Voicu Dynamics of Procurement Auctions

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The Dynamics of Procurement Auctions

Cristian Voicu

Department of EconomicsStanford [email protected]

May 2002

Abstract

Procurement contracts account for a large percentage of the US GDP. They are usually

awarded by auctioning authorities through first-price sealed-bid auctions. Previous work in

auction theory provides answers for the static equilibrium with one auction. This paper ex-

tends the one-auction model to a dynamic model, introducing capacity constraints and con-

tinuation values. I develop a comprehensive model for the dynamics of procurement auc-

tions by proving a series of complex theorems and by analyzing the properties of the

Markov-perfect equilibrium. Furthermore, I obtain a range of new results, relating the ex-

pected markup and the present value to environment variables such as available capacities

and costs of the bidders. Some results may seem counterintuitive and can only be under-

stood in the dynamic setting. For example, there are instances when the present value of a

firm increases when one of its competitors has more capacity. The law of diminishing

marginal returns holds in general, although there is a notable discontinuity point when the

project size is equal to the available capacity.

Keywords: first-price auctions, asymmetric bidders, procurement contracts, independent

private values, capacity constraints, and Markov-perfect equilibrium.

Acknowledgements: I would like to thank Professors Susan Athey and Patrick Bajarifor suggesting the research topic and monitoring my progress.


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INDEX

Page 3 Section 1: IntroductionPage 10 Section 2: Literature ReviewPage 19 Section 3: Empirical ObservationsPage 19 3.1 Advertising, Bidding, and the Award ProcessPage 22 3.2 Description of the DataPage 25 3.3 County by County AnalysisPage 26 3.4 Regressions

Page 30 Section 4: The Theoretical ModelPage 30 4.1 Markov-Perfect EquilibriaPage 32 4.2 Properties of the EquilibriumPage 48 4.3 The Infinite Time HorizonPage 52 4.4 Defining Aggressiveness MeasuresPage 54 4.5 Modeling as Discrete-Time Markov ChainPage 57 Section 5: Computer SimulationPage 57 5.1 Practical ConsiderationsPage 61 5.2 Nonlinear Least Squares OptimizationPage 64 5.3 The GamePage 70 Section 6: Results

Page 79 Section 7: Final CommentsPage 81 BibliographyPage 85 Appendix: Computer Code


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1. INTRODUCTION

Auctions are a part of our way of life. They fill the buying and selling needs for

millions of people and products. Auction buying and selling are so widespread in the

United States and overseas, that few individuals and companies can escape their effects.

Auction theory is important because of its practical, empirical, and theoretical ramifica-

tions.

The practical significance is justified by the huge volume of economic transactions

which are conducted daily through auctions. Houses, cars, agricultural produce and live-

stock, art and antiques are commonly sold by auction. Other economic transactions, like

takeover battles, are auctions in effect, if not in name. The government uses auctions for

selling a variety of assets such as treasury bills, foreign currency, mineral rights including

oil and gas fields, and also for privatizing state-owned firms. Most procurement contracts

are awarded to private firms through auctions. In these cases, the procurement authority is

seeking the bidder who is willing to perform the required goods or services for a minimal

cost. There has recently been an explosion of interest in designing new forms of auctions,

for example to sell radio spectrum licenses, or in using auctions to set up new markets, for

example in electricity and transportation.

The empirical importance of auctions is derived from their simple and well-defined

economic environments. They provide valuable testing ground for many aspects of eco-

nomic theory, and in particular game theoretical models with incomplete information. Ma-

jor empirical research efforts have focused on auctions for oil drilling rights, timber and

treasury bills, and there has been an increasing interest in experimental work on auctions.


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Auction theory has been the basis of much fundamental work. First, it has been im-

portant in developing understanding of the methods of price formation, such as posted

prices, and negotiations in which both the buyer and the seller are actively involved in de-

termining the price. Second, there are close connections between auctions and competitive

markets. An important strand of the auction literature is concerned with the properties of

auctions when the number of bidders becomes large. The question is whether the sale price

converges to the true value, thus fully aggregating all the economys information even

though each bidder has only partial information. If it does, then it would be attractive to

think of an auction model as justifying some of the ideas economists have about perfect

competition. Third, there is also a close analogy between the theory of optimal auctions

and the theory of monopoly pricing, and auction theory can also help develop models of

oligopolistic pricing. Auction-theoretic models also apply to non-price means of allocation.

Examples include queues, lobbying contests, wars of attrition, or rationing.

Bajari and Tadelis (1999) overviewed the building construction industry, and found

there were 2 million establishments in the United States construction industry that com-

pleted $528 billion dollars of work in 1992. The Bureau of the Census of the U.S. Depart-

ment of Commerce documents that these firms directly employed 4.7 million workers and

had a payroll of $118 billion dollars. In 1997, the construction industry comprised 8 per-

cent of the U.S. GDP by itself, and the construction industry worldwide was a 3.2 trillion

dollar market. The vast majority of procurement auctions were awarded by auctions.

Most procurement contracts are variants of either simple fixed-price or cost-plus

contracts. In fixed-price contracts, the buyer offers the seller a fixed price determined by


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auction for completing the project. A cost-plus contract stipulates a base fee, and then the

contractor is reimbursed for costs that exceed projections. Bajari and Tadelis (1999) indi-

cate that most of the simple projects are procured through fixed-price contracts. More

complex projects or projects that need to be finished quickly are procured using cost-plus

contracts and are accompanied by low levels of design completeness, which means that

there is a high probability that adaptations are needed.

There are four basic types of auctions widely used and analyzed: the ascending-bid

auction (also called open, oral, or English auction), the descending-bid auction (tradition-

ally used in the sale of flowers in the Netherlands, from which it derived the name of

Dutch auction), the first-price sealed-bid auction, and the second-price sealed-bid auction

(also called the Vickrey auction.) The model developed in this paper will use the first-price

sealed-bid auction, where each bidder submits a single bid independently, without seeing

others bids, and where the contract is awarded to the lowest bidder. The winner pays his

or her own bid in first price auctions. My choice is consistent with the fact that most U.S.

procurement auctions are awarded by sealed-bid first-price auctions. It should be noted

however that the choice of auction type can generate important differences in bidding be-

havior. For instance, Milgrom (1996) compares auction for the sale of radio spectrum fre-

quencies in the United States, Australia and New Zeeland. Different rules of auctions were

characterized by significantly different prices for similar bandwidths.

A key feature of auctions is the presence of imperfect information. If the partici-

pants had perfect information, the solution of auction model would be relatively trivial.

From a game-theoretical perspective, the types of equilibria that deal with imperfect in-


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formation are Bayesian-Nash equilibria. That is, each firms strategy is a function of its

own information, and it maximizes the expected payoff given the competitors strategies

and given the beliefs about their information.

In my model, bidders have private values. Each bidder knows precisely the cost of

executing a contract, but the value is only privately known. In contrast, in the pure com-

mon-value model the actual cost is the same for everyone, but bidders have different pri-

vate estimates about what their cost will be. For example, the value of an oil-lease depends

on how much oil is under the ground, and bidders may have access to different geological

signals about that amount. In this case, a bidder would change the estimate of the value if

he or she learnt of another bidders signal. In the private-value model, the bidders own

cost would be unaffected by learning any other bidders preferences or information. A

more general model encompassing both cases assumes that each bidder receives a private

information or signal, but allows each bidders value to be a general function of all the sig-

nals. However, it is safe to model the auctions by using the private-value model in simple

procurement contracts. Departures from the private-values model may bring the question

of collusion.

I make a number of simplifying assumptions in my game in order to render the

problem tractable. For example, I assume that it is costless to prepare a bid, and that the

number of bidders is exogenously specified. There are major technical hurdles for theoreti-

cal analysis when these assumptions are dropped. Few results can establish existence, not

to mention uniqueness, outside these restrictive assumptions. Maskin and Riley (1996)

showed that general existence and uniqueness results do not exist when the number of bid-


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ders is endogenous, when information acquisition is costly, or when bidders participate in

more than a single auction. Without uniqueness and existence of the equilibrium distribu-

tion of bids, game theory could not provide satisfactory answers.

The dynamic game will extend existing work by Guerre, Perrigne and Vuong

(2000), Bajari (2000b), and Jofre-Bonet and Pesendorfer (2000). Independent opinions

from these authors suggest the need for further study. Bajari, Jofre-Bonet, and Pesendorfer

suggest exploring the bidding behavior over time in a dynamic setting. This can be done by

introducing capacity constraints. During our discussions, Professor Patrick Bajari ex-

pressed his interest in the auctioneers optimal policy in procurement contracts under a dy-

namic setting. The difficulty of achieving the goals above is of theoretical nature. The

equilibrium equations have not yet been solved in closed form, not even for the single-unit

auction with asymmetric bidders. Several algorithms exist for solving the equations nu-

merically, but they are not very fast in practice. Bajari (2000) proposed a better algorithm,

and I have obtained even faster computer runs by improving his algorithm. My model will

extend Bajaris work to a multiple-period dynamic setting. The static model cannot fully

explain the dynamics of bidding behavior because it ignores capacity constraints and the

associated continuation values. I will explicitly include these variables in the equilibrium

equations.

The model assumptions are congruent with existing work. It describes an infinite-

horizon discrete-time industry, in which risk-neutral firms choose strategies that maximize

the expected discounted value of sealed-bid first-price auctions. The firms have private

valuations of the auctioned contract, and they are differentiated in their cost structure and


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in their available capacity. The information about their available capacity at each decision

node is summarized by a state, which in turn evolves as a Markovian transition process.

The firms are asymmetric in their probability distribution of costs, but have the same cost

interval ],[ maxmin cc (also called the support interval). Each firm observes its own cost, and

then determines the bid which maximizes utility, given the continuation values of winning

and losing, as well as the other firms strategies.

I will focus on the properties of the equilibrium and the comparative statics results

it predicts. As you will see, my model provides theoretical justification for many hypothe-

ses specific to the dynamic setting. I will explore the changes in the expected bid, the ex-

pected markup, and the present value, with respect to changes in the firms cost and capac-

ity, as well as to a change in its competitors changes in costs and capacity. The model is

consistent with economic intuition, since it predicts diminishing marginal returns to in-

creases in capacity. I will also look into the interaction effects between a firms cost and

capacity on the present value. In the light of the foregoing, the study has theoretical and

practical significance. The model can explain intuitive our intuition of the bidding behavior

in a dynamic context and it provides the tools to analyze many other aspects of the dy-

namic of procurement auctions.

Let me briefly describe what you will find in this paper. I will start with a fairly

succinct review of the literature related to the dynamics of procurement auctions. Then, I

will show the results I obtained from a dataset of bids from the Washington State Depart-

ment of Transportation. The dataset inspires the main elements to be taken into considera-

tion in the theoretical model, which is fully developed in section four. The model is pre-


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sented in great detail, so you may feel the need to skip some parts, especially the quite in-

volved proofs given for a series of propositions. If you decide not to follow the proofs, I

would like you to at least acknowledge their importance. The theoretical section discusses

Markov-perfect equilibria, delves into the intricacies of the equilibrium properties, shows

how the model can be turned into an infinite time-horizon game, introduces new measures

for quantifying the dynamics of procurement auctions, and also shows how the system

could be understood as a stochastic process. The computer simulation section outlines the

assumptions made in implementing the very general theoretical results. It gives informa-

tion about the most difficult issue I had to tackle in the computer simulation, the optimiza-

tion of a non-linear objective function with a very large number of real variables. In the

end, I will explain the comparative statics results predicted by the model.


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2. LITERATURE REVIEW

The review of relevant papers is organized by topic. Similar works are clustered

around the topics which emerge in my paper, in order to enhance understanding. I have

also attempted to provide a chronological progression and guide the reader through the de-

velopments in auction theory which led to my interest in this particular subject. The topics

covered in this section are: classical theoretical models, econometrical estimation models,

asymmetrical bidders, competition and collusion, independent versus affiliated private val-

ues, and other attempts towards a dynamic model of procurement auctions.

2.1 Classical Theoretical Models

Vickrey (1961) made an enormous progress in analyzing the game theoretical as-

pects of the auction problems. Griesmer, Levitan and Shubik (1967) analyzed the equilib-

rium of a first price auction in which contestants valuations are drawn from uniform dis-

tributions with different supports. Wilson (1969) introduced the common-value model and

developed the first closed-form equilibrium analysis of the winners curse. The full flower-

ing of auction theory came with critical contributions from Riley and Samuelson (1981),

Milgrom and Weber (1982), Maskin and Riley (1985), and McAfee and McMillan (1987).

In the classical model, a given number of risk-neutral potential buyers of an object

have privately-known information independently from each other, drawn from a common,

strictly increasing, atom-less distribution. The Revenue Equivalence Theorem states that,

under all types of auctions, the object always goes to the buyer with the highest signal,

yields the same expected revenue, and results in each bidder making the same expected


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payment as a function of the signal. The expected revenue from an auction is shown to

equal the expected marginal revenue of the winning bidder. Also, all the standard auctions

are all optimal as long as the seller imposes the optimal reserve price.

Milgrom and Weber (1982) develop a model of competitive bidding in which the

winning bidders payoff may depend upon his personal preferences, the preferences of oth-

ers, and the intrinsic qualities of the object being sold. In this model, the English ascending

auction generates higher average prices than does the second-price auction. Also, when

bidders are risk-neutral, the second-price auction generates higher average prices than the

Dutch and first-price auctions. In all of these auctions, the seller can raise the expected

price by adopting a policy of providing expert appraisals of the quality of the objects he or

she sells.

More advanced models deal with issues like risk-aversion, affiliation, asymmetries,

entry, collusion, multi-unit auctions, double auctions, royalties, incentive contracts, budget

constraints, externalities between bidders, and all-pay auctions. An elementary and non-

technical survey of auction theory is provided by Klemperer (1999).

2.2 Estimation Models

Besides theoretical models, research on auctions has also generated an impressive

body of experimental work, documented by Cox, Smith and Walker (1985, 1988). How-

ever, according to Laffont, Ossard and Vuong (1995), only a few empirical studies have

attempted to validate theoretical auction models using real auction data. Some examples

include Hansen (1985, 1986), Hendricks, Porter and Boudreau (1987) and Hendricks and


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Porter (1988). Instead, most empirical studies have concentrated on testing some implica-

tions of the theory of auctions using reduced-form econometrics models. An exception is

Paarsch (1989, 1992) who estimates econometric models that are closely derived from the-

ory. A possible reason for this gap between theoretical work and empirical work arises

from the computational difficulties due to the non-linearity and numerical complexity as-

sociated with the estimation of structural econometric models.

Laffont, Ossard and Vuong (1995) propose an estimation method to fill this gap.

They focus on first-price sealed bid and descending auctions, where each bidder is as-

sumed to a have a different private value for the object that is auctioned. The method relies

on simulated non-linear least square objective function appropriately adjusted as to obtain

consistent estimates of the parameters of interest. The symmetric Bayesian Nash equilib-

rium of the corresponding game of incomplete information was characterized by Riley and

Samuelson (1981), among others. The equilibrium expresses the optimal bid as a function

of the bidders private value, the reservation price of the object, the number of bidders, and

the distribution of private values. In general, bids are observed, but actual private values

are not observed. The preceding theoretical model leads to a closely related structural

econometric model. Since optimal bids are functions of private values, which are theoreti-

cally random, then observed bids are also random with a distribution that is uniquely de-

termined by the structural elements of the model. Unfortunately, the equilibrium function

relating optimal bids to private values is usually untractable. As a consequence, only very

specific distributions of private values have been considered in empirical work. The esti-

mation methods are based on simulations as proposed by Lerman and Manski (1981),


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McFadden (1989), Pakes and Pollard (1989), Laroque and Salanie (1989), and Gourieroux

and Monfort (1990, 1993).

2.3 Asymmetrical Bidders

Auction situations provide many examples where the symmetry assumption is not

tenable. This can be because of bidders difference in size, in geographic locations as in

Bajari (1998), and Flambart and Perrigne (2000), and in capacity constraints as in Jofre-

Bonet and Pesendorfer (2000). Other examples where asymmetry arises naturally are col-

lusions, asymmetrically informed bidders, or firms bidding jointly. Cartels are described

by Porter and Zona (1993), Baldwin, Marshall and Richard (1997) and Pesendorfer (2000).

Asymmetrically informed bidders in the outer continental shelf drainage auctions are

documented by Hendricks and Porter (1988) and Hendricks, Porter and Wilson (1994), and

joint bidding in the outer continental shelf auctions are studied by Hendricks and Porter

(1992). All these examples illustrate the necessity of developing general structural econo-

metric models to deal with asymmetric auctions.

2.4 Independent versus Affiliated Private Values

Campo, Perrigne and Vuong (2000) confirm that the existence of cartels, collusion

and heterogeneity across firms induces asymmetry in bidding games. A major difficulty

when considering asymmetric auctions is that the equilibrium strategies are solutions of an

intractable system of differential equations. Instead of focusing on the independent private

values model, they make an extension to include the affiliated private values paradigm.


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The attractive feature of their indirect procedure is that is does not require solving the dif-

ferential equations for the equilibrium strategies. The method can be easily generalized to

many types of bidders.

2.5 Competition and Collusion

Bajari and Ye (2000) study a class of econometric models derived from auction

theory. They are useful for modeling both competitive and collusive bidding in many ap-

plied settings. Their models allow for asymmetries and attempt to model the auction dy-

namics. They develop a set of conditions that are both necessary and sufficient to rational-

ize a distribution of bids with a model of competitive bidding. Their conditions can be

tested empirically in a straightforward fashion. As a second stage, they explain how the

choice between structural models of competition and structural models of collusion can be

posed as a statistical decision problem. As an application, they apply their test to a data set

of bidding by firms for construction contracts.

The approach suggested by Bajari and Ye (2000) is Bayesian. To describe the intui-

tion behind the tests, one could imagine the limiting case that the structural cost parameters

for all firms in the market are known with certainty. Then one would be able to compute

the equilibrium distribution of bids that are consistent with the competitive model. If the

empirical distribution of bids agrees with the computed distribution of bids then one cannot

reject the hypothesis of competitive bidding. Porter and Zona (1993), and Baldwin, Mar-

shall and Richard (1997) also propose econometric tests designed to detect collusive bid-

ding. Baldwin, Marshall and Richard (1997) nest both competition and collusion with a


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single model to test for collusion. Their model is applicable for oral or second-price auc-

tions with private values. Porter and Zona (1993) propose a procedure where two models

of bidding are estimated. Bajari and Yes paper sheds now light on Porter and Zona (1993,

1999). Observable differences across firms such as location and capacity play a key role in

their identification. The analysis tightens the link between the economic theory and the

econometrics of how asymmetries can be used to identify collusive bidding.

Another feature of Bajari and Yes model is that the econometric approach can be

used for any parametrically specified auction model. They specify a parametric distribution

of costs and compute the equilibrium inverse bid functions for the structural model in order

to evaluate the likelihood of the observed bids conditional on the covariates of the parame-

ters. By considering asymmetric auction models, they extend the earlier work which dis-

cussed parametric estimation of symmetric auction models such as Paarsch (1992), Laf-

font, Ossard and Vuong (1995), Armantier, Florens and Richard (1997) and Hong and

Shum (2000). The econometric techniques may provide an alternative to the important es-

timation algorithms developed by Guerre, Perrigne and Vuong (2000) and Jofre-Bonet and

Pesendorfer(2000) in at least three ways. Guerre et al (2000) and Jofre-Bonet et al (2000)

use a two-step procedure that first estimates the empirical distribution of bids conditional

on observed distribution of costs. The number of conditioning arguments grows exponen-

tially in the number of bidders, while Bajari et al (2000) suggest an algorithm where the

computational complexity grows linearly. They argue that in applications with a relatively

small number of bidders, there exists insufficient data to apply the techniques of Guerre et

al (2000) and Jofre-Bonet et al (2000). Unlike these two works, Bajari and Ye (2000) use a


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Bayesian rather than classical econometric methods in their estimation. A first advantage

of the Bayesian procedure is that it allows them to impose a priori restrictions on the pa-

rameter space in a natural fashion. It may not be possible to detect collusion with a priori

restrictions of the parameters of the structural model. The Bayesian approach is also ap-

pealing for testing non-nested models such as competition versus collusion.

2.6 Dynamic Models and Capacity Constraints

Jofre-Bonet and Pesendorfer (2000) estimate a repeated auction game under the

presence of capacity constraints. The estimation strategy is computationally simple as it

does not require solving for the equilibrium of the game. It uses a two-stage approach. In

the fist stage the distribution of bids conditional on the state variable is estimated using

data on bids, bidder characteristics and contract characteristics. In the second stage, an ex-

pression of the expected sum of future profits based on the distribution of bids is obtained,

and costs are inferred based on the first order conditions of optimal bids. They apply the

estimation method to repeated highway construction procurement auctions in the state of

California. They conclude that in this market, previously won uncompleted contracts re-

duce the probability of winning further contracts. They quantify the effect of intertemporal

constraints on bidders costs and on bids. Due to the intertemporal effect and also due to

bidder asymmetry, the auctioning mechanism can be inefficient. Based on the estimates of

costs, they quantify efficiency losses.

Even though the works of Paarsch (1992), Laffont et al (1995) and Guerre et al

(2000) develop an empirical approach to quantify informational uncertainty in static auc-


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tion games, there is little empirical work on dynamic auction games or dynamic oligopoly

games. Some examples include Laffont and Robert (1999) and Donald, Paarsch and Robert

(1997) who analyze finitely repeated auctions. Laffont and Robert (1999) consider a se-

quence of auctions in which, at each stage, an identical object is sold. Their model gener-

ates complex intra-day dynamics which are applied to data on eggplant auctions. Donald,

Paarsch, and Robert (1997) consider a model in which a finite number of objects are sold

in a sequence of ascending-price auctions. They estimate the model using data on the sales

of Siberian timber-export permits. The literature on estimation in dynamic games is sum-

marized by Pakes (1994).

Jofre-Bonet and Pesendorfer (2000) propose an estimation method that infers costs

based on the first order condition of optimal bids of the repeated bidding game. Their cru-

cial idea is that the expected discounted sum of future profits can be written looking for-

ward and depending entirely on the distribution of bidders bid choices. The resulting ex-

pression can be written as a linear equation system which can be easily solved numerically,

because they make a number of simplifying assumptions. In my model, the analysis is

much more general.

A related estimation strategy of the value function is employed by Hotz and Miller

(1994). They approximate the value function with discrete choices using estimates of

choice probabilities. Their framework differs from Jofre-Bonet and Pesendorfer (2000) be-

cause the model considers only a single agent dynamic decision problem, attention is re-

stricted to discrete actions, and informational constraints are not modeled. In work in pro-

gress, Berry and Pakes (2000) consider a related estimation strategy for dynamic games.


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The distinguishing feature of their approach is to consider an alternative representation of

the value function in which the expected sum of future profits is replaced with a sequence

of future profit realizations. However, their representation is less attractive in dynamic auc-

tion games, because profits are not observed and cannot be expressed indirectly from ob-

served bids, without knowing the equilibrium bid functions.


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3. EMPIRICAL OBSERVATIONS

This section deals with the results I obtained from a dataset of bids in the Washing-

ton State highway construction industry. I begin by describing the advertising, bidding and

awards process. The analysis gets more involved when the market structure and concentra-

tion are analyzed; important observations are made from the data. I also confirm some of

the previous comparative statics results, using a dataset that covers a different administra-

tive area and has a longer time span relative to datasets in previous work.

3.1 Advertising, Bidding and Award Process

There are a number of steps that must be addressed to effectively advertise, award,

execute, and document procurement contracts. A chronological order of phases is pre-

sented below:

Preparation for advertisement (technical reports, engineers estimate, risk determi-

nation, funding authorization).

Advertisement (usually done in trade journals for 2 to 4 weeks depending on the

nature of the project, after which auction packages are sent to pre-qualified bidders).

Bid opening (public opening and reading of bids, document verification).

Contract execution.

These detailed procedures presented in the Advertisement and Award Manual

published by the Washington State Department of Transportation (August 1998) provide

the framework for the award for all contracts, except emergency projects, which may be


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exempted. However, there are some exceptions for business enterprises owned by women

and minorities.

In Washington State, one of the conditions for awarding the contract is concerned

with the amount of the winning bid relative to the engineers estimate. If the winning bid

exceeds the engineers estimate by 10% or $50,000, whichever is greater, special guide-

lines must be followed. First, the governmental office needs to determine if there was ade-

quate competition. Second, it needs to document the impact of postponing the projects. Af-

ter taking into consideration all pertinent aspects, the procurement authority needs to pro-

vide justification for award or rejection.

The number of project proposal holders and the actual number of bidders provide

an indication of the level of competition. The Washington State Department of Transporta-

tion considers that the winning bid should not exceed the engineers estimate by more than

20% for 5 bidders or more, by 15% for 4 bidders, or by 10% for 3 bidders or less. How-

ever, the department recognizes that these guidelines may not be appropriate for some

specialized contracts and other factors should be considered. Also, it is important to know

if an attempt was made to attract additional bidders, and to find the reasons why some pro-

posal holders did not submit bids..

Postponing and re-advertising the project may have a negative impact. The pro-

curement authority takes into consideration constraints such as the urgency of the project

completion, the ability to complete the work within the current construction season, and

the potential for loss of construction season. There are direct costs related to re-advertising

the project. There may be risks involved with postponement, such as potential damage and


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severe deterioration. Also, the timing of re-advertising may impact the project funding be-

cause of budget constraints.

These regulations have the same effect of setting a reservation price. In a sense, the

auctioneer is a special bidder, whose bid is a direct function of the engineers estimate and

of the number of firms bidding in the project. The amount of the reservation price may be

uncertain because of the special circumstances surrounding a project. It can be modeled

with a probability distribution on a closed support interval. The lower end of the interval

depends on the engineers estimate and on the number of bidders.

It is interesting to watch the efforts that go into insuring that all bidders are serious

and that there is no collusion. To screen out unserious bids, the Washington State Depart-

ment of Transportation requires a bid deposit (also called proposal guarantee or contract

bond) which amounts to 5% of the submitted bid, and which must be sent before the bid

can be taken into consideration. Other clauses are related to subcontractors. If some sub-

contractors will execute more than 10% of the project, they must be mentioned on a sub-

contractor list when the bid is submitted. The constraints I have just mentioned limit the

contractors room of maneuver.

The bidders are required to sign a non-collusion declaration. The issue is so impor-

tant that the US Department of Transportation has a hotline that can be used to report bid

rigging, bidder collusion, or other fraudulent activities. The non-collusion declaration con-

tains the following statement: I, by signing the proposal, hereby declare, under penalty of

perjury under the laws of the United States that the undersigned person, firm, association

or corporation has not, either directly or indirectly, entered into any agreement, partici-


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pated in any collusion, or otherwise taken any action in restraint of free competitive bid-

ding in connection with the project for which this proposal is submitted.

3.2 Description of the Data

The data used here was obtained from the Washington State Department of Trans-

portation at no cost. As a matter of fact, the Department of Transportation saves all the bid

information and can provide it at public request. This fact may reduce the bidders infor-

mational costs of the bidders. With certainty, it helped reducing the costs for my project,

since the other (much more costly) alternative would have been to purchase the data set

from a private firm such as Construction Market Data Group. The advantage of receiving

the data directly from the original source (the Department of Transportation) had the addi-

tional advantage of covering a larger time span, due to thee access to older bid archives.

The data1 covers an almost five year period from January 1, 1997 to July 11, 2001.

The dataset itself included a wealth of information, organized around each project,

each firm, and each submitted bid. Project-specific data included contract number, region

number, work order, bid date, award date, highway number, project description, and pro-

ject county. Firm-specific data included contractor name, contractor number, contractor

address and phone number. Bid-specific data was broken down item by item and included

the following categories: item, standard item, item description, unit of measure, estimated

quantity, estimated unit price, estimated total, contractor unit price, and contractor total.

1 WSDOT provides project by project information online at the following Web address:http://www.wsdot.wa.gov/biz/contaa/BIDSTATS/bidresults.htm


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Among all available projects, I focused attention on those involving asphalt fixing

or resurfacing, because they are smaller in size and do not require the use of subcontrac-

tors. The projects can be finished in a single season and there is no expectation that the

specifications will be changed during the execution of the contract.

Let me briefly provide the main statistics. There were a total of 325 projects involv-

ing repairs of asphalt surface, with an average project size of $1,423,616. The total value

of the projects was $462,675,065 for a period of approximately 5 years. The average num-

ber of bids per project was 2.55, indicating that the level of competition was not too fierce,

especially in the more isolated areas of Washington state. There were 32 firms who sub-

mitted successful bids during that period, with the largest firm Lakeside Industries control-

ling 18.69% of the market with 59 projects own and $86,491,604 in revenues. There were

also a number of firms that submitted bids, but never won any project. It appears that some

of these firms did not have asphalt repair as their core competency. In any case, they are

examples of unsuccessful attempts of entering the asphalt repair industry.

One issue of interest is the level of concentration in the industry measured by the

Herfindahl index. The HH index was 761.90 in Washington State, with the concentration

of each firm being computed as a percentage of revenues from the total volume of con-

tracts. The HH index indicated an adequate level of competition at state level. There were

only four out-of-state firms that submitted bids, and among them only two were able to

win any project at all. The most successful out-of-state bidder was Poe Asphalt Paving

from the neighboring Idaho state, with only 3.62% market share. Viewed from this per-


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spective, it is difficult to enter the highway construction market unless the bidder has its

headquarters and equipment in the state where it does business.

Firm Bids Wins Total Contracts Percentage

LAKESIDE INDUSTRIES 142 59 $86,491,604 18.69%SUPERIOR PAVING CO 47 33 $50,481,739 10.91%ACME MATERIALS & CONSTRUCTION CO 53 19 $33,388,269 7.22%INLAND ASPHALT COMPANY 48 16 $32,064,480 6.93%ASSOCIATED SAND AND GRAVEL CO 21 11 $31,161,610 6.74%

WILDER CONSTRUCTION COMPANY 79 31 $24,693,088 5.34%TRANSTATE PAVING CO 27 15 $19,251,219 4.16%WESTERN STATES PAVING CO 13 10 $17,217,087 3.72%POE ASPHALT PAVING, INC. 35 11 $16,745,262 3.62%CSR WEST, LLC DBA CSR ASSOCIATED 27 13 $15,123,337 3.27%ACE PAVING CO., INC. 16 13 $15,066,175 3.26%BASIN PAVING CO 26 12 $15,026,485 3.25%MORRILL ASPHALT PAVING 38 11 $13,744,653 2.97%TUCCI & SONS, INC. 19 10 $12,894,625 2.79%COLUMBIA ASPHALT & GRAVEL, INC. 36 9 $11,975,311 2.59%VALLEY ASPHALT & PAVING, INC. 8 6 $9,873,917 2.13%CENTRAL WASHINGTON ASPHALT, INC. 35 6 $8,215,847 1.78%

ARC MATERIALS CORPORATION 13 7 $6,995,098 1.51%WATSON ASPHALT PAVING 28 7 $6,943,958 1.50%WOODWORTH & COMPANY, INC. 24 7 $6,505,445 1.41%JC COMPTON CONTRACTOR INC 4 2 $6,388,873 1.38%WILDISH STANDARD PAVING CO. 5 2 $6,355,138 1.37%QUAD-CITIES CONSTRUCTION, INC. 6 3 $4,534,370 0.98%R EXCAVATING, INC 2 1 $2,891,740 0.63%ICON MATERIALS, INC. 6 3 $2,518,306 0.54%OLDCASTLE NORTHWEST, INC. 10 1 $2,390,830 0.52%GOODFELLOW BROS., INC. 2 1 $1,121,627 0.24%KRIEG CONSTRUCTION, INC. 4 1 $646,465 0.14%STEELMAN-DUFF, INC. 2 1 $627,271 0.14%MURPHY BROTHERS, INC. 5 1 $599,727 0.13%

WESTWAY CONSTRUCTION, INC. 2 1 $432,898 0.09%RAY POLAND AND SONS, INC. 4 2 $308,612 0.07%A AND B ASPHALT, INC. 2 0 - -DOOLITTLE CONSTRUCTION CO., INC. 12 0 - -FRANKLIN CONSTRUCTION, INC. 5 0 - -GRADY'S, INC. 1 0 - -KIEWIT PACIFIC CO 8 0 - -M.A. SEGALE, INC. 11 0 - -PACK AND CO. 1 0 - -VANCOUVER PAVING COMPANY 2 0 - -WALKERS' PAVING, INC 1 0 - -


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3.3 County by County Analysis

The study of market concentration by county reveals more about the market struc-

ture in the highway construction industry. I ignored the counties with less than five pro-

jects because of their lack of statistical significance. The remaining counties had an aver-

age of 4.17 firms per county being able to win projects. Thus, the bidding dynamics in a

small region could be modeled as a repeated auction game with s small number of bidders.

The implicit assumption is that if the fringe bidders were unable to win any bid for a period

of almost five years, it would be very difficult for them to enter the industry in that particu-

lar county. Even the county with the largest number of projects, King County with 31 pro-

jects (including the Seattle area), had a Herfindahl index of 2323, which would be more

County Projects Winners HH

Adams 15 5 3101Benton 19 4 3325Chelan 10 5 2309Clallam 6 1 10000Clark 11 3 5349

Douglas 10 3 3641Ferry 9 4 3449

Franklin 11 5 2470Grant 16 5 3536

Grays Harbor 8 2 5381King 31 12 2323

Kitsap 11 3 6901Kittitas 9 2 5809Lewis 9 4 3059

Lincoln 7 3 3616Pierce 14 3 5155

Snohomish 22 6 1872Skagit 11 4 3211

Spokane 15 4 4450Thurston 8 3 5428

Walla Walla 11 4 3395Whatcom 24 5 3924Whitman 12 7 2063Yakima 17 3 5159


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than the maximum of 1800 legally allowed for a reasonable level of competition. In King

county, the leader Lakeside Industries controlled approximately 50% market share. The

industry is even less competitive in other counties. For example, Clallam county had ex-

actly one bidder for all of its 6 projects during the five-year time span; the single bidder

and winner for all of these projects was again Lakeside Industries. The firms bids were

about 10% above the engineers estimate, which was basically the maximum allowed by

the Washington State Department of Transportation. The bottom line is that the Herfindahl

index exceeded 1800 in all of the counties in Washington State in our industry. Although

the FTC guidelines regarding competition refer to the market concentration for the whole

country, it shows that the firms location and the project location are very important in the

highway construction industry. Distance from the firm to the location of the project will be

shown to be a significant factor in bidding behavior. When we focus to a smaller adminis-

trative region such as a county, the industry can be modeled as an oligopolistic or even

monopolistic game. It is thus interesting to see what happens when a few firms compete

against each other.

3.4 Regressions

Two other empirical studies shed light on the basic comparative statics results of

bidding behavior in the highway construction industry. Therefore, I will not insist too

much on this aspect. My regressions use a different dataset, and they reassert the validity

of the previous results. The dataset from Washington State has the advantages of being

more recent and of covering a longer time span.


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Bajari and Ye (2000b) use a data set purchased from CMD which contains bid in-

formation for Minnesota, North Dakota and South Dakota for the period 1994-1998. They

use the following reduced form bid functions for each firm and each project:

++++

++++=

)capacityMax(*)distanceMin1ln(*

Capacity)Own(*)DistanceOwn1ln(*Estimate

Bid

43

210

bb

bbb

Jofre-Bonet and Pesendorfer (2000) use a data set which consists of all contract

awards for highway and street construction in California from 1996 to 1998. The probit

estimates of the bid submission decision followed the following equation:

++++

++++=

)regiontheinCapacity(*)CapacityOwn(*)DistanceOwn(*

)biddersFringe(*)daysWorking(*)Estimate(*Bid

985

4210

bbb

bbbb

I decided to follow the regression model used by Bajari and Ye(2000b), and I inde-

pendently confirmed their results. Some clarifications are necessary to fully understand the

regression. The free capacity is computed as the percentage of projects won before the bid

over the total number of projects won during a season (May to September). The logarithm

of distance was used to account for the possibility that firms may have equipment at vari-

ous sites in the state. The variables Min distance and Max capacity refer to the firms

competitors minimum distance and maximum capacity. Other factors have been taken into

consideration, such as the correlation effects between the variables, and the individual ef-

fect of each bid and each project. Due to the large number of dummy variables for each

firm and each project, the parameter estimates were obtained in SAS. There were 829 ob-

servations and 074742 =R . The parameter estimates are summarized in the table:


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The significant variables at 5% confidence are the own distance to the project site

and the own free capacity. The results imply that an increase of one order of magnitude in

the distance will increase the bid by approximately 2%, whereas an increase of 10% in ca-

pacity would reduce the bid by approximately 1%, keeping everything else constant. The

interaction effects between competitors do not seem to be too strong. We would expect that

competitors with higher distances would reduce the level of competition and increase the

bid (confirmed by the positive sign of the regression), whereas competitors with more free

capacity would increase the level of competition and reduce the bid (again confirmed by

the negative sign of the regression parameter.)

These empirical results indicate that distance, as a factor that affects transportation

costs, is positively correlated with the bid level. The effect caused by the available capacity

is more subtle. At the beginning of the season, firms submit what we could call normal

bids, but at the end of the season they may submit ridiculously high bids since their capac-

ity is already filled up and they are not actively looking for projects. This explains why

bids are negatively correlated with available capacity. The knowledge that a competitor has

a higher distance or a lower capacity level indicates they will have a high bid, which in

Coefficient of Estimate Std Error Significant at 5%

Intercept 1.03424 0.10647 yesLn(1+Own Dist) 0.01743 0.00876 yes

Ln(1+Min Dist) 0.00143 0.01156 no

Own Capacity -0.09466 0.02265 yes

Max Capacity -0.01865 0.03193 no

Parameter Estimates


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turn makes the firm with this knowledge bid higher. This also agrees with the signs of the

parameter estimates in the table.

The purpose of looking at the empirical data was to get a better understanding of

the bidding behavior in the highway construction industry. In the next section, we will de-

velop a comprehensive mathematical model of the dynamics of procurement auctions. Not

only that it will verify the results observed in the real world, but it will also make a set of

new predictions regarding other indicators such as the expected markup and the expected

present value.


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4. THE THEORETICAL MODEL

Before developing the theoretical model, I discuss the Markov-perfect equilibria. It

will be necessary to give complex proofs to a series of propositions, in order to justify the

differential equations I will obtain. The model can be used either for a finite or infinite

game. Several measure for quantifying the aggressiveness level are introduced. At the end

of this section, I develop a framework for understanding repeated auctions as a stochastic

process.

4.1 Markov-Perfect Equilibria

A Markov-perfect equilibrium is a refinement of the Nash equilibrium. Game theo-

rists are interested in refinements because the class of Nash equilibria can be very large in

complex games. In the case of dynamic auctions, the Nash equilibria include many types

of collusive equilibria. These self-sustaining collusions would be interesting to discuss and

analyze, but the solution space is probably huge. The refinements are an attempt to sim-

plify the game and, ideally, reduce it to a unique equilibrium.

The technical definition of the Markov-perfect equilibrium is given by Fudenberg

and Tirole (1991): A Markov-perfect equilibrium is a profile of strategies that are a per-

fect equilibrium and are measurable with respect to the payoff-relevant history.

The graph below shows successive refinements of the Nash equilibrium.

Equilibria

Nash

Equilibria

Perfect(Subgame-)

Equilibria

PerfectMarkov

Markov-perfect equilibria are suited to dynamic games in which the environment or

state changes from period to period. In repeated auctions, the past has a direct influence on


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the present, because it determines the current available capacity. Markov-perfect equilibria

ignore past actions that did not change the environment, and take into consideration only

the current state. This refinement is necessary to deal with very complex games such as the

repeated auction game. Markov-perfect equilibria do not discriminate between different

histories that led to the same present state. Since the current state summarizes the influence

of past play on the strategies and payoff functions for each subgame, there is no profitable

deviation from the Markov strategies if all the other players are using Markov strategies.

Examples of games that are suitable to a Markov perfect equilibrium analysis are bequest

games, extraction of a common resource, or capital accumulation games. In the bequest

game, intergenerational family transfers give raise to situations in which each generation

cares about its own consumption and the consumption of future generations. The succes-

sion of generations does not behave like a single decision maker, because of differences in

preferences among generations. The bequest game is the ideal example for a Markov-

perfect equilibrium analysis.

Markov-perfect equilibria may be inappropriate in environments that are more

psychological in nature, such as those characterized by threats and commitments. It is

perfectly possible that firms care about being consistent to previous course of action. The

idea of forward induction is that past actions will be interpreted as signals of future inten-

tions even when those actions may not influence payoffs in the continuation game.

Markov-perfect equilibria take a somewhat contrary perspective, since backward induction

is used to find the optimal course of action. In Markov-perfect equilibria we start with the

future and end with the current state.


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It should be noted that dynamic games can be modeled with discrete time or with

the analogue continuous time. Due to the technical difficulties in continuous-time games

(also called differential games,) I will focus on the discrete-time game. Another issue is

whether one should consider pure-strategy or mixed-strategy equilibria. I will work only

with pure-strategy equilibria. In a repeated auction game, the optimal response is pure-

strategy if all the other players are also playing pure-strategies.

An important issue is the existence of the equilibrium. Ignoring existence will lead

to nonsensical results, without logical foundation. Fudenberg and Tirole (1991) have

shown that Markov perfect equilibria exist in stochastic games with a finite number of

states and actions. Furthermore, the existence theorem was extended to countable state

spaces by Parthasarathy and Rieder. Existence theorems for uncountable or continuous

state spaces were much harder to obtain, but this is beyond the current paper anyway. In

my model, capacity is represented as an integer. Even for cases that require more granular

capacity intervals (i.e. percentage of available capacity), smaller countable intervals can be

used. Either way, the existence of Markov perfect equilibria can be easily met by setting

the granularity of the intervals at the required level.

4.2 Properties of the Equilibrium

I will develop here a dynamic model for procurement auctions. It describes an in-

dustry in which firms choose strategies that maximize the expected discounted value of

sealed-bid first-price auctions. My approach takes into consideration the dynamic behavior

in auctions, where the number of periods can be either finite or infinite.


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The firms have private valuations of the auctioned contract, and they are differenti-

ated in their cost structure. At each time period, the environment is a state characterized by

the information about the firms available capacity. In an industry withNfirms, the state is

a vector ofN available capacities Ns . The state evolves over time according to a

Markovian transition process.

At the beginning of each period, firms observe their own costs, drawn from an

atomless probability distribution with density )(if and with cumulative distribution

]1,0[],[: max,min, iii ccF . The firms are asymmetric in their probability distribution of

costs. Each firm observes only its own cost, and then determines the bid which will maxi-

mize utility, given the other firms strategies and the continuation values of winning and

losing.

The total utility function is time-additive and exhibits constant time preference

(with discount factor ). The one-period utility function for firm i in state s and at time tis

tsiu ,, . The expected discounted utility for firm i in state s and at time 0t is:

[ ]

=

=

0

0|E ,,

t

,,tt

tsitsi suV t for an infinite number of periods / auctions, or

[ ]=

=

T

tt

tsitsi suV t0

0|E ,,

t

,, for a finite number of T periods / auctions.

The firms are risk neutral. If a firm submits a winning bid, then its utility for the

current state is ii cb . If it fails to submit the lowest bid, the utility of the current state is 0.


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In the model, firm is strategy is a bid function ],[: max,min,,, iitsi ccB which maps

the cost draw to a bid. The bid function depends on three parameters: the identity of the

firm, the current state, and the current time period. Let ))((inf ,,],[

min,max,min,

itsiccc

i cBbiii

= and

))((sup ,,],[

max,max,min,

itsiccc

i cBbiii

= . The bid function can then be written as:

],[],[: max,min,max,min,,, iiiitsi bbccB .

Since the utility function is time-additive, the total utility will be equal to the utility

of the current state plus the utility of all future states. The utility of all future states is

times the expected discounted utility in the following period, contingent on the state. If the

current state is s, I will define is be the state in the next period if firm i wins the auction.

For each particular auction, the firm faces the following dilemma. If it bids too low

for the project, the profit will also be low. However, if it increases the bid too much, the

probability of winning decreases to the point where the firm is guaranteed to lose the auc-

tion. Let )(,, itsj bQ denote the probability that firm j wins, from the perspective of firm i,

and where i could be equal toj. In case of a tie, the winning firm is chosen at random with

equal probability. Firm i assumes that all other firms play their optimal equilibrium strate-

gies. The formula for these probabilities will be spelled out later in this section.

The model allows for firms to have various capacities.

Define=

otherwise0,bidnextfor thecapacityfreeenoughhasfirmif,11 free

ii

Define the profit function as the profit in the current state plus the profit in all fu-

ture states:


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( ) ( ) )(1)|(

ij

free1,,,,,,1,,free,,

++++=

jtsiitsjitsitsiiiiiitsi ji

VbQbQVcbcb

)|,(max)(,,,, iitsibitsi

cbscVi

=

A Markov-perfect equilibrium is the natural solution which would emerge from the

game, assuming no coordination between bidders, and no other commitments to a certain

course of action. The equilibrium is subgame-perfect because the firms maximize in all

future periods. The equilibrium follows the recurrent Bellman equations below:

( ) ( ) )(1max)(

ij

free

1,,,,,,1,,free,,

++++=

j

tsiitsjitsitsiiii

b

itsi jii

VbQbQVcbcV

For the rest of the analysis, I will assume that firm i has sufficient capacity to par-

ticipate in the auction. Otherwise, )()(

ij

free1,,,,,,

+=

jtsiitsjitsi j

VbQcV and the discounted util-

ity does not depend on firm isdecision in period t.

It is very useful to make a few observations about the endpoints of the bid distribu-

tions. I have defined the bid functions as ],[],[: max,min,max,min,,, iiiitsi bbccB . Let

min,min min ii

bb = and max,max min ii

bb = [sic]. Suppose maxmax,such that bbi i > . This shows

that 0)( max,,, =itsi bQ and firm i could change its bid on in the interval ),( max,max ibb without

affecting the profit function

)()|(

ij

free1,,,,,,

+=j

tsiitsjiitsi jVbQcb

This implies that, if an equilibrium solution exists, it is not unique. From now on, I

will focus attention on the particular case ibbi = maxmax, . The second observation refers to


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the minimum bid. There must be at least two firms who bid minb . Assume only firm i bids

minb . Then firm i could strictly increase profits by bidding slightly higher that minb , which

contradicts the optimality of minb , as explained in the proof of Proposition 2. These two

observations show that in the caseN=2 firms, both firms have identical distribution of bids

].,[ maxmin bb

I will make some general assumptions about the cost distribution. The probability

distribution functions+

],[: max,min, iii ccf are atomless, continuous and 0)( >ii cf

],[ max,min, iii ccc . The cumulative distribution functions ]1,0[],[: max,min, iii ccF are dif-

ferentiable. With these considerations in mind, I will provide a series of complex proposi-

tions. The goal is to show that the bid functions are invertible and that the inverses are dif-

ferentiable, which is crucial for the first order conditions of the equilibrium. The proofs

that will follow are quite involved. If the reader feels they are too detailed, the next few

pages could be skipped.

Proposition 1. The probability of winning )(,, itsi bQ is monotonically decreasing,

while )(,, itsk bQ with ik is monotonically increasing.

Proof. After observing its cost independently, each firm chooses and submits a

sealed bid. Firm i doesnt know what its competitors will bid, but it knows their joint dis-

tribution of bids )}({ ijjbg . Consider any set of bids ijjb }{ from the joint distribution and

let 12 ii bb > .


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>

=

==

)}{,min(if,0

bidslowequal),}{,min(if,1

)}{|(,,

ijjik

ijjik

ijjitsk

bbb

pbbbpbbQ

ijjijjitskijjisk bbbQbbtQ }{)}{|()}{|(, 1,,2,

ijjijjijjitskijjijjijjitsk bbgbbQbbgbbQ }{d)}{()}{|(}{d)}{()}{|( 1,,2,,

)()( 1,,2,, itskitsk bQbQ

Therefore )(,, itsi bQ is monotonically decreasing and )(,, itsk bQ is monotonically in-

creasing.

Proposition 2. The functions )(,, itsi bQ are strictly decreasing on the interval

),[ maxmin bb when 1)(0 ,,


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Proof. Let )(,, itsii cBb = for some ic such that ),[ maxmin bbbi and 1)(0 ,, such that no other firms bid in the interval ],[ +ii bb with strictly posi-

tive probability. This shows that )()( ,,,, += itsjitsj bQbQ , including the case ij = .

++++=

free

,,1,,,,1,,,, )()()()|(

j

ijitsjtsiitsitsiiiiitsi

bQVbQVcbcbji

)|()()|()|( ,,,,,,,, iitsiitsiiitsiiitsi cbbQcbcb >+=+

Firm i could strictly increase its profit by bidding +ib instead of ib . However,

this would contradict the optimality condition for ib . Therefore, there is at least one firm

that bids in the interval ],[ +ii bb with strictly positive probability 0> . So

)()( ,,,, +> itsiitsi bQbQ 0> . This indicates that the functions )(, isi bQ are strictly de-

creasing on the interval ),[ maxmin bb .

Proposition 3. The equilibrium bid functions ],[],[: max,min,max,min,,, iiiitsi bbccB are

monotonically increasing on the interval ),[ maxmin bb .

Proof. Let 21 ii cc < and suppose )()( 2,,1,, itsiitsi cBcB >

( ) ( ) ))(()()(1)|)((

ij

free1,,2,,,,2,,,,1,,12,,free

12,,,,

=++=

=

++

jtsiitsitsjitsitsitsiiitsii

iitsitsi

jiVcBQcBQVccB

ccB

( ) )()-())((

)()(1

2,,,,12

ij

free1,,2,,,,

2,,,,1,,22,,free

=++

++=

+

+

itsitsiii

j

tsiitsitsj

itsitsitsiiitsii

cBQccVcBQ

cBQVccB

j

i

( ))()-()|)(( 2,,,,1222,,,, itsitsiiiiitsitsi cBQccccB +=


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By the optimality of the bid function, it must be that

)|)(()|)(( 21,,,,12,,,, iitsitsiiitsitsi ccBccB , so

( ) =+ )()-()|)(()|)(( 2,,,,1221,,,,12,,,, itsitsiiiiitsitsiiitsitsi cBQccccBccB

( ) =++

=

+

+

)()-())((

)()(1

2,,,,12

ij

free1,,1,,,,

1,,,,1,,21,,free

itsitsiiij

tsiitsitsj

itsitsitsiiitsii

cBQccVcBQ

cBQVccB

j

i

( ) ( )

( ) ( )[ ] =+

+++=

++

)()()-(

))(()()(1

1,,,,2,,,,12

ij

free1,,1,,,,1,,,,1,,11,,free

itsitsiitsitsiii

j

tsiitsitsjitsitsitsiiitsii

cBQcBQcc

VcBQcBQVccBji

) ))()()-()|)(( 1,,,,2,,,,1211,,,, itsitsiitsitsiiiiitsitsi cBQcBQccccB +=

Since )(,, itsi bQ is a strictly decreasing function (Proposition 2) and >)( 1,, itsi cB

)( 2,, itsi cB , it follows that ))(())(( 2,,,,1,,,, itsitsiitsitsi cBQcBQ < .

Therefore )|)(()|)(( 11,,,,12,,,, iitsitsiiitsitsi ccBccB > which contradicts the optimality

of )(1,, itsi

cB . It must be that )()(2,,1,, itsiitsi

cBcB 21 ii

cc .


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According to Proposition 1, the probability functions )(,, tsjQ are monotone. This

implies the existence of left and right limits around 0b . A further implication is that one

can take right and left limits of the profit function in the neighborhood of 0b .

++=++

))(1()()()|( 1,,11),2(,11,,11,,11111,,1 1 bQVbQVcbcb tststststs

++=+++ 1,,11,,11,,11,,11111,,1 221

)()()|( tststststs VbQVVcbcb

+++

+++=

++

+++

)()())()(()(

)()()|(

1,,1010,,11,,11,,11,,110

1,,10,,11,,11,,11011,,1

21

221

bQbbbQbQVVcb

VbQVVcbcb

tststststs

tststststs

)()())()(()(

)|()|(

1,,1010,,11,,11,,11,,110

10,,111,,1

21bQbbbQbQVVcb

cbcb

tststststs

tsts

+++

+=

++

Case 1: 1,,11,,101 21 ++ +< tsts VVbc

Since firm 2 bids 0b with positive probability, the following inequalities hold:

0)()(lim 0,,11,,10

01

01

>=

++= ++

.

Case 2: 1,,11,,101 21 ++ += tsts VVbc .

The profit function becomes )()()|()|( 1,,10100,,101,,1 bQbbcbcb tststs +=

Observe that )|()|( 00,,101,,1 cbcb tsts > whenever 01 bb > . By a similar argument,

)|()|( 00,,101,,1 cbcb tsts < whenever 01 bb < . Firm 1 is strictly better off by bidding higher

than 0b : 01,,11,,10,, )( 21 bVVbB tststsi >+ ++ . Again, firm 1 does not bid in the interval

],[ 00 +bb for ))(,0( 01,,11,,10,, 21 bVVbB tststsi + ++ .

Case 3: 1,,11,,101 21 ++ +> tsts VVbc

According to Proposition 1, 01,,11,,10,,1,, )()( 21 bVVbBcB tststsitsi >+> ++ .

The outcome for these three cases completely justifies the assertion that firm 1 does

not bid in the interval ],[ 00 +bb for a small enough 0> . However, this could never be

an equilibrium solution. Firm 2 could strictly increase its profits by increasing its bid from

0b to +0b , following the argument made in the proof of Proposition 2. This contradicts

the optimality of 0b .

In conclusion, no firm bids a certain bid ),[ maxmin0 bbb with strictly positive prob-

ability for the caseN=2.

Proposition 5. The functions ]1,0[],[:)( maxmin,, bbbQ itsi are strictly decreasing

and continuous for the caseN=2.


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Proof: It follows directly from Propositions 2 and 4.

Proposition 6. The image of the bid functions ],[],[: maxminmax,min,,, bbccB iitsi is

connected for the caseN=2.

Proof. Suppose firm 1 never bids ],[ maxmin0 bbb .

Let )(sup 1)(

0

01,1

ccbcB s

+= .

Case 1:+ < 00 cc . Then ),()( 00101,,1 += cccbcB ts contradiction.

Case 2:+

> 00 cc . Then 01,,1 )( bcB ts > and ),()( 00101,,1 +< cccbcB ts . Absurd.

Case 3: 000 ccc == + . If 00,,1 )( bcB ts > , firm 1 never bids in the interval

+

2

)(,

00,,1

0

bcBb

ts. If 00,,1 )( bcB ts < , firm 1 never bids in the interval

+0

00,,1,

2

)(b

bcB ts.

Either way, there exists a closed interval in the neighborhood of 0b where firm 1 never

bids. The contradiction arises by using the same argument as in proposition 2.

Proposition 7. The inverse functions ],[],[: max,min,maxmin,1

,,,, iiitsitsi ccbbB =

ex-

ist and are continuous for the caseN=2.

Proof. Propositions 3 and 5 imply that the bid functions :,, tsiB ],[ max,min, ii cc

],[ maxmin bb are strictly increasing. Propositions 3 and 5 imply they are continuous. There-

fore, these functions are invertible and the inverse functions are also continuous.


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Proposition 8. The functions ]1,0[],[:)( maxmin,, bbbQ itsi are differentiable on the

interval ),[ maxmin bb for the caseN=2.

Proof. The proof is very similar to the one given by Maskin and Riley (1996), ex-

cept for the existence of positive continuation values.

I will prove the proposition for firm 1. The same logic can be used for firm 2. Let

),[, maxmin10 bbbb , 01 bb . Since the bid functions are invertible, let )( 0,,1 bts and

)( 1,,1 bts be the corresponding costs. First observe that:

1,,11,,11,,11,,11,,111,,11,,1 221 )())(())(|( +++ ++= tststststststs VbQVVbbbb and

1,,11,,1max,,1 2))(|(

+= tststs Vbb

The choice of 1b is maximal only if ))(|())(|( 1,,11,,11,,1max,,1 bbbb tstststs

.0)( 1,,11,,11,,11 21 + ++ tststs VVbb

However, the inequality must be strict because otherwise firm 1 could strictly in-

crease its profit by bidding slightly above 1b . But this would contradict the maximal choice

of 1b .

Therefore .0)( 1,,11,,11,,11 21 >+ ++ tststs VVbb

))(|())(|( 1,,11,,11,,10,,1 bbbb tstststs

++

++

+++

+++

1,,11,,11,,11,,11,,11

1,,10,,11,,11,,11,,10

221

221

)())((

)())((

tststststs

tststststs

VbQVVbb

VbQVVbb

+++

)()())()(())(( 1,,1011,,10,,11,,11,,11,,10 21 bQbbbQbQVVbb tstststststs


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1,,11,,11,,10

1,,1

01

0,,11,,1

21)(

)()()(

+++

tststs

tststs

VVbb

bQ

bb

bQbQ

Similarly, I will show that

))(|())(|( 0,,11,,10,,10,,1 bbbb tstststs

++

++

+++

+++

1,,11,,11,,11,,10,,11

1,,10,,11,,11,,10,,10

221

221

)())((

)())((

tststststs

tststststs

VbQVVbb

VbQVVbb

+++

)()())()(())(( 0,,1011,,10,,11,,11,,10,,10 21 bQbbbQbQVVbb tstststststs

1,,11,,10,,10

1,,1

01

0,,11,,1

21)(

)()()(

++ +

tststs

tststs

VVbb

bQ

bb

bQbQ

Take the two equations above to the limit for 01 bb ; this is a legal operation be-

cause tsQ ,,1 and ts ,,1 are continuous functions.

1,,11,,10,,10

0,,1

01

0,,11,,1

1,,11,,10,,10

0,,1

2101

21)(

)()()(lim

)(

)(

++

+++

+

tststs

tststs

bbtststs

ts

VVbb

bQ

bb

bQbQ

VVbb

bQ

This shows that1,,11,,10,,10

0,,1

01

0,,11,,1

2101 )(

)()()(lim

++ +

=

tststs

tststs

bb VVbb

bQ

bb

bQbQ

, which

means the functions ]1,0[],[:)( maxmin,, bbbQ itsi are differentiable on the interval

),[ maxmin bb .

Proposition 9. The inverse bid functions ],[],[: max,min,maxmin,,, iiitsi ccbb are dif-

ferentiable for the caseN=2.

Proof. Observe that ))((1)( 1,,221,,1 bFbQ tsts = ))(1))(( 1,,11,,22 bQbF tsts =


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The right hand side is differentiable by Proposition 8.

Therefore ]1,0[],[: maxmin,,22 bbF ts is differentiable. By Assumption (A2), 2F is

differentiable, so therefore )(,,2 ts is differentiable. By a symmetrical proof, )(,,1 ts must

also be differentiable.

Conjecture: Propositions 4-9 are true in general for 2N . The inverse bid func-

tions ],[],[: max,min,maxmin,1

,,,, iiitsitsi ccbbB =

exist and they are continuous and differen-

tiable.

Comment. I proved the statement above for the particular case of two firms.

Maskin and Riley (1996, 1999) have shown the general result holds for the static problem,

where all continuation values are 0. However, they note that even for the static case, the

problem is significantly more complex for 3N . Accepting that the bid functions are con-

tinuous and differentiable allows me to find the equilibrium conditions.

The equilibrium analysis continues from now on for the general case 2N .

In order to win the auction, a firm must submit a bid that is lower than its N-1 com-

petitors bids. This occurs exactly when )(,, itsjj bc > for all ij , which means that all

firms ij have cost draws greater than )(,, itsj b .

The probability that firm i wins is computed as:

))

=ijj

itsjjitsi bFbQfree,

,,,, )(1)(

From the perspective of firm i, the expected probability that another firm wins is:


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( )( )

=

==

i

iisj

b

b kjtsjjjtskkitsj

b

b

jtsjjjtsjj

b

c

jjjtsjtsjitsj

bdFbFbQ

bdFbQcdFcBQbQ

min

min

,

min

jki,k

free ,,,,,,

,,,,free

)(

,,,,,,

))(()(1)(

))(()(1)())(()(

Lets plug this formula into the Bellman recurrent equations:

( ) ( ) ++=

++)(11max)(

ij

free,,free1,,,,1,,free,,

jitsjjtsiitsitsiiii

bitsi

bQVbQVcbcVji

i

( ) ( )( )( )

( )( )

+

+

+

++=

i

j

j

ii

b

b k

jtsjjjtskk

j

tsi

j

itsjjitsiitsiib

itsi

bdFbFV

bFVbbcV

min,

jki,k

free,,,,

ij

free1,,

ij

free,,free1,,,,,,

))(()(1

11)(max)(

The expected discounted utility in state s is )(,, itsi cV after the cost is known and

== maxi,

mini,

c

c

,,,, )()( iiitsitsi cdFcVV max

mini,

))(())(( ,,,,,,

b

b

itsiiitsitsibdFbV before the cost is known.

( ) ( ) ))(()()(max ,,ij

free1,,,,,,1,,,,

)(,,

max

mini,,,

itsii

b

b jtsiitsjitsitsiitsiitsi bdFVbQbQVbbV ji

tsi

++=

++

( ) ( )( )( )

( )( )))((

))(()(1

11)(

max ,,

jki,k

free,,,,

ij

free1,,

ij

free,,free1,,,,

)(

,,

max

mini,

min,

,,

itsii

b

b

b

b k

jtsjjjtskk

j

tsi

j

itsjjitsiitsii

tsibdF

bdFbFV

bFVbb

Vi

j

j

i

tsi

+

++

=

+

+


47/93

Page 47

It is important to see there are two possible cases here. In the first case firm i has

enough capacity to participate. In the second case, firm i doesnt have sufficient capacity to

participate, but its continuation value depends on the outcome of the auction nevertheless.

The model implicitly assumes that at least one firm has enough capacity to bid and win,

otherwise the auction would not take place and the outcome would be trivial.

If the firm does not have enough capacity, there will be no benefit from the current

period, and the discounted present value depends only of future auctions. There is no

maximization problem for firm i in case 2; the discounted value is inferred from other

firms strategies.

+=

ij

free1,,max,,,, )(

jtsitsjtsi j

VbQV

( )( )

+

=

ij

free

jki,k

free,,,,1,,,,

max

min,

))(()(1j

b

b k

jtsjjjtskktsitsi

j

jbdFbFVV

The first order conditions will help establish the optimal bid function when firm i

has sufficient capacity. I worked on the equation by equivalence and I am showing only the

main steps below:

( )( )( )

( ) ( )( )

+=

=

++

)(1)('))(()(

1

ij

free

jki,k

free,,,,,,1,,1,,,,

ij

free,,

j k

itskkitsjitsjjtsitsiitsii

j

itsjj

bFbbfVVbb

bF

ji


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Page 48

( )( )( )( )

0))((1

)()('))((11

ij

free ,,

1,,1,,,,,,,,

ij

free,, =

+

++

j itsjj

tsitsiitsiiitsjitsjj

j

itsjjbF

VVbbbbfbF

ji

( ) ( )),[1

))((1

)()(')(maxmin,

ij

free ,,

1,,1,,,,,,,, bbbbF

VVbbbbfii

j itsjj

tsitsiitsiiitsjitsjj ji =

+

++

The bordered formula above will play a crucial role in computing the equilibrium

solution. After carefully analyzing all the mathematical assumptions that went into the

model, I found the first order conditions of the equilibrium solution. Notice that the condi-

tion does not apply to the maximum value of the bid maxb because the fractions would not

be correctly defined.

4.3 The Infinite Time Horizon

I analyzed the Markov perfect equilibrium by looking at successive time periods

and by finding relationships between current and continuation values. The model can be

used directly to a game with a finite number of periods or auctions. I will show here that it

can also be used to get an insight into the bidding behavior when the number of periods is

infinite.

In the infinite time horizon problem, it is crucial to understand the behavior of the

continuation values0,, tsi

V in the limiting case when the number of periods Tgoes to infin-

ity T . For a fixed player i and state s, three possible cases are conceivable: (1) the

sequence1,, 0 Ttsi

V is convergent and thus we can write the limit as0,,

lim tsiT

V

, (2) the se-


49/93

Page 49

quence1,, 0 Ttsi

V leads to some sort of cyclical behavior in the long run, or (3) the se-

quence fails to converge or lead to a cyclical behavior.

The computer simulations that were run for this project suggest that the sequence

1,, 0 TtsiV is strongly convergent. For terminal values of 0 in the last period T, sequence

converged to %1 accuracy after less than 100 iterations. In other words, strong conver-

gence is expected for 100T . As I will immediately show, convergence of the sequence

1,, 0 TtsiV has profound implications on the repeated auction game with infinite periods,

and it is an important milestone in getting a better understanding of the underlying dynam-

ics.

Besides the convergence results mentioned above, the simulation has also shown an

interesting situation. If sufficiently large termination values are chosen, it is possible to

achieve a different solution in the limit. A careful analysis suggests the existence of multi-

ple Markov-perfect equilibria in the infinite time horizon case. Various beliefs of the con-

tinuation values can be sustained in equilibrium. However, the equilibria generated from

non-zero terminal values do not exhibit an unusual degree of degeneracy. As a matter of

fact, they share the same common results in terms of bidding aggressiveness. It almost

seems as if two sets of equilibrium continuation values differ only by a constant. The fact

that various long run equilibria exhibit common characteristics allows me to study the case

of zero terminal values as being representative. From now on, I will focus on the long run

equilibrium that is an outcome of the repeated game with zero terminal values. At least

from an empirical point of view, computer simulations suggest the Markov-perfect equilib-


50/93

Page 50

rium of this particular game exists and it is unique. Let0,,,

lim tsiT

si VV

= for any player i and

state s. This is great, because it implies a series of other results related to bidding behavior

in the long run.

One of the important findings relating continuation values was that

( ) ( )( )( )

( )( )))((

))(()(1

11)(

max ,,

jki,k

free,,,,

ij

free1,,

ij

free,,free1,,,,

)(,,

max

mini,

min,

,,itsii

b

b

b

b kjtsjjjtskk

jtsi

jitsjjitsiitsii

tsi bdF

bdFbFV

bFVbb

Vi

j

j

i

tsi

+

++

=

+

+

After I replace the continuation values for period t+1 with their continuation values

in the limit, I achieve the following result:

( ) ( )( )( )

( )( )))((

))(()(1

11)(

max ,,

jki,k

free,,,,

ij

free,

ij

free,,free,,,

)(,

max

mini,

min,

,,

itsii

b

b

b

b kjtsjjjtskk

jsi

jitsjjisiitsii

si bdF

bdFbFV

bFVbb

Vi

j

j

i

tsi

+

++

=

Thus, the maximization problem in the long run is the same for all time periods.

Since the same problem is solved every period and it generates the same maximal value,

the repeated auction game becomes time-homogeneous in the long run. Observe that the

sequence of functions1,,

)(0

Ttsi converges to a unique value for T , because it is the

argmax of the maximization problem. Thus, convergence of the current continuation val-

ues for T implies the convergence of bidding behavior over time, with the limit

)()(lim ,,, 0 = sitsiT . Please keep in mind we are not taking a sequence of real numbers to


51/93

Page 51

the limit, but a sequence offunctions with their entire structure: the domains and the actual

transformation.

Since the inverse bid functions are known to be stationary in the long run, lets take

a look at the measures that quantify the probability of winning. From the perspective of

firm i, the probability of winning after the costs are revealed are:

( )( )

=ijj

itsjjitsi bFbQfree,

,,,, )(1)( for i, and

( )( )

=ib

b kjtsjjjtskkitsj

bdFbFbQ

min

jki,k

free,,,,,, ))(()(1)( for ij .

Since the probabilities of winning are direct functions of the bid functions, they are

also stationary in the long run. Let )(lim)(0,,,=

tsi

Tsi QQ and )(lim)( 0,,, = tsjTsj

QQ for

ij . With the implications shown so far, the convergence of the continuation values im-

plies that the auctions are time homogeneous for any given player i and state s, in other

words they are stationary over time.

For reasons that will become clear immediately, I am interested in the probability

of winning not only after the costs are revealed, but also before the observation is made.

This follows easily from the previous results:

( )( )= ijj

itsjjitsibFbQ

free,

,,,, )(1)(

( )( ) ( )=

)(d)(1 ,,free,

,,,,

max

min,

itsii

b

b ijj

itsjjtsibFbFQ

i

( )( ) ( ) iitsiitsiib

b ijj

itsjjtsi bbbfbFQ

i

d)()()(1 ' ,,,,free,

,,,,

max

min,

=


52/93

Page 52

Since )()(lim ,,, 0 = sitsiT for any firm i, it follows that the long run probability of

winning before the costs are revealed is:

( )( ) ( ) iisiisiib

b ijj

isjjtsiT

si bbbfbFQQ

i

od)()()(1lim ',,

free,

,,,,

max

min,

==

Just as an exercise, I show analytically that the probabilities add up to one. That is,

11

,, ==

N

i

tsiQ for the finite game and 11

, ==

N

i

siQ for the infinite game. First, define the func-

tion ( ) =free

,,, 1i

tsiits F with the following derivative:

( ) ( )

=

free free,

,,,,,,, 1i ijj

tsjjtsitsiits Ff . Then observe that:

1)()()(d)( max,min,,,1

,,

max

min

max

min

==== =

bbbbbQ tstsb

bts

b

b

ts

N

i

tsi .

An analogous proof leads to 1

1

, ==

N

i

siQ for the infinite game.

4.4 Defining Aggressiveness Measures

Aggressiveness in procurement auctions can be under

Documents

Voicu Dynamics of Procurement Auctions