Incompletely specified combinatorial auction

MISSING IMAGE

Material Information

Title:
Incompletely specified combinatorial auction
Physical Description:
Book
Creator:
Jones, Joni Lynne, 1958-
Publication Date:

Record Information

Rights Management:
All applicable rights reserved by the source institution and holding location.
Resource Identifier:
aleph - 24889184
oclc - 45341142
System ID:
AA00020431:00001

Full Text










INCOMPLETELY SPECIFIED COMBINATORIAL AUCTION:
AN ALTERNATIVE ALLOCATION MECHANISM
FOR BUSINESS-TO-BUSINESS 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 well-lit 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 Anti-Clutter 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 Conflict-Directed 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 Arc-Consistency (M AC) ............................................... 60
4.4.3 Hybrid Backtracking/Forward Checking Algorithms................................ 60
4.4.4 Im proving Perform ance............................................................................. 61
4.5 Arc-Consistency 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 Sub-Problem 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 BUSINESS-TO-BUSINESS 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 multi-criteria 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 non-auction

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 multi-object 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 price-item 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

(Englbrecht-Wiggans, 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 multi-criteria

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 multi-criteria bids. The bids must extend beyond the current price-item

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 NP-complete (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 long-term

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 real-time

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 web-based 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 short-term 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 business-to-business 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 multi-dimension combinatorial auction will accommodate

business models that currently use negotiations as their primary sales mechanism and

sales for products and or services that require multi-criteria 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 (Engelbrecht-Wiggans, 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

(Engelbrecht-Wiggans, 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 (Engelbrecht-Wiggins, 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 (Engelbrecht-Wiggins, 1980). Single objects are the common focus

of classical auction theory, however objects may be single divisible or indivisible items, a

package of non-identical 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 (Engelbrecht-Wiggins, 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 rational-expectations) strategies maximize the expected

utility of the outcome and is used routinely as the strategy of choice in classical auction

theory (Engelbrecht-Wiggins, 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 sealed-bid 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

(Engelbrecht-Wiggans, 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 multi-unit sales. How







15
the auction(s) are conducted within the multi-unit environment (i.e., sequentially or

simultaneously) propagate a variety of alternative designs. Additionally, recent progress

has been made in facilitating package bidding within a multi-object 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 budget-constrained 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 multi-object 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 "set-packing" 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 Web-base 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 ascending-bid 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 semi-closed 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 non-conflicting 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 NP-complete (Rothkopf et al.,

1998). Some geographic structures such as an interval of consecutive assets or items that

could be organized into a k-dimensional matrix, thereby permitting row or column-wise

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; Ortega-Reichert, 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 risk-averse 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 risk-averse, the seller prefers the

Dutch or first-price 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 Engelbrecht-Wiggans

(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 second-price auction.

Dutch and first-price 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 low-value

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 non-cooperatively, 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 ascending-bid

auctions were more susceptible to collusion than were sealed-bid 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 re-auctioned







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-

your-bid, 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; Engelbrecht-Wiggans & Kahn, 1998). Engelbrecht-Wiggans and Kahn

(1998) discovered that with uniform price auctions (a common form of simultaneous

multi-unit auction in which high bids win the units, but all units are sold for the same

price) tend to encourage zero bids. Pay-your-bid 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 multi-unit 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 first-price 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 budget-constrained 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 (Engelbrecht-Wiggans, 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 multi-object 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 non-combinatorial 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 two-sided 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 large-scale 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 "hyper-rational" by

Varian (1995), may account for some of this heterogeneity. Yet another study suggests

the revenue equivalence theorem is not supported for on-line auctions (Lucking-Reiley,

1999). A field study of 100 on-line 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, E-Bay is an on-line

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 on-line auction mechanisms. For example, one of the most popular on-

line auction forms is the multi-unit 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 on-line 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,

"up-front "or long-term, "scatter" or short term and "opportunistic" or last minute buys

(Katz, 1995). The majority of sales are conducted during "up-front" 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 "up-front" 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 15-second 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 15-second spot.

All national television time is priced based on a 30-second 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 non-standard commercial lengths

to discourage a large number of 15-second 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 15-second (60-second) commercial is one-half (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 60-second commercials, or any

combination of these lengths in a single campaign.







34

Although it would appear most cost effective to purchase all 15-second

commercials, since reach, or the number of viewers exposed to a commercial, is relatively

the same regardless of commercial length, the 30-second commercial is most prevalent.

It has been suggested that 30-second 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 30-second equivalents, where a 15-second commercial represents a half a

unit, with 30-seconds being the base unit and a 60-second commercial is comparable to 2

units. We simplify our calculations by redefining the base unit as a 15-second

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.

15-second 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 15-second commercials. If, for example, an

additional 15-second spot is required to complete a pod then a single advertiser's 15's may

be "book-ended" 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 air-dates, 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 30-second equivalized cost per million (CPM) for the

demographic. Proposals are iteratively modified until the parties reach an agreement.

Our auction model accepts multi-dimensional 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 semi-closed 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 semi-closed format,

proposed by Tiech (1999), will satisfy the need for non-disclosure 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 (non-placement) 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 15-second unit in show s.
D Vector of 15-second demographic values for show s.
U P.S Number of 15-second 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 15-seconds 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 -(I-r) 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~ (l-r{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 "up-front" 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 two-minute 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 60-second spots must eliminate any collection

of 15-second 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 15-second units from the same advertiser may be placed

in the same pod to fill an empty 15-second 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 15-second 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 Anti-Clutter 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 15-second commercials. To reduce clutter, Equation

(1.6) allows at most two 15-second ads to appear in each pod.

B
(1.6) -'Ip.s.b 2 p=l ,...,P, s = 1,...,S anti-clutter
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 25-54, Men 18-49, Men 25-54, Adults 18-49 and Adults 25-54. 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 30-second calculations that

differentiate based on the length of commercial. For example, the number of gross

impressions for a 15-second spot is 1 unit of demographics while the unit gross

impressions for a 30, 45 or 60-second 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 (p-1 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 a-d) implements this notion of"pod protection."

Pod protection is normally not given to 15-second commercials, therefore we need only

investigate anti-competition 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 15-second 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 30-second or

longer commercial. Equations (1.1 la-c) 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 30-second spot in the pod and buyers wants a 15-second

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 30-second or 15-Second 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 30-second 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 15-second commercial in the same pod, has only

his zp,.j set to 1. Equation 1.1 d uses the fact that only 30-second or greater

commercials have both fp",b = 1 and Zp,,,b = 1, while 15-second 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 + t-t
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 semi-sealed 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 multi-criteria

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 NP-Complete 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 large-scale

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 Constraint
D,={1,...,5} Dj={1,...,5}
(a) Original Domains

/^ ^ X,
D.={1,2} Dj=(1...,5}
(b) Domains with (X,,X.) arc-consistent

^T^ ~Xi D=(1,2} Dj={4,5}
(c) Domains with (XX) and (X,X) arc-consistent

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 arc-consistent but (X ,XJ) is not, while in








Figure 4. Ic the variables are fully arc-consistent, 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) arc-consistent

(Brailsford et al., 1999). However, the simple fact that none of the arc-consistent

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 arc-consistency 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. Arc-consistency 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 Sam-Haroud & 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 arc-consistent.

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 depth-first instantiation technique.

Another alternative involves enforcing arc-consistency, 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

dead-end 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 "dead-end."








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 dead-end.

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 n-queens problem described by

Kondrak and van Beek (1997) that shows how the backjump technique skips the circled

nodes. (N-queens 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 Conflict-Directed Backiumping (CBJ)

By tracking previous failures, Conflict-Directed 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 dead-end, are backtracking techniques employed. If a dead-end

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 dead-end, 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 Arc-Consistency (MAC)

Similar to Forward Checking, Maintaining Arc-Consistency (Sabin & Freuder,

1994) focuses on checking future variables for arc-consistency. 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 arc-consistency

algorithm for re-establishing arc-consistency 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 (FC-CBJ) 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 dead-end. 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 Conflict-Directed Backjumping, Backmarking with Backjumping and

Backmarking with Conflict-Directed Backjumping tend to outperform the original

algorithms. In fact, FC-CBJ 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 "fail-first 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 Arc-Consistency Algorithms


Arc-Consistency algorithms complement, rather than substitute for, backtracking

algorithms. Arc-consistency 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, arc-consistency 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 AC-3

and AC-4. Unlike the previous algorithms that evaluated every arc, AC-3 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 AC-7

(Bessiere, Frueder & Regin, 1999) takes advantage of the bi-directional 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 bi-directionality properties allow the algorithm to perform

fewer consistency checks and thus improve computational efficiency. Specifically, AC-7

can avoid checking if(J,b) supports (I,a) since it knows that the inverse, (I,a) supports

(J,b), is true. Another arc-consistency algorithm, AC-8 (Chen, 1999) breaks the problem

into smaller sub-problems 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 hill-climbing (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 constraint-based 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 30-second campaign must has either zero units in a show or a multiple of
two.
* Anti-Clutter Restraints: No more than two 15-second 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 30-second 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 sub-problems 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 upper-bound 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
Sub-Problems
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 Sub-Problems


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 =(l-r)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 15-second 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 non-linear 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 sub-problem, 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 sub-problem, 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 ) s=l


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 15-second commercials in buyer b's campaign

indicating that the anti-clutter constraint is not applicable to this campaign. If the

buyer is running 15-second commercials and not 30-second 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 15-second 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 15-second slot in that pod then the pod is "weekly owned"

and only 1 unit is considered open. This methodology ensures pod-protection 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 60-second

commercials they could only have a 30-second (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 15-second units are allowed in the same pod if y(N,,b ) = 1.

The aggregate over each show is X,.b = U. This enforces the anti-clutter
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 60-second 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 top-most bid. See sorting criteria 1 and 2 below.

Solve the aggregate sub-problem for the top-most 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 s-l s= p=l u=l) )

Sort the remaining feasible bids by some criterion of interest with the least
desirable bid designated as the top-most bid. See sorting criteria 3 below.

Set the top-most active bid's aggregate sub-problem 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 sub-problem 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 pre-defined.

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 top-most 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 bffi
with
B S
>lbYb < I C,
b=t $=I


5.7 Determining an Upper Bound to PI:


The overall problem (P 1) is upper-bounded 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 bounded-variable 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 18-49 and 25-54. 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 Kolmogorov-Smimov one-sample 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
Kolmogorov-Smirnov One-Sample Test
Significance Level = .05 (2-tailed)
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 15-second 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.