961). Problems exist with sequential auctions making them less attractive than simultaneous auctions. First, they reduce the bidder's ability to efficiently aggregate. Once an auction is complete the items are allocated, if in subsequent auctions the bidder is unable to obtain a complementary product, the value of the previously acquired object diminishes. With the sequential auction you can not modify earlier bids. McAfee and McMillan, (1996) and Ashennfelter (1989) also discovered that sequential auctions produce various prices for identical items depending on their position in the sales sequence (later sales tend to fetch greater prices). Finally, bidders that are budgetconstrained can be eliminated by one or a group of bidders driving up the price in the early rounds, thus effectively exhausting the constrained budget (Benoit & Krishna, 1998). Revenue comparisons conducted by Benoit and Krishna (1998) show that sequential auctions have appropriate applications. Sequential auctions were discovered to outperform simultaneous if objects are substantially different or if the complementarities are significant (Benoit & Krishna, 1998; Krishna & Rosenthal, 1995). Alternatively, a collection of auctions equal to the number of objects could be held simultaneously. Simultaneous ascending auctions are covered in detail by McMillan (1994), Krishna and Rosenthal (1995), Crampton (1995), McAfee and MccMillan (1996), 16 and Milgrom (1997). With simultaneous ascending auctions bidding is open for multiple items at the same time and remains open as long as there is active bidding on some unit. Bidding occurs in rounds with the results posted at the end of each (Milgrom, 1997). Information access allows participants to analyze their position with respect to the bundle of items they hope to acquire. Keeping all auctions open gives a bidder the flexibility to aggregate products in the manner she chooses and reconfigure those choices should the combination become too expensive (McAfee and McMillan, 1996). Arguably the most widely known form of multiobject simultaneous auction are the FCC Spectrum auctions. The government allocated PCS narrowband licenses using a simultaneous ascending auction, dubbed "the greatest auction in history" by the New York Times which was designed specifically to facilitate the aggregation of multiple licenses by a single buyer (Benoit & Krishna, 1998). The positive synergy between licenses in adjoining areas was acknowledged by the architects, but combination or package bidding was not allowed due to its perceived complexities (Cramton, 1995). Simultaneous auctions are subject to an exposure problem, a phenomenon causing bidder losses (DeMartini, et al. 1999). Losses result from "mutually destructive bidding" where bidders unable to obtain a complete set of goods due to competition are left holding goods priced at more than their value (Bykowsky, Cull & Ledyard, 1995). To protect themselves, bidders may bid less aggressively precipitating reductions in efficiency. To overcome this problem investigators have suggested allowing package bids (Ausubel, Cramton, McAfee & McMillan, 1997). Very little work had been done involving bundling of auction goods until the FCC auction design debate highlighted an assignment void involving auctions for multiple 17 heterogeneous goods with synergies. Palfrey (1983) had analyzed bundling decisions for multiple heterogeneous objects where demand was uncertain and found that seller surplus diminishes as the number of bidders increase. Information asymmetries led the seller to bundle items for which higher individual prices could have been obtained. Kim (1996) later derived an equilibrium model in which bidders incorporate their individual and complementary valuations so as to ensure that the auction is efficient with respect to the bundled value while not actually auctioning the bundle as a whole. Enlisting the help of electronic agents, Fan, Stallaert and Whinston (1998) introduced "bundle trading" to effectively construct investment portfolios. If complementarities exist among the items being sold, evidence suggests that it may be more appropriate to permit bidders to bid for packages, rather than simply bidding item by item such as in the FCC auction (Banks, Ledyard & Porter, 1989; Bykowsky, Cull & Ledyard, 1995; McMillan, Rothschild & Wilson, 1997; Ledyard, Porter & Rangel, 1997; DeMartini, Kwasnica, Ledyard & Porter; 1999). Unlike the work of Palfrey (1983) where the seller creates the bundle of goods, bidders form the bundles by submitting package bids in what is termed a "combinatorial auction." One of the first investigations into this auction format was conducted by Rassenti, Smith and Bulfin (1982) and allowed package bidding to allocate airport time slots to competing airlines. They employed a "setpacking" algorithm to determine resource shadow prices later used to ensure package prices fell at or below bid amounts. Empirical tests showed the mechanism to be efficient and demand revealing. Rothkopf, Peked and Harstad (1998) describe this type of auction as one that facilitates a bidder's desire to submit bids for combinations of assets. 18 Three newly designed implementations of the combinatorial auction, the Adaptive User Selection Mechanism (AUSM), the Resource Allocation Design (RAD) auction and a Webbase implementation are being investigated to meet multiple item allocation needs when synergies between or among objects exist. The Adaptive User Selection Mechanism is a modification of the English ascendingbid auction that allows both package bids and individual item bids. Described by Banks et al. (1989) as a decentralized mechanism, with continuous bidding communicated via an electronic bulletin board. Initial implementations suffered from a new phenomenon, the threshold problem. It was discovered that when package bids are allowed, small bidders may not be able to dislodge a large but inefficient package bidder (see DeMartini et al.(1999) for an excellent description). A "standby" queue was added to facilitate coordination among bidders to form a large enough collection of small bids to displace the current winners) (Banks et al., 1989). Although the queue solved the threshold problem, it added complexity to the mechanism. The Resource Allocation Design (RAD) Auction simplifies the AUSM by introducing a new pricing rule eliminating the need for the standby queue and thereby overcoming the AUSM's complexity issue. A vector of single item prices is internally computed from bids and used to check minimum bid increment requirements and convey information to bidders. Bid opportunities are presented to bidders combating the formation of thresholds (DeMartini et al., 1999). Finally, Teich Wallenius, Wallenius, & Zaitsev (1999) offer an electronic auction for multiple homogeneous units that allows the bidders to specify if they will accept partial fulfillment of their package bid. Additionally, due to its semiclosed nature, sellers can establish various reservation prices for different quantity levels thus facilitating price 19 discrimination. The auction is sealed in that bidders do not have access to the bids of others but the mechanism recommends entering bids to inactive bidders, or bids that based on current conditions would be a potentially winning bid. This design has been shown to overcome the "winners' curse" prevalent in common value auctions. Other problems faced in this arena stem from the sheer number of possible bundle combinations. In light of the complexity, Rothkopfet al. (1998) investigate how to determine a revenue maximizing set of nonconflicting bids and identify structures that are computationally manageable. Placing certain restrictions on the family of permitted bids formed the basis of their analysis. Nested combinatorial bids that form a single tree structure provide, through "rolling back" the tree, a straightforward way to determine the revenue maximizing outcome (Rothkopfet al., 1998). Additionally, if bid combinations are composed of either at most doubletons or at least a large proportion of the number of available assets, maximizing algorithms could be found that are mathematically tractable. Bids with the intervening cardinalities are considered NPcomplete (Rothkopf et al., 1998). Some geographic structures such as an interval of consecutive assets or items that could be organized into a kdimensional matrix, thereby permitting row or columnwise bids, were proven by Rothkopf et al. (1998) to be computationally manageable. 2.4 Framework: The Benchmark Model The next logical question to be answered is which of the auction forms is optimal? Bulow and Roberts (1989, pp. 1060) define an optimal auction as a "bidding mechanism designed to maximize a seller's expected profit." This complex topic has received a great deal of scholarly attention including work by Vickrey (1961), Myerson (1981), Bulow & 20 Roberts (1989), Riley and Samuelson (1981), Milgrom and Weber (1982). We will begin our analysis by presenting a simple framework. The following standard assumptions, gathered by McAfee and McMillan (1987) from early game theory models such as those defined in Vickrey's 1961 seminal paper, define the "Benchmark Model." The assumptions are common knowledge among the participants. 1. All participants are risk neutral (bidders and sellers). 2. Bidder valuations are independent and private (Independent Private Value Model). The value (vi) that bidder i places on an object is independently drawn from a distribution Fi. Note: although the individual values are private, all players know the distribution governing their valuation. 3. Bidders are symmetric, every buyer has the same cumulative distribution denoted by F. 4. There are no fees associated with the auction. The price paid by the winning bidder is dependent entirely on the bids themselves. Based on these fairly restrictive assumptions, each of the four basic auction types results in the same expected revenue. This notion is the basis of the Revenue Equivalence Theorem which states that for the benchmark model, the four standard auction forms yield the same price on average (Das & Sundram,1997; Milgrom, 1989; Vickrey, 1961; OrtegaReichert, 1968; Myerson, 1981; McAfee & McMillan, 1987). Fueling the field of optimality study are modifications to the basic or benchmark assumptions. 21 2.5 Modifying The Benchmark Model Assumptions There are several modeling issues that warrant consideration when choosing particular auction rules. Most, but not all, correspond to the benchmark model assumptions of risk aversion, value formulation, bidder asymmetry, fees and the number of items to be sold. Strength of bidding competition and the potential for collusion also must be addressed. Uncertainty is central to choosing to utilize an auction as the sales mechanism. Should the seller possess perfect information, the need for price discovery would be moot and posted prices would optimize the seller's surplus (McAfee & McMillan, 1987). How the participants deal with this uncertainty determines their degree of risk aversion. Studies by Hanson and Menezes (1968), Baron (1972) and McAfee and McMillan (1987) confirm that varying the degree of risk aversion affects bidders' behavior. The buyer's surplus received by a bidder i with valuation v, bidding bi is v, b1 if she is awarded the item and zero otherwise. To enhance the probability of winning, the riskaverse bidder increases the size of her bid, reducing her surplus and increasing the seller's surplus (Das & Sundaram, 1997). If either the seller or buyer is riskaverse, the seller prefers the Dutch or firstprice auction (Harris & Raviv, 1981; Holt, 1979; Milgrom & Weber, 1982). When bidders are asymmetric (valuations are no longer based on a common distribution F), the Revenue Equivalence Theory does not hold (Das & Sundaram, 1997; McAfee & McMillan, 1987). Competition from the now disparate bidders has an effect on the determination of bids for the first price sealed bid auction since its bidding strategy 22 considers both the individual's value and that of the second highest competitor. Within this environment two bidders valuing the item equally may evaluate their nearest rival's value for an item differently, thereby leading to incongruous bids. The inconsistent bidding may award the item to a bidder without the highest value making this auction form inefficient (Das & Sundaram, 1997; McAfee & McMillan, 1987). Auction mechanisms with bidding strategies dependent on the participant's individual valuation (i.e., English, Dutch and Vickrey) remain efficient. The benchmark model assumes independent private values. Relaxing this assumption gives rise to another extreme, the common value model. Here bidders guess an item's unique true value (McAfee & McMillan, 1987). A phenomenon inherent with the common value assumption is the "winner's curse;" defined by EngelbrechtWiggans (1980 pp. 133) as "when the individual to whom the object is awarded tends to be the one who most overestimated the true value of the object." Information asymmetries account for the variation in bids. Here, Milgrom and Weber (1982) show that the English auction performs best in generating the greatest revenue followed by the secondprice auction. Dutch and firstprice auctions are equivalent as least effective under the common value assumption. Value uncertain bidders can acquire additional information by observing the behavior of other bidders in the English auction. Additionally, Milgrom and Weber (1982) propose that the seller can increase his expected revenue by providing the bidders with information correlated to the item's true value. This phenomon is referred to as the linkage principle is based on the fact that with additional information initially lowvalue bidders will raise their estimates, thereby promoting more aggressive bidding (Milgrom & Weber, 1982). More recent literature has found that this principle does not hold beyond single item auctions (Perry & Reny, 1999) So far we have based bidder payments entirely on the bids. Relaxing this assumption the seller can obtain additional information about valuations (McAfee & McMillan, 1987). There are many auctions, such as for book publishing and mineral rights, where payment depends on both the bid and information revealed ex post (via royalty rate or sharing parameter) (Das & Sundaram, 1997). The price paid p is a combination of the bid price, a royalty rate r, and the value V of the object unknown at the time of the auction p = b + rV. Introducing a royalty rate reduces the impact of the inherent variances in bidder valuations, thus inducing bidders to act more aggressively. Seller revenue rises with aggressive bidding (McAfee and McMillian, 1987). Most auction theory literature assumes that bidders act noncooperatively, that is they do not agree to modify their competitive bidding behavior to manipulate equilibrium pricing (Kagel, 1995). In reality, collusion exists in the form of cartels or rings; agreements between bidders regarding bidding aggressiveness and/or predetermination of winners (McAfee & McMillian, 1987). Mead (1967) hypothesized that ascendingbid auctions were more susceptible to collusion than were sealedbid auctions. Milgrom (1987) confirmed these results using the case of two bidders agreeing to alternate wins. The intuition for this result rests with being able to hide secret price concessions using a sealed bid mechanism, which is impossible in an open ascending auction. Bidding rings or cartels operate on the premise that without competition from the other ring members a designated member can obtain the item at a reduced price. The item is then reauctioned 24 among the cartel members, with members sharing in the proceeds resulting from the difference in original auction price and the cartel auction price (McAfee and McMillan, 1987). To combat these activities, Cassady (1967) recommends establishing a reservation price that increases with the number of potential cartel members. Another consideration for auction models is the potential strength of bidding competition. Holt (1979) and Harris and Raviv (1981) have shown that increasing the number of bidders increases seller revenue on average. They propose that the greater the number of bidders, the smaller the gap, on average, between the value of the highest and second highest bidder (the winning price). In independent private valuation first price auctions the uncertainty of the number of participants can be exploited. McAfee and McMillan (1987) show that if the number of bidders is unknown and they have constant or decreasing absolute risk aversion, then concealing the number of bidders enhances revenue. The results so far have applied to auctions for a single item. The impact of varying the number of items for auction is gaining a great deal of attention. Both Vickery (1961) and Weber (1983) look at how the "benchmark model" holds in this setting. They discovered that using the Independent Private Values model (IPV) and Nash equilibrium bidding strategies, sustains the Revenue Equivalence when bidders take only one item. Kim (1996) and Lengwiler (1999) look at the issues raised when auctioning more than one item. When the bidder can purchase more than one unit, Engelbrecht and Kahn (1998) claim striking differences emerge. Namely, although a weak form of revenue equivalence holds, the various auction formats (i.e., uniform price, discriminatory or pay yourbid, and Vickrey) allocate units differently. Attempts have been made to establish 25 which auction performs better for the seller (Back and Zender, 1993; Noussair, 1995; Katzman, 1995; EngelbrechtWiggans & Kahn, 1998). EngelbrechtWiggans and Kahn (1998) discovered that with uniform price auctions (a common form of simultaneous multiunit auction in which high bids win the units, but all units are sold for the same price) tend to encourage zero bids. Payyourbid auctions, requiring the winners to pay the individual winning bid, outperform uniform price auctions according to Katzman (1995) while the Vickrey auction provides the seller with the most revenue. The complexity of multiunit auctions has also affected the study of mechanism optimality. Armstrong (1999) considers the optimality of heterogeneous multiple unit auctions with package bids. Although his analysis was limited to two objects, he found that bundled auctions are efficient but generate different revenue which are strictly optimal in some circumstances. Revenue equivalence does not generally hold within this environment if values are discretely distributed (Armstrong, 1999). The presence of financial constraints introduces important differences into traditional auction theory. Most notably the revenue equivalence theorem fails (Pitchik & Shotter, 1988; Che & Gale, 1996, 1998). In the case of private information and absolute spending limits, Che and Gale (1996, 1998) found that firstprice auctions yield higher expected revenue and social surplus than the other standard auction forms. Also a subsequent bidder's payoff is influenced by the price paid by rival bidders (Benoit & Krishna, 1998; Pitchik & Shotter, 1988). When faced with budgetconstrained bidders, the order that items are presented for sale in a sequential auction is important. Benoit and Krishna (1998) demonstrated that it is not always optimal to sell the more valuable object first. Budget constraints likewise impact the strategic bidding behavior of auction 26 participants. Pitchik and Schotter's 1988 experiments using sequential auction with complete information discovered that the trembling hand perfect equilibrium is more representative than Nash equilibrium in predicting prices. Trembling hand perfection allows for bidders to "tremble" or make mistakes but eventually equilibrium will be achieved by taking their rival's mistakes into consideration in the limit or in this case at the completion of a sequence of auctions (Fudenberg & Tirole, 1991). This equilibrium is robust enough to compensate for the possibility that some players may not play their dominant strategies (Rasmussen, 1989). Financial constraints may be absolute with a preset upper bound, be limited to a certain amount on average, or may be determined endogenously as part of the strategic auction decisions (EngelbrechtWiggans, 1987; Benoit & Krishna, 1998). To combat the negative effects of budget constraints Che and Gale (1996) propose a policy that alleviates indivisibility of the good, allows joint bidding, and offers seller provided financing. Most of what we know about combinatorial auctions has been discovered through empirical studies. Recent experimental work by DeMartini et al. (1999, pp. 22) comparing multiobject auctions reveals "the option to bid for packages clearly improves performance in difficult environments, and does not degrade performance in simple environments." They also presented evidence that the RAD combinatorial auction outperforms both noncombinatorial models and earlier combinatorial (i.e., AUSM) models. Performance was judged on efficiency, auction length, and bidder losses. From the seller's perspective, package bidding was shown to reduce revenues as a percent of the maximum possible (but not average seller revenue). An interesting caveat of their investigation looked at the tradeoff between bidder profits and seller revenue and 27 suggested that high seller revenue is driven by high bidder losses (DeMartini et al., 1999). This posses two concerns, that of possible bidder default when faced with copious expenses as well as potentially decreasing consumer goodwill. The RAD design was able to achieve revenue gains while at the same time yielding high seller revenue, implying the mechanism is effective in minimizing the tradeoff between the opposing objectives (DeMartini et al., 1999). Solving a large problem with this mechanism could prove computationally intractable requiring some sort of heuristic to reach an acceptable conclusion (Rothkopf et al., 1998). Recently twosided versions of these combinatorial auctions have been deployed, one for trading environmental emissions permits and another to support bond trading. The bond trading mechanism was built to handle 2,000 bonds (commodities) and 50,000 bids. Given the complex bids allowed, a lot of non convexities, this means about 200,000 variables and 300,000 constraints. A fully relaxed linear program solution takes about 20 minutes to solve. The heuristic algorithm produces a solution which exceeds 85% of the best known bound, 90% of the time (Personal correspondence Ledyard, 1999). This is the first largescale introduction and will prove an interesting test of the tractability of the auction form. 2.6 Electronic Auctions The introduction of auctions on the Internet has heralded a resurgence in their popularity as a selling mechanism and as an area of academic research. Much of the classic auction theory is being reevaluated in light of the new medium and plentiful data. An empirical investigation by Bapna, Goes and Gupta (1998) revealed that classical assumptions might not hold with electronic auctions. Namely, they found heterogeneity 28 among bidders, characterized by different bidding motivations such as the entertainment value of participation and other rational and irrational bidding strategies. Introduction of software agents, or automated bidding mechanisms, described as "hyperrational" by Varian (1995), may account for some of this heterogeneity. Yet another study suggests the revenue equivalence theorem is not supported for online auctions (LuckingReiley, 1999). A field study of 100 online auctions by Beam and Segev (1998) recommends a set of criteria for "good" auctions and provides an overview of current practices. Traditional auctions differ from electronic auctions in several ways that may account for the departure from classical theory. For example, a traditional auction is held at a physical location, is conducted by an auctioneer and lasts a few minutes. In the classical setting this leads to a great deal of expense to establish a site, employ an auctioneer and gather the potential customers. Goods must be transported to a central location and may not be easily examined due to time or physical limitations (Turban, 1997). Conversely, electronic auctions close on average once per week, with many closing daily or hourly, they can be conducted anywhere, use an electronic agent as the auctioneer, and multimedia and database facilities allow for extended complexity of the trade object description (Beam and Segev, 1998; Klein and O'Keefe, 1998). Additionally, the Internet provides a global pool of potential bidders suggesting more aggressive bidding resulting from increased participation. Electronic auctions have lowered entry barriers for all auction participants including auctioneers, suppliers or sellers and consumers (Klein & O'Keefe, 1998). For example, EBay is an online auction that provides a forum where any seller can submit items for sale and reap the benefits of worldwide exposure. An opportunity associated with electronic auctions is the 29 potential to establish more complex trading rules through utilization of the environment's computing power (Klein & O'Keefe, 1998). One of the most noteworthy departures from the classical auction format is the emergence of various multiple object auctions. The strategy of distributing the multiple objects amongst winning bidders and preferences given to those who bid in bulk have formed different online auction mechanisms. For example, one of the most popular on line auction forms is the multiunit English auction. A common version of this auction type is the "Yankee Auction" in which a specified number of identical items are offered for sale simultaneously. At the close of the auction, the highest bidders win and pay their bid price (Vakrat & Seidmann, 1998). Bids are ranked in order of price, then quantity, then time of initial bid. Specifically, if two or more bids are for the same price, the larger quantity bids take precedence over smaller quantity bids, while if bids and quantity are the same, then the earlier bid takes precedence over later bids (Beam & Segev, 1998). Bapna, Goes and Gupta (1998) describe a modification of the Vickery Auction used on line which adopts a uniform pricing scheme over the collection of items where each winner pays a price equal to the highest rejected bid. Modeling the emerging online auction forms has proven a formidable task. Vakrat and Seidmann (1998) establish a model that incorporates the "time dimension" of online auctions or the impact of extending bidding over extended periods of time and space. Beam et al. (1999), using a Markov Chain, were able to model a typical online single item auction and extend their model to allow for the sale of multiple identical items where each bidder wants at most one item. The lack of adherence to classical auction theory assumptions and the variety of online mechanisms have hampered the quest for a 30 single concise model. The most recent advances, combinatorial auctions, have only been investigated experimentally and due to the heuristic nature of their solutions have yet to be modeled definitively. 2.7 Summary In this chapter we presented a synopsis of classical and emerging auction theory. A great deal of scholarly research exists in the field providing well established guidelines on everything from bidding strategies to optimality of various auction forms. New discoveries are emerging due to the changing environment heralded by auctions migrating to the Internet and the evolution propagated by harnessing computing power. Unique, previously unworkable, auction forms have been introduced that do not readily conform to classical theory therefore revitalizing this exciting area of study. CHAPTER 3 APPLICATION ENVIRONMENT AND MODEL 3.1 Network Television Practices Television advertising sales provides an interesting arena in which to investigate the use of the Incompletely Specified Combinatorial Auction (ISCA). Sissors and Bumba (1989) describe network television as a negotiated medium; similar to commodities bought and sold on the commodities exchange. There are three markets for television, "upfront "or longterm, "scatter" or short term and "opportunistic" or last minute buys (Katz, 1995). The majority of sales are conducted during "upfront" where contracts usually involve campaigns spanning an entire broadcast year. Scatter buys are generally for an upcoming quarter while the sale of excess last minute inventory is referred to as opportunistic buys (Sissors & Bumba, 1989). Advertisers seek to maximize the number of exposures to their desired demographic group per dollar or to minimize the cost per thousand viewers (CPM) while networks attempt to maximize the dollars received per show. Currently all sales are negotiated with no fixed price for placement in individual shows and evaluated on the CPM for a single designated demographic category. Media buyers must meet a predetermined amount of exposures to satisfy campaign goals. Supply and demand plays a significant role in negotiations for what can be considered a perishable good, since any 32 commercial time that is unsold at airtime can never be recovered. Advertiser demand for show placement, viewer demand indicated by audience delivery estimates, network overhead and expenses, and the proximity to airdate combine to form the basis for network reservation prices (Sissors & Bumba, 1989). Heterogeneity in campaign length and size, the advertised brand's buying power and estimated show ratings determine an advertiser's valuation for individual units and collectively entire campaigns. These valuations can vary greatly, as do their budgets. Auctions, due to their ability to facilitate price discovery and maximize seller surplus for items with widely dispersed values, are logical sales mechanisms for this environment. The combinatorial auction with its ability to accept package bids is best suited, among the current auction designs, to accommodate the synergies between products as well as the need to aggregate or bundle goods to meet buyer's campaign exposure requirements. In this environment there are also ample substitutes so buyers are not restricted to obtaining a specific item but may be satisfied with any number of substitutes available. The current versions of combinatorial auctions do not allow for substitutes or bids that do not precisely specify the desired objects. Our auction will be employed to accommodate the following "upfront" sales practices. Commercial sales are negotiated by daypart, i.e., specific time slots within the day such as primetime, daytime, kids and sports. We will concentrate on primetime where approximately 250 to 300 buyers compete for a finite amount commercial airtime in the upcoming season. Primetime extends from 8 p.m. to 11 p.m. with shows varying 0 POD POD SPOE POD 15 451 POD POD 30 Figure 3.1 Example of Pod Placement in an Hour Show in length from 30 minutes to 2 hours. An hour show generally contains 5 to 7 commercial pods and roughly 4 to 8 15second slots per pod. See Figure 3.1. A pod is a collection of commercials normally lasting one to two minutes and includes a number of commercials of various lengths. The base unit in our auction will be the 15second spot. All national television time is priced based on a 30second spot. Advertisers wishing to utilize longer or shorter duration commercials (i.e., 15 second or 60 second) can expect the amount charged to be adjusted according to the length (Katz, 1995). Rarely, networks will charge a premium for handling of nonstandard commercial lengths to discourage a large number of 15second commercials that contribute to clutter. Clutter, generated by an overwhelming number of ads appearing in a show or pod, dilutes the strength of the advertiser's message (Sissors & Bumba, 1989). Our model assumes the price for a 15second (60second) commercial is onehalf (twice) that of a 30. Campaigns can consist of either a single length commercial or a mix of lengths. For example, an advertiser may run strictly 15, 30, 45 or 60second commercials, or any combination of these lengths in a single campaign. 34 Although it would appear most cost effective to purchase all 15second commercials, since reach, or the number of viewers exposed to a commercial, is relatively the same regardless of commercial length, the 30second commercial is most prevalent. It has been suggested that 30second commercials better satisfy the creative needs of advertisers who are seeking to gain both consumer attention and convey the product message. To allow comparison between packages of various length commercials, advertisers use 30second equivalents, where a 15second commercial represents a half a unit, with 30seconds being the base unit and a 60second commercial is comparable to 2 units. We simplify our calculations by redefining the base unit as a 15second commercial representing 1/2 the reported demographics and the longer commercials a multiple of this new base unit. This corresponds with our treatment of unit pricing. Although, as noted earlier, the amount of exposure is not effected by the length of commercial, in this scenario the demographics are scaled by the length of the commercial. The show's list price is then divided by the total number of exposures to determine the Cost Per Million (CPM). This equivalized CPM allows the buyer to analyze the best mixture of commercial lengths. 3.1.1 Environmental Constraints There are constraints imposed on the placement of ads. It is common practice to guarantee that competing products do not appear in the same pod, referred to as "pod protection." Two similar products can advertise in the same show but every effort is 35 made to ensure that they do not appear in the same commercial break. However, 15 second commercials are excluded from this protection. Media buyers often express preferences for placement in particular shows and occasionally sales are conditional on acquiring those specified shows. In addition, buyers may require that their advertisements not appear in selected shows due to what may be deemed inappropriate content. 15second commercials require special handling. To avoid clutter, industry practice allows at most two 15's in the same pod. Duplicate ads, or more than one advertisement of any length from any one buyer, are not allowed to appear in the same pod. However, this restriction is relaxed for 15second commercials. If, for example, an additional 15second spot is required to complete a pod then a single advertiser's 15's may be "bookended" or placed at the beginning and end of the same pod. Airdates are also a critical consideration. Although, advertisers may not have specific tastes for individual programs, they may require that an ad appear on certain dates to coincide with other media campaigns (i.e., radio, billboard, cable television, etc). Campaigns are scheduled as continuity (continuous over a length of time), bursts (ads placed at a specific frequency over an extended period such as twice a month all year), or flights where ads aired for specific periods are followed by periods of inactivity (Katz, 1995). 3.1.2 Negotiation Strategies Individual show pricing and commercial availability statistics are considered proprietary and jealously protected by both the network and advertisers. Although 36 television stations may publish "rate cards," the rates shown are viewed as the starting point for negotiations and do not reflect the ultimate prices (Merskin, 1999). Both parties exploit these information asymmetries during mediation. There is considerable negotiation back and forth in terms of what the media buyer is willing to pay for a particular offering or collection of show placements and what the network representatives feel is a fair and acceptable price. In this industry there is no after market where buyers can sell directly to other advertisers. Network representatives calculate their package prices based on a discount rate to the list price. This rate varies with supply and demand but has an explicit upper limit established by the daypart manager. A variety of facts are exchanged between parties in the negotiation. They usually include the buyer's stated budget (not necessarily his true budget), a minimum reach requirement for designated demographic, the lengths) of commercials for the campaign. Flighting information is also provided which often, but not always includes, a set of desired and/or forbidden shows or airdates, the length of commercials allowed in each show and a maximum number of commercials per show. The seller also knows the type or brand of service/product advertised. Collectively, the information provided allows the network to generate a package that is presented to the media buyer for approval. The deal is evaluated based on a 30second equivalized cost per million (CPM) for the demographic. Proposals are iteratively modified until the parties reach an agreement. Our auction model accepts multidimensional bids from all buyers and allows market forces to determine the allocation and prices that generate equilibrium. 37 3.2 Auction Description We propose a progressive semiclosed auction format that allows the media buyer to dynamically create individual bundles from a selection of commercial slots upon which they then bid. Our research will provide a new and innovative way of using an auction mechanism by allowing inexact bidding with multiple evaluative criteria as well as providing for constraints unique to the television sales environment. Bidders are given the flexibility to change and or modify their bids and bundles until a stopping criterion has been reached. Suggestions are provided to the buyers to help them formulate successive bids, but active pricing will not be disclosed. This semiclosed format, proposed by Tiech (1999), will satisfy the need for nondisclosure of market prices that is required by both buyer and seller. Additional constraints that will be modeled include separation of competing ads, meeting bidder's demographic group exposure requirements while ensuring the seller receives his reservation prices, accommodating specific show placement (nonplacement) requests with commercial length specification while not exceeding an upper bound on the number of ads allowed per show. Our model assumes that campaigns are continuous throughout the season thus does not provide for fighting nor does it contain provisions for special programming that may displace regularly scheduled shows during the course of the year. We model our auction as an integer program. A summary of the notation is presented in Table 3.1. Our main decision variable is x ,p,s,b, a binary variable set to 1 if a particular buyer b is allotted a unit u in podp for show s. 3.2.1 Notation Table 3.1 Summary of Notation General u,p,s,b,i Subscripts s=show, b=buyer, p=pods, u=pod part, i= allowable ad campaign length (using this order). DV Signifies a decision variable. Shows S Number of shows. Ps Number of pods in show s. L, List price for each 15second unit in show s. D Vector of 15second demographic values for show s. U P.S Number of 15second portions in pod p for show s. C, Maximum number of total units in show s available to sell. Buyers: B Number of buyers. Tb Target Vector of desired demographic impressions for buyer b. hb Vector of desired shows for buyer b (hsb = 1 if show s is desired by buyer b, 0 otherwise) Note that hb can be a zero vector. Ni, Set of specified commercial lengths) in show s for buyer b. Nsb Set of allowable commercial lengths) in show s for buyer b. Hb, Hb Min/max number of desired shows that buyer b must have ( Hb < hb < Hb). KSb Number of correct length commercials allowed in show s by buyer b. Mb Type of merchandise advertised by buyer b. ab hbHbHb, .Is,b, N, Mb, K,,b bid from buyer b. ab is the amount bid. Vb, (d) value of buyer b's advertising given a cumulative demographic vector d. Seller: fplsb DV: 1 if bidder b has more than 15seconds in pod p in show s. Yb DV: 0,1 variable. If 0, buyer b can't buy any pods p. X ups,b DV: 0,1 variable. If 1, buyer b, has spot u in podp in show s. I p,s,b,i DV: 0,1 variable. If 1 pod p of show s for buyer b uses an allowable number of advertising slots. z ps,b DV: 0,1 variable. 1 if Ipsbi = 1 otherwise 0. notationall simplicity) ie N., b J,b DV: 0,1 variable. If 1, buyer b has any unit in shows. r budget discount rate applied to the list price. 3.2.2 Objective Function This problem focuses on a fixed time horizon of some specified number of weeks, each week of which is a repeat of the pattern sold in the auction. The seller solves B PI: max .abyb b=l The objective function maximizes the total revenue from accepted bids, ab. The variable Yb is an indicator variable that is set to 1 if the bid is accepted subject to the following constraints. The objective in this case is to maximize revenue. This approach is selected, over maximizing profit because it achieves a stated goal of satisfying a predefined budget. Additionally, the product involved is considered perishable and therefore the seller is more concerned with depleting inventory than selling at the greatest profit. Should profit be the motivating factor our objective function can be restated as follows: B (B ( S (LU,, \Y (API) max ELayb (Ir) S Z Xup ^ With this formulation the seller reservation price is incorporated into the objective and therefore can be dropped from the constraints, i.e. Equation (1.1). 3.2.3 Constraints Rarely are sales predicated solely on price and availability. Incorporating the constraints of the environment may be more challenging than determining the highest 40 bids in the television industry. The following constraints are required to adequately represent the restrictions inherent in commercial airtime sales. 3.2.3.1 Reservation requirement The reservation requirement in the television industry is based on a discount rate, r that is applied to the list price for each show, Ls. Daypart managers must meet an annual budget that is a proportion of the total possible revenue available based on the list price. The discount to list reflects this proportion, however, discount rates for individual sales may be above or below the discount as long as ultimately the aggregate meets or exceeds the budget. Therefore the sum of the accepted bids must be greater than the discounted commercials purchased where xp,,b = 1 indicates that a commercial has been placed in pod p of show s by buyer b. The following formula first determines the non discounted revenue per show over all the shows and buyers and then applies the discount rate. The resulting discounted revenue requirement is compared with the total amount bid by all buyers to insure that the minimum reservation price is received. (1.1) B r) S ( PZx,.L, reservation requirement a~ (lr{E@Z EI EUsbLj b=l b=1 s1 pI=1 u=I 3.2.3.2 Maximum seller coverage Although there is a finite amount of commercial airtime available, not all is targeted for sale during "upfront" sales. The maximum coverage constraint insures that at most a specific number of commercials are sold per show as noted by C . B P U,, (1.2) Y xup,sb < Cs = 1,...,S maximum coverage b=1 p=1 u=l 3.2.3.3 Maximum spot availability Each show is broken into pods, usually one to twominute blocks of airtime reserved for commercial placement. The number of pods per show and their length vary from show to show. The following guarantees that the number of commercial placements per pod does not exceed the number of spots available to accommodate them. The number of individual units in each pod summed over all buyers must be less than or equal to the total number available in that pod. B Up. (1.3) XUpsb b=1 u=l 3.2.3.4 Buyer selection indicator If a particular bidder b does not obtain any airtime the following constraint forces all his x values to zero and thus drops her bid from consideration. (1.4) Z"s' xU,,p,,b < UI, Yb b=l,...,B can't buy if not selected S=l p=l U=l s=l p=l Equation (1.4) can also be written as (A1.4) Xu^p,.,b <Yb u =l,...,Up,, P=1,...,P, s=l,...,S b=,..., B This alternative formulation is an example of constraint disaggregation for binary variables and has been shown to provide for a stronger LP relaxation (Johnson, Nemhauser & Savelsbergh, 2000). The original formulation will cause B constraints to be included in the formulation, one for each bidder in the problem. The alternative expression enumerates each combination of bidder, show, pod and unit. Adding more constraints and thus growing the size of the problem seems counterintuitive to the goal of improving the LP relaxation. However, Johnson, et al. (2000, pg. 5) suggest that "to 42 obtain strong bounds, it may be necessary to have a formulation in which either the number of constraints or the number of variables (or possibly both) is exponential in the size of the natural description of the problem." 3.2.3.5 Campaign commercial length constraint Advertising campaigns may consist of commercials of varying lengths. A campaign composed of only 15, 30, 45 or 60second spots must eliminate any collection of 15second units in an individual pod that will not form the desired commercial length. Buyers may also have mixed campaigns, or campaigns that consist of a combination of lengths. Ns,b is a set of permissible commercial lengths for each show s supplied by buyer b. Note that the permissible collection of lengths can vary by show facilitating a buyer's need to change the lengths) of the campaign over time. Industry practice dictates that no more than one commercial per buyer appears in the same pod. An exception to this rule allows that at most two 15second units from the same advertiser may be placed in the same pod to fill an empty 15second slot and complete the pod. To account for this exception we define N {Nb U {2}ifl ENb and 2 o NSb Sb\ = Nsb otherwise Together Equations (1.5a) and (1.5b) require the selected units in each pod to correspond to one of the allowable lengths listed in Nb. The unit slots are numbered for convenience but the number does not correspond to a specific location in a pod, therefore there is no need to insure that the units are consecutive when forming a 30, 45 or 60 second commercial. Up.., (1.5a) ZXUp,,b = i,.,,bi p=l,..,P s=l,...,S b=l,...,B campaign length u=1 'ieN,. Equation (1.5b) prevents more than one correct length commercial for buyer b from appearing in podp of show s. Note, two 15second commercials are allowed in one pod if 1 E NVb. Note that zp,,b is equivalent to E p.,,b.i indicating if set to 1 that buyer b iEN,., has a correct length commercial in podp of show s. zp.,b is used within our formulation to simplify the notation but will not appear as a variable in the implementation of our problem. (1.5b) zp,,b 1 p=l ,...,P, s =1,...,S b = 1,...,B commercials per pod 3.2.3.6 AntiClutter Control Placing a large number of different commercials in the same pod weakens the impact of all commercial messages within that pod. This phenomenon results from the "clutter" exacerbated by the use of 15second commercials. To reduce clutter, Equation (1.6) allows at most two 15second ads to appear in each pod. B (1.6) 'Ip.s.b 2 p=l ,...,P, s = 1,...,S anticlutter b=l 3.2.3.7 Frequency: Max commercials per show Controlling the number of commercials appearing in each show will provide the buyer with the ability to spread or aggregate commercials over the length of a campaign week. K,,b indicates the number of correct length commercials that are allowed in show s by buyer b. Should a buyer want to place all of her inventory early in the week she would set K,.b high for shows during the desired days and other days to zero. A buyer can 44 identify a forbidden show, say show s', by setting K,,b, = 0. Should a buyer forbid a show, during implementation of the model, the corresponding variables will not be generated. Normally, buyers want their ads to appear only once per show to enhance the number of different viewers that are exposed to their spot, in this case all K,., 's would be set to 1. Equation (1.7) provides for this constraint. P, (1.7) _zp,,,<: Ks.b s = 1,...,S b= 1,...,B maximum spots per show p=l 3.2.3.8 Demographic gross impression guarantee Media buyers desire a specific amount of demographic reach, or number of people exposed to their commercial during their campaign. There are a variety of demographic categories upon which a show is rated. Each show's gross impressions per category forms the demographic vector D, and indicates the seller's estimated reach for that particular show in the upcoming season. D, is ordered by the categories: Women 18 to 49, Women 2554, Men 1849, Men 2554, Adults 1849 and Adults 2554. Tb represents the vector of demographic reach or gross impressions that the buyer needs to meet the product's campaign goals and is ordered with the same categories in D,. Note that although total number of gross impressions per show, D,, in reality does not change with the length of the commercial our model uses equivalized 30second calculations that differentiate based on the length of commercial. For example, the number of gross impressions for a 15second spot is 1 unit of demographics while the unit gross impressions for a 30, 45 or 60second commercial is 2, 3 and 4 respectively. Bidders normally evaluate their package on a single demographic category. The sum of the total 45 number of 15 second units, times the specified demographic gross impressions over all selected shows must meet or exceed the required reach for the specified demographic groupss. Equation (1.8) assures that the minimum demographic requirement is achieved. (1.8) xXU,,b D, > TbYb b = 1,...,B demographic reach required S=l (p1 p =l 3.2.3.9 Show placement requirement In addition to the actual dollar amount bid and demographic requirements, a buyer may specify desired shows within which they would like their commercials placed. Setting hbto 1 indicates that buyer b wants placement in show s. She can further indicate her willingness to deviate from her program choice by setting the upper Hb and lower Hb bounds to the number of shows required. j,b identifies which shows a buyer has been allocated and is determined with the following formulas. jb,,'s is set equal to 1 if buyer b has any slot in show s and 0 otherwise. (1.9a) zp.b j,,b s =,...,S b = ,...,B p=1 Show allocation (1.9b) Z>,bP s=l,...,S b=l,...,B p=i Equation (1.10) ensures that a buyer is allotted at least H,, shows and no more than Hb of the shows requested. To satisfy this constraint at least H,, j,. 's will have to be set to 1 and no more than Hb. Ss (1.10) Hb> hbj,,b ;j>Hbyb b =I,...,B desired shows $=I 3.2.3.10 Pod protection constraints Networks routinely guarantee that competing advertisements do not appear in the same pod. The group of equations (1.11 ad) implements this notion of"pod protection." Pod protection is normally not given to 15second commercials, therefore we need only investigate anticompetition when a buyer has two or more units in a particular pod. The decision variable fp,,b is set to 1 if a bidder b has two or more 15second units in a particular pod p of show s. When the number of units a bidder has per show is 0 there is no competition and Equation (1.1 la) forces both fp_,b and z P,,b to zero. up., (l.ll1a) xu ,p,,b > z,, +fpb p=l,...,P, s=l,...,S b=l,...,B, u=l In the case where one unit is assigned in a particular show to buyer b, pod protection is not enforced and the requirement that zp,,, equal or exceed fp,,b in Equation (1.1 Ib) sets f .,b to zero, and zpb to at most one. This corresponds with the fact that buyer b has a single unit in any pod of show s. (l.llb)fp,,b Zp,,,b p=l,...,P, s=l,...,S b=l,...,B, Equation (1.1 ic) will force z p,,,b to one in this case. Pod protection only becomes an issue when two or more units are assigned within the same pod of a show to a single bidder thereby generating a potential 30second or longer commercial. Equations (1.1 lac) will force zpb to one and f,,,b to one when two or more spots are bought by buyer b in show s, pod p. u=\ p=l 47 Finally, Equation (1. 1 d) will keep competitors away from a protected pod in show s for buyer b. (l.1ld)fp/,s+zp.sl i An example will help clarify this fairly complicated methodology. Suppose we have two buyers i andj who are being considered for placement in podp of show s. Further suppose that buyer i wants a 30second spot in the pod and buyers wants a 15second U U ' commercial in the same pod. Therefore the .XU,pSi = 2 and x,,, = 1. Table 3.2 u=l u=l lists the possible values of the z andfvariables in equations 1.1 a through 1.11 c. If buyer Table 3.2 Variable Value Allocations for Pod Protection Variable 30second or 15Second No > Commercial Commercial Commercial Equation'x, fps~i zP.,.i fps z,.s fx.. z7p., 1.11a Oorl Oorl O0orl Oorl 0 0 l.1 lb 0 Oorl 0 1 0 0 _____1 1___ l.11c 1 1 0 1 0 0 i has a 30second or greater commercial in a pod both fp.,,i and z p, are set to 1 through the series of equations. Buyer j, with a 15second commercial in the same pod, has only his zp,.j set to 1. Equation 1.1 d uses the fact that only 30second or greater commercials have both fp",b = 1 and Zp,,,b = 1, while 15second commercials have only Zp s,b = 1 to prevent both from being placed in the same pod. If a buyer does not have a commercial in a pod both variables are set to zero. In our example 48 fpi = 1)+ (zpsj =1)> 1 is a contradiction to what is allowed by Equation 1.1 Id. Therefore, since Equation 1.11 d is only applicable when the products advertised are the same between two buyers, buyers i andj could not appear in the same pod if they were selling like products. 3.2.3.11 Bid specification and ordering The action starts with buyers placing bids (ab ,hH ,Hb ,Hb, Ip,,NbMb,K Sb). If a buyer submits more than one bid, the most recent one is used. If more than one bid is for the same amount, priority is given to the earlier bid through a lexicographic ordering imposed by altering bids with a timestamp, tb, as a,, b + ab + tt M where t is the current time and M is sufficiently large. 3.2.3.12 Bidder reservation price The auctioneer solves P 1 and presents the solution to the buyers. An infeasible solution may also be announced in which case, nothing is accepted by the seller. A rational buyer will only accept a feasible solution if vb D, x( Pub, > ab, where ab < Bidder's Budget. If everyone accepts the solution, the auction is over. Otherwise, new bids can be submitted. When there is only one show and one pod with only one pod part, this reduces to a normal first price English auction. 49 3.3 Summary This chapter defines the network television advertising sales environment. The description includes the product characteristics, environmental constraints imposed on the allocation of goods and current negotiation strategies. A detailed integer program was developed to incorporate industry practices into a semisealed progressive combinatorial auction designed to replace the current negotiated environment. The problem objective is to maximize seller revenue while satisfying all constraints imposed by both the buyers and the seller. Buyer constraints and requirements are conveyed via a multicriteria incompletely specified package bid. To accommodate industry practices the auction problem becomes quite complex. A few of the many constraints incorporated in the model include those designed to separate competing commercials, retain a portion of inventory for later markets and achieve specified demographic exposures in a particular demographic category while satisfying individual show placement requests. The complexity and combinatorial nature of the auction suggests the need for a heuristic solution method, which is explored is subsequent chapters. CHAPTER 4 CONSTRAINT PROGRAMMING 4.1 Introduction Constraint Satisfaction Problems (CSP) involve finding values for all problem variables that simultaneously satisfy all problem specific constraints. Constraints can be viewed as a relationship between variables that restrict their possible instantiations. The paradigm has been studied since the 1960's and 1970's when the artificial intelligence community applied it to picture processing (Montanari 1970; Waltz 1975). The ability to achieve solutions to these complex problems rests on the notion of eliminating impossible alternatives from consideration as early in the allocation as possible. Early elimination reduces domains from which values are chosen and thus facilitates expeditious results. Recently a great deal of attention has focused on Constraint Programming (CP) due to its ability to solve combinatorial problems such as the one described in this research. Constraint Programming takes the solutions to a standard CSP and applies them to an objective function which is successfully tightened to find an optimal solution (Bartak, 1999). Constraint Programming has various advantages over other methodologies. Of major importance is the time it takes to achieve a solution. CP algorithms can often achieve solutions more quickly than can integer programming methods. Additionally, CP representation corresponds more closely with the entities of the original problem. Thus making formulations simpler to compose, heuristics more readily developed and the solutions less difficult to interpret (Bartak, 1999). By using constraint programming as a basis for our heuristic we hope to capitalize on these advantages. This chapter begins with a brief introduction to constraint satisfaction problems. Section 4.2 defines the constraint satisfaction problem and methodology. Various CSP solution techniques are described in sections 4.3 through 4.6. Section 4.7 introduces Constraint Programming. The remaining section explores how these methodologies apply to the auction problem presented in this research. 4.2 Constraint Satisfaction Problems The auction problem developed in Section 3.2 is a combinatorial auction and thus achieving an optimal solution has been shown to be an NPComplete problem (Rothkopf et al., 1998). As such, a heuristic is required to allow a satisficing solution to be reached in real time. Combinatorial problems are found in areas such as planning, scheduling, generalized assignment and resource allocation and have been effectively formulated as constraint satisfaction problems (CSP) (Nonobe & Ibaraki, 1997). Constraint Satisfaction is a general term describing a class of problems involving a set of variables that are to be instantiated from an associated domain while satisfying a set of constraints that limit the assignment (Mackworth, 1992). CSPs use a variety of search methodologies to find a feasible solution. The goal of a constraint satisfaction problem (CSP) is the assignment of values to its variables that will satisfy all constraints. More formally, the finite CSP is defined by its three components. V: a finite set of variables {X1 ,X,...,X}, D: the set of corresponding domains {D,, D2,..., D,, } where Di is the finite domain of Xi, and C is a finite set of constraints or relations {CK, C2 ,..., Cr } restricting the assignment of values to variables. A constraint Cij,k between the variables X,,Xj,Xk... is any subset of the possible combinations of values of X,X, XXk .... For example the cross product Cik, c Di x Dj x Dk x ... indicates the possible combinations of values that the constraint allows (Brailsford, Potts, & Smith, 1998). If there exists an assignment of a value from a variable's domain for all variables that satisfy every constraint then there is a feasible solution to the problem, otherwise it is said to be unsatisfiable. Solving the CSP can be accomplished by either constructing a solution by iterative variable assignments leading to a feasible solution or by starting with an initial solution (but not necessarily feasible) and subsequently modifying or repairing the solution until it becomes feasible. Known as the systematic search or "constructive" approach, the former methodology applies backtracking techniques and is usually designed to solve a given problem instance exactly. While the stochastic search or "repair" approach gradually repairs an initial solution in order to reduce the infeasibility until all constraints are satisfied. Several greedy algorithms, such as tabu, genetic algorithms or neural networks have been shown to be effective in generating initial solutions. The repair approach has been found to be particularly effective for largescale problems (Nonobe & Ibaraki, 1997). 4.3 Arc Consistency In a binary CSP, constraints involve only two variables and are visually depicted by a constraint graph (Figure 4.1). The nodes of the graph represent the variables and the XVariable X D,={1,...,5} Dj={1,...,5} (a) Original Domains /^ ^ X, D.={1,2} Dj=(1...,5} (b) Domains with (X,,X.) arcconsistent ^T^ ~Xi (c) Domains with (XX) and (X,X) arcconsistent Figure 4.1 Constraint Graph (Brailsford et al., 1999) lines or arcs connecting the nodes represent the constraints between them. Arc consistency is achieved by reducing the domains of the problem variables until the remaining values are all supported; a value is supported if every constraint on the variable includes a tuple in which the variable takes this value and all other variables take supported values. For example if a constraint Cid exists between variables X, and X. the arc (Xi,Xj) is arc consistent if for every value a in the domain D, of variable X, there is a value b in the domain Dj of variable Xj that satisfies the constraint Cj (see Figure 4. lb & c). In Figure lb, (Xi Xj) is arcconsistent but (X ,XJ) is not, while in Figure 4. Ic the variables are fully arcconsistent, b E Di is called a supporting value for a e D,. Any values of a that do not satisfy the constraint, i.e. have no supporting value, are removed from the domain DA and in so doing makes the arc (Xi, Xj) arcconsistent (Brailsford et al., 1999). However, the simple fact that none of the arcconsistent domains are empty does not imply that the CSP has a solution. A solution is achieved only if all variables can be assigned a specific value such that they all support each other. There are several arcconsistency algorithms employed to establish consistent domains. The importance of the algorithms rests in their ability to reduce the size of the problem and save computational processing time. Arcconsistency has been widely used as a preprocessing step to eliminate local inconsistencies before any attempt is made to construct a solution. Several algorithms have been developed that capitalize on the knowledge about constraint properties to reduce the cost of consistency checking, see Chen (1999) for an overview. CSPs are frequently a subpart of a larger application. In these cases it is often important to compute all possible solutions which can be systematically explored to find the best configuration for a given situation. Optimization problems can assume this approach, thus delaying optimization criterion development until a set of solutions has been discovered. This technique is tractable when the possible value assignments are discrete but faces challenges with continuous values since continuous domains admit an infinite set of values. To overcome the complexity of continuous value representation in a binary constraint environment SamHaroud & Faltings (1996) suggest descretizing variable ranges into one or a small collection of intervals that roughly approximate a constraint by an enclosing box whose borders represent the unary outer projections of the variables involved. In this case values falling within the confines of this box are considered arcconsistent. 4.4 Systematic Search Algorithms Systematic search algorithms involve attempting to achieve a consistent solution by repeatedly extending partial solutions. The most basic algorithm called Generate and Test (GT) randomly selects a variable to instantiate and then checks that the labeling is consistent. It is inefficient in that the random selection of variables does not capitalize on problem specific information and thus must perform an exhaustive search. Backtracking improves on this technique and is best described as a depthfirst instantiation technique. Another alternative involves enforcing arcconsistency, an elimination approach ruling out all solutions containing local inconsistencies (Mackworth, 1992). A branch and bound type search tree is typically used to graphically represent the current state of the search. A node represents a partial solution and the branches different values that could be assigned to some variable. Past variables are those that have already been assigned a value, while future variable has not yet been assigned a value. Choosing a branch of the tree to explore instantiates the variable with the value associated with the chosen branch. Should the domain of a future variable become empty the problem has reached a deadend or has become annihilated. Note that mathematical programmers would use different terminology to describe the elements of the problem. For example, they would say, "fathomed" instead of "deadend." 4.4.1 Look Back Algorithms The most common algorithm for performing systematic search is backtracking; an approach that after variable instantiation "looks back" to ensure the assignment is consistent with previously instantiated variables. Backtracking incrementally attempts to reach a complete solution from an intermediate partial solution by repeatedly assigning values consistent with the partial solution. If a consistent value cannot be found the algorithm backs up to a point where successful choices can be made. The method that is used to choose which previous variable to return to in the event of inconsistencies defines the various backtracking algorithms. 4.4.1.1 Chronological Backtracking (BT) Chronological Backtracking (Bitner & Reingold, 1975) is the generic backtracking algorithm. At every stage of backtracking search, there is some current partial solution that the algorithm attempts to extent to a full solution. The process begins with the current variable being assigned a value from its domain. Then consistency is checked between this instantiation and the instantiations of the current partial solution. If any constraint between this variable and the past variables is violated the assignment is abandoned and the next domain value of the current variable is tried. If there are no more domain values left, BT backtracks to the most recently instantiated past variable, assigns it a new value and the process repeats. If all checks succeed, the branch is extended by instantiating the next variable to each of the values in its domain. If a value has been assigned to every variable a complete solution has been found otherwise the problem is infeasible (Mackworth, 1992). 4.4.1.2 Backiumping (BJ) Backjumping (Gaschnig, 1977) is similar to, but more intelligent than, Chronological Backtracking. BJ identifies the latest instantiated variable causing a constraint failure and proceeds directly to that variable when it reaches a deadend. Instead of chronologically backtracking to the preceding variable, BJ jumps back to the Chronological Backtrack {2,5,3,1} 1 2 3 {2,5} 4 5 2, "5,3 5 {2,5,3} 6 {2,5 S 1 3 2 Q 1 1 1 1 1 1 Q 2 3 3 1 3 0 2 1 2 2 0212 2 1 3 1 2 3 4 5 6 5,3,6}) Backjump: S\ Skips A** circled nodes \/ / Figure 4.2 Partial Backtrack Tree (Kondrack & van Beek, 1997) deepest past variable that was checked against the current variable. For example, Figure 4.2 represents a partial backtracking tree from an nqueens problem described by Kondrak and van Beek (1997) that shows how the backjump technique skips the circled nodes. (Nqueens is a classic problem used in artificial intelligence to demonstrate difficult problems. The goal is to assign chess queens positions the on an n by n game board such that no queen can capture another.) BJ reduces the number of consistency checks by skipping search tree nodes thus it behaves more efficiently when all instantiations are inconsistent for the current variable. Changing the value assignment of the failure causing past variable may allow a consistent instantiation to be found for the current variable. Backtracking to any of the intervening variables will have not effect since they have not impact on the reason for the failure. 4.4.1.3 ConflictDirected Backiumping (CBJ) By tracking previous failures, ConflictDirected Backjumping (Prosser, 1993) demonstrates more sophisticated backjumping behavior than BJ. Every variable has its own conflict set that lists the past variables that have failed consistency checks with this current instantiation. Every time a consistency check between the instantiation of the current variable and an instantiation of some past variable fails, the past variable is added to the conflict set of current variable. When all possible values for the current variable have been exhausted, CBJ backjumps to the deepest past variable in its conflict set, this variable becomes the current variable and a new value assignment is attempted. Note that the variables in the conflict set of the variable that could not be instantiated are propagated up the tree and added the conflict set of the past variable so that no conflict information is lost. Figure 4.3 depicts a conflict set that would be formed for this example. 4.4.2 Look Ahead Algorithms A disadvantage of"Look Back" algorithms is late discovery of conflicts. The "Look Ahead" approach attempts to overcome this problem by looking at future variable assignments and eliminating impossible values from consideration earlier in the process. , {2,5} I _Conflict Set for CBJ Figure 4.3 Partial Backtrack Tree with CBJ Conflict Set 4.4.2.1 Forward Checking (FC) Forward Checking (Haralick & Elliot, 1980; McGregor, 1979) performs consistency checks from the current instantiation to future variables. The algorithm assigns a value to current variable from its domain then propagates the effect of that assignment to future variables by removing inconsistent values from their domains. Only when the future domain is annihilated (becomes empty), indicating that the current assignment has lead to a deadend, are backtracking techniques employed. If a deadend is reached the domains of the future variables are returned to their original state, and the next value is tried. If all values have been exhausted for the current variable domain, FC backtracks chronologically to the most recent successfully instantiated variable. This process continues until a complete solution is found or until all possible assignments have lead to a deadend, in which case the problem has no solution. Forward Checking, in contrast with backward checking algorithms, visits only consistent nodes, although not necessarily all of them. 4.4.2.2 Maintaining ArcConsistency (MAC) Similar to Forward Checking, Maintaining ArcConsistency (Sabin & Freuder, 1994) focuses on checking future variables for arcconsistency. However, MAC not only checks the consistency of all potential future variables and deletes any values that are not supported by the current variable, it also checks for consistency between the newly identified future variables and their values. This type of incremental arcconsistency algorithm for reestablishing arcconsistency after each assignment reduces the size of the overall problem and thus has been shown to be efficient (Van Hentenryck, Deville and Teng,1992). 4.4.3 Hybrid Backtracking/Forward Checking Algorithms Various combinations of the previously described basic algorithms have been proposed to combine their advantages. For example Forward Checking and Conflict Directed Backjumping (FCCBJ) tracks information about inconsistent variables and subsequently uses this information to determine the backtracking point. This algorithm has the advantage of establishing a conflict set to more efficiently direct the backward movement of the Forward Checking algorithm when it encounters a deadend. Another extension, Backmarking (BM) improves the efficiency of the backtracking algorithms by adding a marking scheme (Gaschnig, 1977). Without the marking scheme consistency checks are performed to determine if the current instantiation of variables satisfies the constraint between the variables without regard to any historical checks that may have already determined the consistency between these same two variables. The BM marking scheme reduces the number of consistency checks by employing the notion that if at the most recent node where a given instantiation was checked, the instantiation failed against some past instantiation that has not yet changed, then it will fail again. Therefore, all consistency checks involving it need not be investigated. It can also be assumed that a successful instantiation of some past instantiation that has not yet changed will succeed again. By marking the instantiations that have already been tested we avoid redundant consistency checks. The implication is that we need only check past instantiations that have changed or are "unmarked." Imposing a marking scheme on an algorithm does not change the nodes visited and therefore can extend any of the basic algorithms. Kondrak and van Beek (1997) evaluated the efficiency of several backtracking algorithms with respect to the number of nodes visited and the number of consistency checks performed. They found that the hybrid backtracking algorithms such as Forward Checking and ConflictDirected Backjumping, Backmarking with Backjumping and Backmarking with ConflictDirected Backjumping tend to outperform the original algorithms. In fact, FCCBJ has been shown to be among the best for solving hard problems (Kondrak & van Beek, 1997; Smith & Grant, 1995). 4.4.4 Improving Performance The order in which variables are chosen for instantiation can play a significant roll in the performance of the algorithm. Variable ordering can be either static or dynamic. With a static variable ordering the order of the variables must be established prior to the constraint network being passed to the backtracking algorithm. A static order is in contrast to a dynamic order of instantiation in which the decision of which variable to instantiate next is based on the current state of the search. Large portions of the search space can be pruned by employing the "failfirst principle" which chooses the most constrained variable first thereby forcing failures higher in the backtrack search tree (Van Hentenryck & Saraswat, 1996). A dynamic ordering algorithm that chooses the variable with the minimum remaining values (MRV) in its domain has been developed for use with both backtracking (Sabin & Frueder, 1994) and forward checking (Bacchua & van Ran, 1995) algorithms and has been shown to perform well on specific problems. The order in which the values are chosen will likewise determine how quickly the algorithm achieves a solution by allowing the most promising value to be assigned first. What constitutes "promising" is problem specific, for example if you are attempting to maximize profits, ordering the values for largest to smallest may achieve the best results. 4.5 ArcConsistency Algorithms ArcConsistency algorithms complement, rather than substitute for, backtracking algorithms. Arcconsistency algorithms remove inconsistencies from the network generated by an instantiation that can never be part of a global solution. Removal of inconsistencies reduces thrashing (Mackworth, 1977). Mackworth (1992, p287) describes thrashing "as the repeated exploration of subtrees of the backtrack search tree that differ only in inessential features such as the assignment of variables irrelevant to the future of the subtree." By analyzing the various basis of thrashing behavior in backtracking, arcconsistency algorithms can eliminate the source. Instantiating a variable impacts the domains of all prior variables and consistency algorithms must determine if the instantiation has violated any constraints or caused prior instantiations to be in violation. The most widely used consistency algorithms are AC3 and AC4. Unlike the previous algorithms that evaluated every arc, AC3 rechecks consistency of only those arcs that could have been affected by current instantiation. AC 4 uses the same approach but maintains a special data structure that prevents repeated reexamination of pairs of values (Mackworth 1977). Other algorithms exploit problem specific knowledge. For example AC7 (Bessiere, Frueder & Regin, 1999) takes advantage of the bidirectional property of binary constraints to remove redundant checks. This algorithm works on the simple notion that a value a at node I (I,a) supports a value b at node J (J,b) if and only if (J,b) supports (I,a). Constraint bidirectionality properties allow the algorithm to perform fewer consistency checks and thus improve computational efficiency. Specifically, AC7 can avoid checking if(J,b) supports (I,a) since it knows that the inverse, (I,a) supports (J,b), is true. Another arcconsistency algorithm, AC8 (Chen, 1999) breaks the problem into smaller subproblems then solves them sequentially. 4.6 Stochastic Search Algorithms Stochastic search algorithms begin with a solution that may or may not be feasible and repairs it using a variety of techniques to achieve feasibility. This class of algorithms will normally achieve a feasible solution more rapidly than their systematic counterpart. The quality of the solution is predicated on the initial solution and the technique used for repairs. Stochastic algorithms have been proposed that use hillclimbing (Minton, Johnston, Phillips & Laird, 1992), neural networks (Popescu, 1997; Kurgollus & Sankur, 1999), and genetic algorithms. Kanoh, Matsumote, Hasegawa, Kato & Nishihara (1997) suggest that genetic algorithms (GAs), due to their global search characteristics, provide effective solutions to CSPs that have many local optima. In fact, GAs have been used to effectively seed CSPs designed to solve ship maintenance scheduling (Deris, Omatu, Ohta, Kutar & Samat, 1997) and timetable planning problems (Deris, Omatu, Ohta, & Saad, 1999). Combining the two methods takes advantage of the strength of both constraint satisfaction methodologies and GA techniques. The genetic algorithm plays its role as a tool to generate promising solutions while constraintbased reasoning processes the constraints to ensure that the solutions are legal and valid. Kanoh et al. (1997) further modify the mutation process of the standard GA by substituting "viral infection" for standard mutation. A virus is defined as a partial solution to the CSP and is generated by the GA along with other candidate solutions. Crossover and infection then generate new candidate solutions. Infection gives direction to the evolution by substituting the genes of the virus for those of the individual generating a new candidate solution based on partial solutions proven to be consistent. 4.7 Constraint Programming Constraint Programming finds variable instantiations that simultaneously satisfy all specified constraints while optimizing a stated objective. One strategy used for Constraint Programming is to model the problem as a CSP. After a feasible solution to the CSP has been found an additional constraint is added to represent the objective function. The new constraint requires that the objective strictly improve over the objective value of the current CSP solution. This process is repeated until no feasible solution can be found. The last solution obtained prior to the problem becoming unsatisfiable is an optimal solution (Nonobe & Ibaraki, 1998; Potts & Smith, 1999). 4.8 Advertising Sales Application Defining the television commercial time allocation problem as a constraint programming problem involves specifying the variables, domains and constraints as well as the ordering of variable instantiations and value assignments. Variables in the problem represent the airtime assignment to each bidder and the values assigned form a tuple indicating the shows that have been allocated. The allocation is subject to the following constraints: * Maximize Seller Revenue: The overall objective function that must increase at each iteration. * Reservation requirement: The aggregate sum of the all accepted bids must be greater than the sum of the discounted list prices for the commercials purchased. * Maximum seller coverage: A seller specified maximum number of commercial slots must be sold in each show. * Maximum spot availability: The number of commercial placements per pod cannot exceed the number of spots available to accommodate them. Assignments must be within the range of the domain. * Buyer selection indicator: A buyer cannot be assigned units if his bid has been rejected. * Campaign commercial length: The number of units assigned to a buyer in each show must be a multiple of 15 that corresponds with the campaign length. For example a buyer with a 30second campaign must has either zero units in a show or a multiple of two. * AntiClutter Restraints: No more than two 15second commercials can appear in the same pod. * Maximum Commercials per Show: The number of correct length commercials in each show must not exceed the bidder specified maximum. This could be as small as zero which effectively eliminates that show from consideration. * Demographic gross impression guarantee: The demographic gross impressions summed over all selected shows and equivalized by commercial length must meet or exceed the required reach for the specified demographic group for each bidder. Show placement minimum: Winning bidders should be placed in the shows they requested. The minimum number of requested show assignments should correspond with the lower bound specified by the bidder. Show placement maximum: Winning bidders should not be placed in an identified show more than the maximum number of times indicated by the upper bound specified by the bidder Pod protection: No two buyers advertising the same category product can be placed in the same pod if at least one has a 30second commercial. Bid amount not exceeded: The sum of the discounted list prices for the allocation of units to each bidder must be less than or equal to their amount bid. The constraint programming methodologies will be employed to determine variable instantiation and manage the large domains of the problem. Variable instantiation ordering involves sorting the bidders by some criteria of interest. Both variable instantiation and value ordering will be discussed in detail in subsequent chapters. 4.9 Summary This chapter presented an overview of constraint satisfaction problems and constraint programming. The former involves finding a set of values that simultaneously satisfies all constraints while the later extends the feasible solution to the CSP by including an objective function that is iteratively tightened to find an optimal solution. We looked a various algorithms designed to discover satisfying allocations. Arc consistency techniques, algorithms employed to control the consistency of domains were reviewed. They are important to our research as they provide an efficient means of managing the large domains of our problem. Finally, we introduced our auction mechanism in constraint programming language. CHAPTER 5 HUERISTIC DEVELOPMENT 5.1 Introduction A direct attack on solving problem P1 is probably doomed. For example, in a representative problem with 325 bidders competing for 587 units in 109 pods across 24 shows P1 generates approximately 278,000 binary variables and 587,000 constraints. A heuristic is clearly needed. The heuristic used to solve our combinatorial auction in real time is developed in this chapter. The overall approach is outlined in Section 5.1. It incorporates a mixture of problem aggregation with linear, constraint and dynamic programming methods. Once we give the overall approach, we focus on the particulars. Many of the decisions necessary to discover an optimal allocation of goods can be determined at an aggregate, or show level rather than at the unit or pod level. Working with the aggregate problem dramatically reduces the size of the problem. Descriptions of the aggregate subproblems are presented in section 5.2. The results of these sub problems are incorporated into the master problem in section 5.4. The aggregate problems use constraint programming methods as an efficient way to collapse the size of the original search space. Section 5.3 defines the constraint programming aspect of the heuristic that will manage the domains from which the allocations are chosen. Constraint programming employs simple computer programming logic to replace complicated 68 equations thus provides a more efficient method of ensuring that constraints are enforced. Finally, branch and bound methodology is used to search for an optimal allocation, search guiding heuristics and fathoming criteria are presented in section 5.5. 5.2 Overview The solution methodology employed to allocate units in our Incompletely Specified Combinatorial Auction is fairly complicated incorporating several techniques. Figure 5.1 presents an overview of the procedures. The auction begins with the collection of bids. Each bid is subjected to an initial feasibility check to ensure that it meets minimum requirements for entry. This is accomplished by solving an aggregate problem defined below. Once all bids have been tendered, an initial feasible solution is generated with the use of heuristics that will be defined in detail in section 5.4. An upperbound is established using a linear relaxation of a second type of aggregate integer program. This bound is used to judge the quality of our solutions. The best solution to date is then used to start a branch and bound search. At the end of each auction round, the selected stopping criterion is checked. If stopping conditions are not met, bidders are informed of the results. Loosing bidders will have the opportunity to change their bids commensurate with their behavior profile. When all stopping conditions have been met, the current bid amounts are replaced with bidder reservation prices (but otherwise unaltered) and are used to compute a solution to be used for the efficiency calculation. 69 Start I For All Agents q 14 No Get Next Bid Agent _ Participating? Yes Get Bid No I  i  IN o Get Next Bid 'heck feasibility by solving SubProblems APb & Nb < Feasible? > All Bids Processed? Generate Initial Feasible Solutions Heuristics A and B Yes Determine Efficiency Figure 5.1 Auction Overview Auction Flowchart No Suggest Bid Modification 70 5.3 Aggregate SubProblems The majority of constraints involved in the auction problem can be determined by examining the allocations of each show rather than a pod or unit level allocation. The overall driving heuristic is a greedy allocation of show slots to bidders. Each bid is considered sequentially, conditional on the tentative allocations made to other bidders. By aggregating to the show level we reduce the size of the problem and thus enhance our ability to achieve a solution. Let 8(x) be the normal Kronecker delta, i.e., (xsb)= Ixxb =>0 Also, let xsb be the number of allowable units that bidder b may purchase in show s and Xs.b be the current domain of x,,b. The domain of x ,,b will change as other variables associated with earlier accepted bids are instantiated either because units become unavailable or some constraint such as pod protection or maximum spots per show would be violated. As we show in Section 5.4, Xs,b incorporates all the constraints given in Equations (1.2)(1.11 d) except for demographic reach and the desired shows constraints. These latter two are handled directly in the following aggregate problem. We define the aggregated problem, (APb) for each bidder as s (APb) ,b =(lr)min ELSxb S Dxs*b : Tb$ Hb E hS(Xb ) H b S=I 71 When there are no current other assignments, the objective value is labeled crb which gives the minimum discounted show costs needed to satisfy all problem constraints (1.2) (1.11 d). When there are current assignments meeting (1.2) (1.1 d), then (APb) provides an assignment for this bid (if it has a feasible solution) that, together with the current assignments, meet all constraints (1.2) (1.1 ld). By minimizing the discounted costs, we hope to also, in total, satisfy the one remaining constraint, constraint (1.1), the seller reservation price constraint. If there is no feasible solution, then set the xUpSb variables to zero. Otherwise, a solution to (APb) can be expanded to yield x ,psb values by recovering a combination yielding the correct entry in X,,b. There may be many such combinations. No currently protected pod will be violated by these xUP,,,b. If (APb) has a feasible solution, then we set Yb = 1. If not, we set Yb = 0. The resulting y vector indicates the eligible participants for this round. The remaining decision variables in the original problem (P1) can be recovered by analysis of the expanded solution X,,p,,,b 's. A simple count of the ultimate allocation ofx's for each bidder b in each show s and podp will determine the value to assign fp.,b. If the sum of the x's is greater than 1 in a pod then that bidder has more than a 15second commercial in that pod and fp,,b is set to 1. This same number when compared to the set of allowable commercial lengths, N,,b, yields the values to assign the Ip,,,bi variable. If a commercial of length i appears in the final allocation in a particular podp of show s for this buyer p,,b,i is assigned a value of 1 otherwise it is set 72 to 0. This same counting technique aggregated to the show level will identify the appropriate ji, 's to set tol indicating that buyer b has a presence in show s. The aggregate problem (APb) is solved using dynamic programming. Notice that, with the exception of the last constraint, this is a straightforward Knapsack program. The final constraint can make this a nonlinear problem (because of the Kronecker operation) if either or both Hb and Hb are greater than zero and the h's have values necessitating the consideration of these constraints. We utilize one of several dynamic programming routines designed to solve the subproblem, the choice of which depends on the values of Hb and Hb and the nature of the h's that have been selected. The dynamic programs provide exact optimal solutions to (APb). However, these can take some time to solve since the Tb values may be large. At various points, to be discussed, we use a heuristic based on linear programming ideas to give good (often optimal) solutions to the aggregate problem (APb). We call these methods, FastAP. Just as we utilize one of several dynamic programming routines designed to solve the subproblem, the choice of which depends on the values of Hb and Hb and the nature of the h's that have been selected, we also have different FastAP approaches. Overall, however, they are based on linear programming relaxations. When the show selection constraints aren't needed, a straightforward LP Knapsack problem is solved. Otherwise, a slightly more complicated version is employed to satisfy the show selection constraints. Each solution is refined using problem reduction methods which shrink the domains based on simple dominance tests. 73 A problem similar to (APb) is given below and proves useful in several situations. When no assignments have been made, let s (MNb) 7b = min Xs,b X.bEX'b s Z Dx ,.b > Tb s=l S s . Hb I>LhS(X.,b ) Using the same techniques deployed to solve (APb) we are able to determine the minimum number of units, 17, needed to satisfy all of the constraints (1.2) (1.1 Id) for bid b. 5.4 Domain Management Constraint Programming Effectively managing the x,,b domains is extremely important to the heuristics ability to reach a timely solution to this problem. Due to its combinatorial nature and the large number of available units, the problem size can quickly become insurmountable. We employ constraint programming concepts as a means of coping with these sizes. Simple programming logic can replace complicated logic equations providing a more efficient method of ensuring that constraints are enforced. By dynamically reducing the size of the domain as units become unavailable and only generating the combinations that satisfy an individual buyer's constraints we avoid total enumeration. We utilize a "greedy" assignment methodology were we assign one bidder at a time and then adjust the remaining domain to reflect the resulting slot assignments. 74 In the previous section we developed an aggregate problem used as a basic component for solving P1. This formulation relies heavily on the domains X,,. The following procedure is used to determine the domain for each bidder given the current available slots and previous allocation of bid. Step 1. Let X,,b = and r(Nb )= le ONb^ 2Nb y(Nb) is set to 0 if there are no 15second commercials in buyer b's campaign indicating that the anticlutter constraint is not applicable to this campaign. If the buyer is running 15second commercials and not 30second spots, this parameter will be set to 1. When r(N,,b ) = 1 special consideration must be given to ensure that no more than 2 15second commercials for this bidder appear in the same pod. Since the buyer is not running 30's two units in a pod must be individual 15 second commercials. Step 2. Let g,, be the remaining number of open slots in pod p of show s. Open slots are defined as those that are not yet owned. Additionally, if a competitor owns 2 or more slots in a podp of shows then no units in that pod are deemed available. If a competitor owns a 15second slot in that pod then the pod is "weekly owned" and only 1 unit is considered open. This methodology ensures podprotection and gives priority to current owners of weakly owned pods. Step 3. For each pod define Xp,,,b { n e Nb : gs.b > n}. These are the campaign lengths feasible for each pod. The lengths allowed in each pod are entirely dependent on the number of units available and the allowed campaign lengths. For example if 75 there are 3 units available in a pod and a bidder is running 30 and 60second commercials they could only have a 30second (2 unit) spot in that pod. Step 4. Let Ei be the set of all combinations of size i = 1,...,Kb of the sets Xp,Sb (used no more than once in a combination when y(NS,b) = 0 or no more than once with the exception that two 15second units are allowed in the same pod if y(N,,b ) = 1. The aggregate over each show is X,.b = U. This enforces the anticlutter i=l,...n constraint. Step 5. Remove each x,.b E X,b,, from X,,b where the number of currently committed slots plus xsb exceeds C,, thus limiting the number of assignments in each show to no more than the maximum allowable. An example will help clarify how the domains are computed. Assume for some show s and buyer b we have the following: * C, = 24, and the current number of committed slots for that show is 17, leaving seven open units. * Nb,, = {l,4} or the campaign consisting of only 15 and 60second commercials. * K,, =4 (any combination of the allowable 15 or 60 second spots totaling at most 4 correct length commercials are allowed in show s). * P = 4. There are 4 pods in show s * The number of open slots in each of the four pods are as indicated, g. =5, g2,, = 0, g3, = 7, g4, = 3 The above states that this bidder is running campaigns of length 1 and 4 (N,,b= {l,4}), can have at most 4 commercials in this show (K,,, = 4), that there are 4 pods in the show (P = 4) having 5, 0, 7 and 3 remaining slots available to this bidder. The zero availability in pod two may have resulted from pod protection given to another bidder in a prior step 76 of the solution methodology or it may simply have been completely used by prior assignments. Step 1 gives Xsb = ( and y(Nb)= 1. Step 2 is specified above by the g values. Step 3 gives X,,b {1,4}, X2sb X3,b {1,4}, X4,s. {l}. Step 4 gives the following. The examples illustrate possible assignments. i = {1,4} (different combinations consisting of 1 correct length spot) 02 = {2,5,8} (e.g., 5 = a length 1 in pod 1 and length 4 in pod 3) 03 = {3,6,9} (e.g., 6 = a length 1 in pods 1 and 4 and length 4 in pod 3) 04 = {4,7,10} (e.g., 7 = two length 1 's in pod 1, a 4 in pod 3, and a 1 in pod 4) Then X,,, = {1,2,3,4,5,6,7,8,9,10}. Step 5 requires us to remove 8, 9 and 10 since we have at most 7 units available to assign. Thus the final X,b = {1,2,3,4,5,6,7} is the domain for show s from which we select allocations to satisfy buyer b's requirements. Although this example might suggest the contrary, a domain need not have all the integers between the upper and lower element. 5.5 Master Problem An overview of the master problem is presented in Section 5.2, Figure 5.1. The goal is to find a solution that maximizes seller profits as specified in problem P1 of Chapter 3 while satisfying constraints (1.1) to (1.1 1d). The approach employed utilizes the heuristics described in previous sections and methods defined here to determine an allocation that approaches optimality. 77 To establish a good initial solution to the allocation problem, consider the following two greedy algorithms. Assume we are given B bids. The two algorithms differ only by the sort criteria used in step 1. Step 1: Repeat the following until all bids have been processed Sort the remaining bids by some criterion of interest with the most desirable bid designated as the topmost bid. See sorting criteria 1 and 2 below. Solve the aggregate subproblem for the topmost remaining bid and make the appropriate assignments to the variables of P1. Step 2: While the amount bid by the selected bidders is less that the seller's reservation price for the collection of allocated units, i.e. B ( B ( S ( P IabYb <(r) :E t xup, L, b=l sbl sl s= p=l u=l) ) Sort the remaining feasible bids by some criterion of interest with the least desirable bid designated as the topmost bid. See sorting criteria 3 below. Set the topmost active bid's aggregate subproblem solution to infeasible and remove any current allocations to this bidder. Furthermore, the above procedure yields a feasible solution to the auction problem as is now shown. First, Step 2 guarantees the feasibility of the reservation requirement (1.1). The construction of the domains for each aggregate subproblem plus the constraints of(APb) assures the feasibility of all the remaining constraints. See Figure 5.2 for an overview of this heuristic. Wi Start ) While Bids Still Unselected Figure 5.2 Heuristic Flowchart 5.5.1 Sorting Criteria 1 The first sorting criteria is designed to order the bidders in such a manner that those that contribute the most to maximizing seller revenue are assigned first. To accomplish this we find the bidder that solves the following ma a, TotTDJ ((TI )dTotD, where TotT = V remaining demographics L. and TotD = V remaining required demographics a , and d is the demographic required by bidder b. We assume, as is industry practice, that each bidder's demographic requirement Tb is in only one demographic category. Simply stated, the bidder with the highest bid per thousand demographic requirements normalized by the cost of the specific demographic category and demand within that category is selected. 5.5.2 Sorting Criteria 2 The ratio of the actual bid amount, ab, and oCrb,, the minimum possible cost allocation for that bid b, defines sorting criteria 2. The sorting equation is as follows max/aj 5.5.3 Sorting Criteria 3 The following sorting criteria will force those bidders with the largest actual cost to bid ratio to be removed first, enhancing the auction's ability to achieve a feasible solution. ma(Xbs 3 max ` \ b 80 5.6 Branch and Bound Branch and bound techniques are employed to investigate the various combinations of bids that will maximize seller revenue. Total enumeration of the various combinations is impossible in any reasonable amount of time, thus we utilize heuristics to guide our branching behavior. At each branch, we take the partial solution from predecessor branches and solve (APb) (or FastAP). The amount of time allotted to computation in each round is predefined. Therefore we use time remaining as a guide to the search process. After preprocessing is complete, an initial solution determined and an upper bound on P1 computed, the remaining time is used as follows. Thirty percent is spent in a Breadth First Search (BFS), the rest is dedicated to a Depth First Search (DFS). 5.6.1 Breadth First Search (BFS) The Breadth First Search extends to three levels. This means that it looks at all orderings of all combinations of 3 bidders time permitting. Below level three, depth first search is used but is limited to a relatively small number of branchings (we use five times the number of bidders). Bids are initially ordered by the heuristics previously described so as to rank them such that the topmost bids contribute the most to maximizing revenue. However, conflicts between these bids may prevent all of these most desirable bids from achieving an allocation. The order in which the bids are processed affects the allocation so all permutations of the three bids are explored. During the Breadth First phase, the FastAP heuristic is used. 81 For example, the BFS systematically explores all permutations of ordered bids {1,2,3} then bids {1,2,4}, {1,3,4}, {2,3,4}, etc, expanding each of the permutations with a DFS. This process continues until 30% of the remaining computational time has been exhausted at which point, the final 70% of computing time is dedicated to a strictly Depth First Search. 5.6.2 Depth First Search (DFS) A Depth First Search is employed during the final 70% of computational time to seek out the best combination of bids. This search is conducted in two stages, the first solves (APb) exactly using dynamic programming and lasts for 60% if the time allotted. Stage 2 utilizes FastAp and runs until the conclusion of the computational time. 5.6.3 Fathoming Criteria Some fathoming criteria that are used to limit the branch and bound search follow. 1. Based on reservation price. Recall that ob, is the objective value of a solution to the aggregate problem (APb) where no prior assignments have been made. Then any branching exploration should be restricted to cases where the amount bid by the feasible bidders meet the minimum cost allocation that satisfies all constraints. In other words, don't consider any selection of bids where B B abYb < 'bYb b=1 b=1 2. Based on best feasible solution value to date. 82 Don't explore any assignment ofy's giving an objective value to P1 that is less than the current best feasible solution subject to the amount of inventory available. That is, don't explore any y's satisfying: B SabYb with B S >lbYb < I C, b=t $=I 5.7 Determining an Upper Bound to PI: The overall problem (P 1) is upperbounded by: B U= maxzabyb yeY b= B Y, (orb ab)Yb 0 b=1 B S Z rtbYb < C, b=1 3=l O This formulation incorporates the results of the aggregate problem (APb) and the minimum number of units problem (MNb) defined earlier together with constraint (1.1). The bound provides a way to judge the current best solution to all constraints (1.1)  (1.1 Id). This problem is simple to solve since a boundedvariable Simplex method with only two constraints is trivial. 5.8 Summary To cope with the size and complexity of the problem and facilitate reaching a solution in real time a heuristic was developed and described in this chapter. The 83 heuristic solution to the combinatorial optimization problem (P1) is by no means trivial. Various techniques were employed to guide our quest for an optimal allocation, including a mixture of problem aggregation with linear, constraint and dynamic programming methods as well as branch and bound search. Subsequent chapters will test the efficacy of the methods engaged. CHAPTER 6 SIMULATED BIDDING AGENT DEVELOPMENT 6.1 Introduction Our experiment consists of simulating the execution of our auction under various conditions to analyze the mechanism's performance. To facilitate this we generate players that reflect the characteristics of the real world environment. An analysis of data received from a representative of a major television network provides a statistical basis for player typing. Each player or agent represents an individual bidder with defined parameters that reflect media buying practices within the industry. The parameters include bidder product requirements. To establish product requirements necessitates defining the desired demographic category and gross rating points (GRP) required as well as bidder reservation prices. Show selection for each agent includes a list of desired shows and an upper and lower bound to the number required. The number of commercials allowed in each show and the type of product being sold will also be specified. Finally, a bidding strategy is defined for each agent type that governs the agent's behavior during the auction's execution. A visual summary of the entire process of generating a bid agent is presented in Figure 6.1. We describe each process in detail in the remaining sections of this chapter. Every attempt was made to depict as many types of bidders as necessary to accurately represent the behavior of the market participants. Determine # of bidders Randomly assign a Demographic Category Determine Total Demographic GRP Required (Gamma dist. for each Demo) Calculate Bid Amount (Regression equation for each Demo Category based on GRP) Choose Product Type Establish Number of Commercials in Campaign Establish Commercial Lengths for Campaign* (Single Frequency) Set Maximun Allowable Commercials Per Show = 1 Establish Exceptions to Max Allowable Commercials Per Show H Establish Commercial Lengths for Campaign* (Multiple Frequency) Set Maximun Allowable Commercials Per Show* (70% = 1, 28% =2, 2% =3) Figure 6.1 Flowchart of Initial Agent Generation 86 Classical auction theory assumes homogeneity among auction participants, however new evidence suggests that in electronic auction several types exist. Bapna, Goes and Gupta (1998) describe three distinct bidder categories, Evaluators, Participators and Opportunists, these will be used as a basis for our bidder definitions. 6.2 Data Analysis The data analyzed for this research was provided by one of the major television networks. The source prefers to remain anonymous. Within this industry information, especially pricing and inventory availability, is extremely proprietary and jealously protected, therefore we have attempted to disguise the results of our analysis while at the same time benefit from the discovery of patterns. Our analysis consisted of reviewing two representative weeks of actual airtime allocations. The data consists of 150 unique bidders representing 209 purchases to acquire 1290 units of airtime across 48 shows. Not all information needed to formulate our buyer typing was known either because of our inability to access the appropriate data or the lack of representative data. We were required in some instances to make assumptions. Some of our assumptions stem from anecdotal evidence provided by our information source while others are based on expectations of a rational buyer in a competitive market. During the following discussion of buyer typing we will identify and justify the assumptions. The patterns and frequencies identified will be used as a basis from which we randomly generate characteristics of each agent and thus may not be exactly duplicated in our experiments. 87 6.3 Number of Buyers The number of bidding agents to generate was our first consideration. The two weeks of data consisted of roughly 90 to 120 buyers who were awarded "upfront" allocations in each week. Our industry source indicated that 300 to 350 buyers participate in "upfront" with their allocations distributed across the weeks of the year. As a conservative estimate we generate 325 bidding agents. 6.4 Demographic Category and Total Gross Rating Points Media buyers must achieve a certain amount of demographic exposure or gross rating points within a specific demographic group to satisfy their campaign needs. The A. C. Neilson Ratings for network television is the measure used to determine the number of people exposed to a particular program and hence to the commercials appearing in that show. The ratings are broken down in various categories representing the gender and age of the viewing audience. Six of these categories are typically associated with primetime advertising sales; Women, Men and Adults within age groups 1849 and 2554. Category assignments for each agent were randomly chosen from a uniform distribution over the values 1 to 6. An analysis of the data revealed that the amount of demographic gross rating points required by each buyer followed a gamma distribution. Figure 6.2 shows the distribution of the buyer frequencies over the total Gross Rating Points (GRP) averaged across all categories. The individual category distributions appear in Appendix A. Appropriate MLE estimates of Gamma distribution parameters a and / 's were generated for each category. Simultaneously satisfying the following two equations by 25 S20 S15 0 I 5 L 0 I lTnTmm n, rmn nn 0 10000 20000 30000 40000 50000 Average Demographic GRP Figure 6.2 Average Gross Rating Points varying & gives us the estimated parameters. 1nX n and af = X(n), where o(&) = ["(&)/ F() is the digamma function with F' denoting the derivative of F (Law & Kelton, 1991). The ft values were then varied to maximize the goodness of fit measure. The KolmogorovSmimov onesample test was used to determine the goodness of fit between the estimated Gamma distribution and the sample values. The cumulative distribution for the theoretical Gamma distribution is compared to the cumulative distribution of the actual data. The maximum deviation from the theoretical distribution must be less than or equal to a critical value defined for the test. Results of the test are presented in Table 6.1 and validate at a 0.05 significance level that the sample came from a population having a Gamma distribution (Siegel, 1956). Using Phillips 89 (1971) gamma variate generator and the estimated t and /f for the appropriate category we are able to generate a representative random demographic value to assign an agent. Table 6.1 Goodness of Fit Test KolmogorovSmirnov OneSample Test Significance Level = .05 (2tailed) Critical Value = .09407 Max <= Critical Difference Value? Demo 1 0.0675 Yes Demo 2 0.0657 Yes Demo 3 0.0777 Yes Demo 4 0.0721 Yes Demo 5 0.0696 Yes Demo 6 0.0751 Yes 6.5 Bidder Reservation Price The data analyzed for our study includes the amount that each successful buyer paid for each of their allocated commercials with the associated placement information. We can safely assume that the buyer's reservation price over all their units is at least the sum of the individual unit values. We also know the network's estimated gross rating points for the shows within the sampled weeks. The seller's rating estimates are not common knowledge among the buyers, instead buyers base their demographic requirement calculations from approximations discovered from historical ratings of prior seasons and market research on new programming. These approximations are fairly accurate in reflecting the seller's figures, therefore we apply the known seller rating estimates to determine roughly how many demographic gross rating points the buyers in the data desired by summing the demographics over the shows they were allocated. We 90 then equivilize the demographics to reflect the length of commercials. For example, a 30 second commercial would receive twice the demographic exposure of a 15second spot. A strong linear relationship was found to exist between the price paid for the units and the equivilized demographic GRP for the associated shows but showed signs of heteroscedasticity. A natural logarithm transformation of both variables remedied the increasing variance. See Table 6.2 for a summary of the fit. Table 6.2 Regression Fitness Statistics LNPrice = Constant + (Coeficient LNDemo) Demo R Square F t Sig. 1 .836 1056.437 32.503 .000 2 .837 1065.432 32.641 .000 3 .835 1047.752 32.369 .000 4 .834 1038.267 32.222 .000 5 .844 1118.842 33.449 .000 6 .846 1135.407 33.696 .000 By regressing logarithmic price against the logarithmic equivilized total demographic GRP within each demographic category we were able to establish equations for determining a representative reservation price, see Appendix B for the detailed results. Each category's regression equation was determined from 209 observations. Based on the demographic category and the total demographic GRP's formulated in the previous steps we can calculate, from the appropriate regression equation, individual reservation prices to assign the agents that reflect the amount and type of product desired. For the purposes of this study we assume buyers' valuations can be represented by a step function 0 if constraints are not met, b = Reservation Price if constraints are met 91 An allocation has no value to the buyer unless all constraints have been met, while any allocation that satisfies the constraints is valued at his reservation price. Buyers are trying to satisfy explicit campaign goals at a minimal cost thus adding additional units to the minimal constraint satisfying allocation does not add value. The regression gives us the reservation prices representing those buyers who were successful in achieving an allocation. We recognize that there may have been participating buyers that were not successful in securing an allocation and others that were not forced to pay their true valuations, therefore we increase the variance of the actual reservation price assigned by a random amount uniformly distributed between 20% above or below the calculated figure. 6.6 Commercial Lengths and Frequency An analysis of the historical data provided indicates that approximately 76% of bidders aired multiple commercials within the campaign week, while 24% placed only one spot. Our agent demands reflect these statistics. Within the two groups, multiple or single placement, we were also able to determine a frequency of the lengths of commercials aired. Table 6.3 details the breakdown upon which we programmed our agents. Within the multiple commercial campaigns there were instances of mixed commercial lengths where a single buyer uses a variety of commercial lengths within the same campaign that we reflect in defining our agents. 