Auction Models When Bidders Make Small Mistakes:Consequences for Theory and Estimation.

Patrick Bajari and Ali HortacsuStanford University and University of Chicago.1

August 15, 2001.

AbstractIn this paper, we explore the consequences of using equilibrium models of auctions in making policy

recommendations, such as the design of real world markets, or as a basis for structural estimation whenbidders make small errors in optimization. We consider two types of error prone behavior that nest Bayes-Nash equilibrium as a special case. The rst is the logit equilibrium model of McKelvey and Palfrey(1995) where bidders measure their payoffs imperfectly. The second is a submission error model wherebidders submit an unintended bid with positive probability. First, we establish that when the number ofbidders is sufciently large, even if bidders maximize expected prots to within a few cents, the predicitonsof the logit equilibrium model can differ greatly from the predictions of a Bayes-Nash equilibrium model.Second, we demonstrate that if we structurally estimate the model assuming that bidders are playing aBayes-Nash equilibrium when instead they are acting according to the error submission model, we willtend to overestimate markups. Third, we use standard methods to non-parameterically estimate structuralauction models on an experimental data set. We nd that in experiments where the average valuation for theobject being auctioned is fteen dollars, bidders are within twenty cents of maximizing prots on average.However, in one of the experiments, the non-parameteric estimate of average markups is 40 percent whilethe true value is 20 percent. We conclude that it is important to conduct sensitivity analysis to determinehow robust policy recommendations and parameter estimates are to a priori plausible amounts of error pronebehavior.

.

1 We would like to thank Garrett Summers for outstanding research assistance. Bajari would like to thank the NationalScience Foundation grant numbers SES-0112106, SES-0122747, the National Fellows Program at the Hoover Institution and theStanford Institute for Theoretical Economics for research support. Seminar participants at the Harvard-MIT EconometricsWorkshop, Stanford, UCLA and Pittsburg gave very useful comments on a much earlier draft of this paper. All remaining errors are our own.

1

1 Introduction.

Recently, a large body of research has applied equilibrium models of auctions to real world problems such as the

design of markets and structural estimation. The market design literature has used insights from auction theory to

privatize the spectrum in many countries, to establish markets for electricity generation, and to build markets on the

Internet. See for instance, Cameron, Crampton, andWilson (1997), Crampton (1997), Crampton and Schwartz (1991),

Economist (2000), McAfee (1998), McAfee and McMillan (1996), and Milgrom (2000). In the structural estimation

literature, the econometrician assumes that bidders behave rationally and estimates the bidders’ private information

conditional on the observed distribution of bids. Once the distribution of private information has been recovered, it

is possible to use estimates of structural parameters to design the auction, simulate the effects of mergers, or compute

the damages from collusion. Examples of structural estimation are Paarsch (1992), Baldwin, Marshall and Richard

(1997), Armantier, Florens and Richard (1997), Froeb et al. (1997), Bajari (1997), Hendricks, Pinkse, and Porter

(1998), Hong and Schum (1998), Guerre, Perrigne, and Vuong (2000), and Hortacsu (2001) among others.

Few economists who apply auction models to real world problems would argue that the rationality assumptions of

these models hold perfectly. Bidders in real world auctions are not perfectly rational and do not maximize prots to an

arbitrarily large number of signicant digits. However, many economists who have applied these models to real world

problems argue that bidders are approximately rational and come close in some metric to maximizing their expected

payoffs. For real world applications, it is important to know how quickly we converge to equilibrium as the amount

of error in decision making tends to zero and how sensitive our parameter estimates or policy recommendations are to

empirically plausible amounts of error in the decision making process.

In this paper, we shall demonstrate that at least in some cases, the policy conclusions and estimation results allow-

ing for empirically plausible amounts of error are surprisingly different than those based on assumptions of perfect

rationality. We attempt to identify the scenarios in which results will be especially sensitive to error prone behav-

ior and how conclusions might be biased. While the assessment of whether the results generated from a rational

equilibrium model are “far” or “near” to results generated from a model with error will depend on an investigator’s

personal loss function, we believe that our analysis indicates that many investigators will nd it important to check the

robustness of their conclusions to error prone behavior.

We shall consider two models of deviations from rational behavior both of which nest Bayes-Nash equilibrium as

2

a special case when the amount of error in decision making tends to zero. The rst is the logit equilibrium model

of McKelvey and Palfrey (1995) which we can interpret as a model where bidders measure their own prots with an

independent and identically distributed error term. The second model we refer to as the submission error model. In

this model, there is a discrepancy between the bid that a rm intends to submit and the bid the rm actually submits.

This could occur from simple clerical errors that happen in practice all the time (see Bartholomew (1998), Chapter

11). We shall compare the Bayes-Nash, logit and submission error models in the context of a rst price sealed bid

auction. However, as we shall discuss in the paper, many of our results generalize to other auction models and indeed,

to parts of economics outside of auctions.

The rst contribution of this paper is to compare inference and prediction in the logit equilibrium model with the

Bayes-Nash equilibrium model as the number of bidders increases. We demonstrate that as the number of bidders

becomes large a bidder’s prot function will typically become very “at” near the maximum. If the number of bidders

is sufciently large, almost any bid will result in prots that are arbitrarily close to the prot maximizing bid. If

bidders only maximize to within a few cents, they will be indifferent between almost all bids. This implies that

the logit equilibrium model predicts signicantly different bidding behavior than the Bayes-Nash equilibrium model

and that structural estimates based on the Bayes-Nash equilibrium model will not be robust to empirically plausible

amounts of error in decision making.

As a simple example, consider a procurement auction where 20 rms bid for a contract to build a single and

indivisible public works project. Suppose that all rms are ex ante identical and have cost draws uniformly distributed

between one and two million dollars. The median cost draw for any rm is 1.5 million dollars. A rm with the

median cost draw will only win the contract if all other rms have higher cost draws. The probability of this event

is :519 = 0:00000191 or slightly less than 2 parts in one million. It can easily be shown that any bid between

the equilibrium bid for this rm and a bid of 2 million will not change expected prot by more than a few dollars

because the prot function is very at near the maximum. The logit equilibrium of this game will typically be

quite different than the Bayes-Nash equilibrium and any predictions about the equilibrium of this game or structural

parameter estimates will depend critically on assumptions about how bidders make mistakes.

The problem of prot functions that are at near the maximum in equilibrium is present in most auction models

when the number of bidders is sufciently large. It is also present in other economic models such as median voter

models where the odds of being pivotal are small, or contribution games where everyone decides on a donation level

and the public project is implemented if the total contributions sum to at least some xed amount.

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In some real world auctions, prot functions will be at near the maximum due to the large number of bidders.

For instance, the seminal work of Laffont, Ossard and Vuong (1995) estimates the number of bidders in their eggplant

auctions to be 18. Porter (1995), in summarizing his inuential work with Hendricks, nds that the number of bidders

in OCS wildcat tract auctions is between 10 and 18 for 180 of their 2,510 auctions. Gerard van den Berg and Bas van

der Klauuw (1999) study bidding for dutch ower auctions where the average number of bidders is approximately 50.

The second contribution of this paper is to study the sensitivity of parameter estimates based on a Bayes-Nash

equilibrium model when the true model is the submission error model. We demonstrate that if bidders submit an

unintended bid even with small probability, structural parameter estimates will be biased and typically, we will over-

estimate markups.

As a simple example, consider a procurement auction where the lowest cost draw that has positive probability

is zero. In a Bayes-Nash equilibrium, it will clearly not be prot maximizing for a rm with a cost draw of zero

to bid zero and therefore the lowest bid that has positive probability in equilibrium must be bounded away from

zero. Suppose, however, that in our data we see a bid that is arbitrarily close to zero. How could we reconcile this

observation with a Bayes-Nash equilibrium model? As we will show in the paper, to rationalize the observed bids is

we must infer that the a prior distribution of costs for all rms is concentrated near zero, since a rm with zero cost

only nds it rational to bid close to zero if he expects all other rms to have costs close to zero. In this fashion, a

single bid near zero in the data set can cause us to infer that all costs are near zero and that markups are near innite.

The non-robustness of estimates from auction models to submission errors comes from the fact that the support

of the distribution of observed actions depends on the structural parameters of the distribution of private information.

This property holds in many other structural models. For instance, job search models and some production frontier

models display this property. In both cases, outliers generated from mistakes will bias parameter estimates. For a

more detailed discussion of these examples, see Hong (1997).

Thirdly, we test whether our results are empirically relevant by examining the bidding behavior of subjects in

experiments. In the experiments, the average valuation for the object being auctioned is fteen dollars and bidders

maximize prots to within twenty cents on average. However, in at least one experiment, the observed bids differs by

$2.55 on average from the Nash equilibrium bid. This is consistent with our results that suggest that “small” errors in

decision making can lead to “larger” deviations from equilibrium behavior.

Next, we structurally estimate the auction model using the non-parametric techniques of Guerre, Perrigne and

Vuong (2000). We nd that in one experiment, the estimated markups are on average 40 percent while in the average

4

markups actually made by the subjects are 20 percent. The results from the experiments indicate while subjects

maximize prots to a number that would be conventionality regarded as small (20 cents), our estimates of markups

are at least in one case biased upwards by 20 percent. We conclude that while our bidders are approximately rational,

their bidding behavior differs signicantly from the predictions of the Nash equilibriummodel, and structural estimates

based on a rational model are also signicantly biased.

Our research builds on methods and insights in at least four other literatures. As we mentioned previously, the

applications that motivated our paper are from the literature on market design and structural estimation. In addition,

our research is related to Harrison (1989,1992) who argued that anomalies from equilibrium in could be explained

by the fact that the opportunity cost for non-equilibrium behavior was by any reasonable standard very small. See

also Kagel and Roth (1992), Cox, Smith and Walker (1992) and Merlo and Schotter (1992) for further discussion.

Our analysis builds on Harrisons’s insight and considers the implications of at prot functions for policy analysis

and structural estimation. Also, we examine the sensitivity of results in structural estimation to the error submission

model. We also build on the literature that examines logit equilibrium in games. See McKelvey and Palfrey (1995),

Goeree, Holt, Palfrey (1999), Goeree and Holt (1999) and Bajari (1998).

Section 2 of this paper develops a simple model of bidding in a Bayes-Nash equilibrium and in a logit equilibrium.

Section 3 compares the logit equilibrium to the Bayes-Nash equilibrium when there are a large number of bidders.

Section 4 develops the submission errors model and offers an explains why estimates of markups can be too large

when we ignore submission errors and estimate a Bayes-Nash equilibrium model using standard methods. In Section

5, we use experimental data to estimate a structural auction model with standard methods. Section 6 concludes.

2 The Logit Equilibrium Model.

In this section, we develop a simple model of competitive bidding for a single and indivisible procurement contract.

The contract is awarded to the lowest bidder. We begin by considering the benchmark case where bidders are sym-

metric, risk neutral, have private values and the equilibrium concept is Bayes-Nash. We then generalize the model to

allow bidders to make errors in the spirit of the logit equilibrium model of McKelvey and Palfrey (1995).

2.1 The Bayes-Nash Equilibrium Model.

In the model,N rms compete for a procurement contract to construct a single and indivisible project. Before bidding

starts, each rm i forms an estimate of its cost to complete the project. The cost estimate is rm i’s private information,

5

that is, rm i knows its own cost estimate but does not know the cost estimates of the other rms. The cost estimate for

rm i will be denoted as ci. The cost draws for all of the bidders are independently and identically distributed random

variables with a cumulative distribution function F (c) and a probability density function f(c): We shall assume that

the random variable ci has support [c; c].

Let bi denote the bid submitted by bidder i. If bidder i submits the lowest bid then i’s von Neumann-Morgenstern

(vNM) utility is bi ¡ ci; and if i fails to submit the lowest bid then her utility is 0. If a tie occurs, the contract will be

awarded at random among the set of low bidders. Letting ui(b1; :::; bn; ci) denote the vNM utility for bidder i with a

cost draw of ci, we get:

ui(b1; :::; bn; ci) ´½

bi ¡ ci if bi < bj for all i 6= j0 otherwise. (1)

Bidder i is said to have private values since her prot depends only on ci, and not the cost estimates of other rms. If

only bidder specic cost factors account for the differences in cost estimates, then the assumption of private values is

appropriate.

We study the symmetric equilibrium to this game where the equilibrium bid function is b = b(c): It has been

established by LeBrun (1995) and Maskin and Riley (1996) that under weak regularity conditions the equilibrium bid

function is strictly increasing and differentiable so that its inverse Á(b) is also strictly increasing and differentiable.

Therefore, rm i is the low bidder if and only if cj ¸ Á(bi) for all j 6= i, that is, if all rms have cost draws greater

than Á(bi). Since the cost draws of all rms are i.i.d., the probability of this event is (1 ¡ F (Á(b)))N¡1 where N is

the number of bidders in the auction. We will let ¼i(bi; ci) denote the expected prot of bidder i when she has a cost

draw of ci and bids bi; this can be written as:

¼i(bi; ci) ´ (bi ¡ ci) (1 ¡ F (Á(b)))N¡1 (2)

In equation (2), bidder i’s expected utility is its markup over cost bi ¡ ci times the probability that rm i actually wins

the auction.

By standard arguments (see McAfee and McMillan (1987), Milgrom and Weber (1982)) the Bayes-Nash equilib-

rium bidding strategy in this model can be shown to satisfy:

b(c) = c +

R c

c(1 ¡ F (z))N¡1dz

(1 ¡ F (c))N¡1(3)

6

2.2 The Logit Equilibrium Model.

In this subsection, we shall generalize the Bayes-Nash equilibrium model along the lines of McKelvey and Palfrey’s

(1995) logit equilibrium. In McKelvey and Palfrey (1995), the authors develop an equilibrium concept for games

where players make decisions in a similar fashion to random utility models. Bayes-Nash equilibrium can be viewed

as a limiting case of the logit equilibrium model where the errors are very small. Similar applications of the logit-

equilibrium concepts to the auction setting are also used in Bajari (1998), Goeree, Holt, Palfrey (1999), and Goeree

and Holt (1999).

In the logit equilibrium model, the set of possible costs, ci and bids bi will be large, but nite. Let B =

fb1; b2; :::; b#Bg be the set of bids the agents can choose and let C = fc1; :::; c#Cg be the set of possible cost es-timates. In the logit equilibrium model, since agents are identical, we will search for a symmetric equilibrium in

which agents use the same bid strategy. Unlike the Bayes-Nash model, agents will not use pure strategies in the logit

model. A (symmetric) strategy B(bjc) is a probability measure that assigns a probability to every bid b conditional

upon every cost draw c. In order for the strategy to be well dened, the following must be satised:

For all c 2 C and b 2 B; B(bjc) ¸ 0 (4)

For all c 2 C,Xb2B

B(bjc) = 1: (5)

That is, no bid can receive less than 0 probability and conditional upon any possible cost draw c, the probabilities of

all the bids must sum to one.

Suppose that all agents follow the bidding strategyB(bjc), then the probabilityQ(b) that player i wins the auction

with a bid of b satises2:

Q(b) =

"Xc2C

Xb02B and b0<b

B(b0jc)f(c)

#N¡1

(6)

The term inside the bracket is the probability that any given player submits a bid strictly less than b. Since players

make their bids independently, the term inside the bracket is raised to the N ¡ 1st power to compute the probability

that all other players submit bids strictly less than b.

2 We assume that a bid strictly less than all other bidders is required to win the auction, and thus avoid considerationof ties between bidders. This is to simplify exposition of the problem and the computations. With a sufciently large setof types and bids this assumption will not fundamentally change our analysis.

7

In a Bayes-Nash equilibrium, the prot to rm i from bidding bi with a cost of ci would be ¼(bi; ci) = (bi ¡ci) ¤ Q(bi): In a logit equilibrium model, players are assumed to measure the prot to submitting a bid of b with a

cost of c with an i.i.d. error term "(b; c): We will assume that "(bi; ci) follows an extreme value distribution, so that

"(bi; ci) has a cumulative distribution function F (²) = exp(¡ exp(¡¸²)): This distribution has a mean of °¸ where

° is Euler’s constant (0.577) and variance ¼2

6¸2 so that ¸ proportional to the precision, or inverse of the variance. In a

logit model, we will let b¼(bi; ci) be the utility that the agent i receives from bidding bi when she has a cost draw of ci;

this is a sum ¼(bi; ci) and "(bi; ci) :

b¼(bi; ci) ´ (bi ¡ ci) ¤ Q(bi) + "(bi; ci) = ¼(bi; ci) + "(bi; ci) (7)

We can think of the decision process for a single agent as follows: rst, each bidder i learns her private information ci,

next, for every bi 2 B bidder i draws an error term "(bi; ci), lastly, each bidder chooses the strategy which maximizesb¼(bi; ci); her expected prot accounting for errors in measuring prot.

Let ¾(bi; ci; B) be the probability that agent i bids bi conditional on a cost draw ci and that theN ¡ 1 other agents

bid using the strategyB. By well known properties of the extreme value distribution it follows immediately that:

¾(bi; ci; B) =exp(¸¼(bi; ci; B)))P

b02B exp(¸¼(b0; ci; B)): (8)

An equilibrium in the logit model is a bidding functionB(bjc) that is a xed point of (8) meaning that for all c and all b,B(bjc) = ¾(bi; ci; B). That is, if all bidders follow the symmetric bidding strategyB(bjc), then ¾(bi; ci; B) = B(bjc):

Denition. A Logit Equilibrium is a bidding strategyB(bjc) such that for all c and all b,B(bjc) = ¾(bi; ci; B).

As we let ¸ ! 0 the variance of the error term becomes innite so that actions consist of all error. If ¸ ! 1then the variance of the error term is zero and the equilibrium of the game tends towards a Bayes-Nash equilibrium.

Therefore, the logit equilibrium nests Bayes-Nash equilibrium as a special case when ¸ ! 1. It is straightforwardto extend the concept of a logit equilibrium to a richer class of games. A logit equilibrium is always dened as a xed

point of the best response function (8). The existence of the logit equilibrium is obtained by using xed point methods

such as in McKelvey and Palfrey (1995) and Anderson, Goeree and Holt (1999).

2.3 Discussion.

In this section, we shall discuss some differences between the behavioral predictions of the logit model and the Bayes-

8

Nash equilibrium model. The primary difference between the two is that in the Bayes-Nash equilibriummodel, agents

are assumed to optimize without error, whereas in the logit model, we allow for errors in optimization.

The assumption that agents make at least small errors in optimization in real world economic problems is often

appropriate. Suppose that a bidder attempted to learn about its prots using standard tools from econometrics by

regressing its prots on the bids it actually submitted. Even after observing a large number of bids, the rm would

forecast its prots from bidding bi with at least some error. In the logit equilibrium model, we assume that the error

is distributed extreme value and is i.i.d. This may not be true for the forecast error a rm has in assessing its own

prots. However, the extreme value assumption makes the problem of incorporating the error into an equilibrium

model computationally tractable because the best response correspondence (8) has a closed form.

In addition to forecast error, bidders may not choose their best response because they lack the cognitive ability to

think through the consequences of their actions. Indeed, economists who have studied auctions computationally are

only able to solve the games with limited precision. It seems natural to allow for some degree of error in optimization

even among experienced players. The random utility framework of the logit equilibrium model is a convenient tool

that captures this dimension of behavior, although we believe that it is possible to develop many alternative models of

limited cognitive capacity.

Another attractive feature of the logit model is that all bids have positive probability, as can be gleaned from (8).

This is not true of the Bayes-Nash equilibrium model. Suppose, that c > 0, that is, the lowest possible cost in the

auction is greater than zero. In the Bayes-Nash equilibrium model, the lowest bid observed in equilibrium b(c) will

be greater than zero. This causes a problem for empirical work. In some empirical studies, such as Bajari and Ye

(2001) and Hortacsu (2001), there exist some bids that are too low to have been generated by positive costs and an

assumption of equilibrium behavior. While these bids may not occur very frequently, they are inconsistent with the

assumptions of the econometric model. Clearly a richer framework is desirable. Also, as we shall demonstrate later,

the estimates in econometric models can be very sensitive to bids less than b(c) while the presence of such outliers is

consistent with the logit equilibrium framework.

3 Competitive Bidding With A Large Number of Bidders.

In this section, we compare the Bayes-Nash equilibrium model and the logit equilibrium model as the number of

bidders becomes large. We rst demonstrate that as the number of bidders becomes large, the prot function in a

Bayes-Nash equilibrium will typically be very at near the maximum. Therefore, with a sufciently large number

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of bidders, almost all bids will approximately maximize prots. We characterize the distance between the logit

equilibrium model and the Bayes-Nash equilibrium model analytically and study the difference using simulation.

Second, we demonstrate that when bidders are playing the logit equilibrium, standard techniques used in structural

estimation will lead to biased results. The size of the bias increases as the number of bidders becomes large because

the distance between the logit and the Bayes-Nash equilibrium increases.

3.1 Comparing the Logit and Bayes-Nash Equilibria.

Let ¼(b(ci); ci) be the expected prot in a Bayes-Nash equilibrium for a rm that has a cost draw of ci and bids her

equilibrium strategy b(ci). By equation (2) it follows that:

¼(b(ci); ci) = (b(ci) ¡ ci)(1 ¡ F (Á(bi))N¡1 (9)

= (b(ci) ¡ ci)(1 ¡ F (ci))N¡1 (10)

· (c ¡ ci)(1 ¡ F (ci))N¡1 (11)

· c(1 ¡ F (ci))N¡1 (12)

Equation (10) follows from the fact that in equilibrium ci = Á(bi) and that rm i wins the auction precisely when all

other rms have higher cost draws. Equation (11) follows from the fact that in equilibrium all bids are less than c

and (12) follows from the fact that costs are always positive. Equations (9)-(12) imply that for a xed value of ci,

prots decline at an exponential rate as N , the number of bidders, increases. This follows directly from the fact that

the probability of winning, holding ci xed, falls at an exponential rate inN .

In Table 1, we summarize the probability of winning the auction as a function of the percentile of ci and the number

of bidders.

Table 1: The Probability of Winning For Selected Costs.Percentile/N 5 10 20 30.9 1.00e-04 1.00e-9 1.00e-19 1.00e-29.8 0.0016 5.12e-07 5.24e-14 5.37e-21.5 0.0625 0.002 1.91e-06 1.86e-09.3 0.2401 0.0404 0.0011 3.22e-05.1 0.6561 0.3874 0.1351 0.0471.05 0.8145 0.6302 0.3774 0.2259

From Table 1, we see that holding the percentile of the cost draw xed, the probability of winning falls rapidly in the

number of bidders. For instance, the median cost draw when there are 20 bidders wins the auction slightly less than 2

10

out of every one million times! If the upper bound on cost is c = $2; 000; 000:00 then by (9)-(12) and Table 1, $4 is

an upper bound on prots for a rm with the median cost draw. We believe that the optimization assumptions made

in the Bayes-Nash equilibrium are by any reasonable standard too strong as the number of bidders becomes large. We

believe that in a real world application, a rm bidding on a million dollar contract does not optimize to within 4 dollars

(6 signicant digits!) and that an equilibrium based on such an assumption is implausible.

The following gure displays the payoff function for a bidder with cost equal to the median cost draw. The

model used to generate this gure assumes that costs are uniformly distributed on [0; 1], which leads the Bayes-Nash

equilibrium bids to be distributed on [ 1N ; 1].3 As can be observed from the gure, the payoff function is quite at for

cost draws around (and especially above) 0:5 – hence a multiplicity of bids would be “consistent” with optimization if

bidders maximized with even a small amount of error.

Figure 1: Expected Payoff as a Function of the Number of Bidders.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-0.45

-0.4

-0.35

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

Bid

Expe

cted

pay

off

N=5 N=10N=20

In Figures 2 and 3, we use simulation to compare the equilibrium distribution of bids predicted by the Bayes-Nash

equilibrium model and the logit equilibrium. Consider a simple example where there are N = 8 players who have

3 Using equation (3.2), the equilibrium bid can be calculated to be: b(c) = 1N

+ N¡1N

c

11

costs that are distributed truncated normal on the interval [0; 8] with mean 4 and standard deviation 1. The parameter

¸ is set equal to 25, which implies that the standard deviation of the error term "(b; c) is approximately 0.05, or about 1

percent of the mean of the distribution of valuations. In gures 2 and 3 we draw 2000 random costs from the truncated

normal distribution and solve for the 2000 bids associated with these costs in the Bayes-Nash equilibrium. Figure 4

repeats this exercise for the logit equilibrium.4

A comparison of Figures 2 and 3 reveals that, even with 8 bidders there can be a signicant difference between

the equilibrium of a model that permits a small amount of error in measuring or computing payoffs and a model that

does not. Both the mean and the variance of the distribution of bids increase noticeably when error is included. The

intuition behind this result is simple. In a logit equilibrium, when a rm draws a costs sufciently larger than c = 2, it

will win the auction with very low probability and the amount that it bids will not matter very much because the prot

function is at near the maximum. High bids are more likely to be observed in the logit equilibrium and the mass of

the distribution of bids therefore shifts to the right in a logit model as compared to a Bayes-Nash equilibrium model.5

Theorem 1 formalizes how agents become indifferent between almost all bids as the number of bidders becomes

sufciently large and bidders make errors in measuring their payoffs as in equation (8). First, we construct a discrete

approximationBBN(bjc) to the Bayes-Nash equilibrium of the game. As in Section 2, let C µ [c; c] be a large, discrete

grid of cost draws. To construct the nite set of bids B and the discrete approximation to the Bayes-Nash equilibriumBBN(bjc), let b 2 B and BBN(bjc) = 1 if b = b(c) where b(c) is the Bayes-Nash bidding strategy in the game with

a continuum of types.

Theorem 1. Let c and c0 be any two elements of C and let b = b(c) and b0 = b(c0): Dene the real number r as

r =¾(b; c; BBN))

¾(b0; c; BBN))(13)

then log(r) · 2¸ maxfb; b0g(1 ¡ minfF (c); F (c0)g)N¡1:

Proof: By the denition of ¾ it follows that:

4 The equilibrium to the Bayes-Nash model is computed using the methods described by Bajari (2001). To compute the equilibriumin the logit model, we start with an initial guess that the bid functionB(0)(bjc) is the uniform distribution. We then recursively dene:

B(n)(bjc) = ® ¤ B(n¡1)(bjc) + (1 ¡ ®)¾(bi; ci; B(n¡1))

When we set ® = 0:95 and evaluated the above expression until sup jB(n)(bjc) ¡ B(n¡1)(bjc)j < " for a prespeciedtolerance of epsilon (usually " = 0:01): We found that this algorithm typically converged in 20 to 150 iterations.5 For smaller numbers of bidders, such as 3 or 4, the Logit Equilibrium in numerical simulations was found to be muchcloser to the Bayes-Nash equilibrium.

12

2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.50

2

4

6

8

10

12Figure 2: Bid Distribution in Bayes−Nash Equilibrium

Num

ber

of O

bser

vatio

ns

Value of Bid

4 4.5 5 5.5 6 6.5 7 7.5 80

5

10

15

20

25

30

35N

umbe

r of

Obs

erva

tions

Value of Bid

Figure 3: Bid Distribution in Logit Equilibrium

r =

µexp(¸¼(b; c; BBN)))Peb2B exp(¸¼(eb; ci; BBN))

¶¤

µexp(¸¼(b0; c; BBN)))Peb2B exp(¸¼(eb; ci; BBN))

¶¡1

(14)

By straightforward algebra, the above equation simplies to:

log(r)

¸= ¼(b; c; BBN) ¡ ¼(b0; c; BBN) (15)

By the denition of prot ¼ in equation (2) it follows that:

log(r) · ¸ ¢ ¡¼(b; c; BBN) + ¼(b0; c; BBN)

¢(16)

= ¸ ¢ (b ¡ c)(1 ¡ F (c))N¡1 + ¸ ¢ (b0 ¡ c)(1 ¡ F (c0))N¡1 (17)

· 2¸ maxfb; b0g(1 ¡ minfF (c); F (c0)g)N¡1 (18)

Q.E.D.

As N ! 1, (1 ¡ minfF (c); F (c0)g)N¡1 approaches zero (for c and c0 not at the left end of the support of F ), so

r ! 1. This implies that if the bidder makes errors in measuring her prot, a bidder with cost c will place near equal

weight on bids b and b0. This formalizes how the Bayes-Nash equilibrium is not robust if N is sufciently large.

3.2 Estimation With a Large Number of Bidders.

It is not uncommon for auction data sets to have more than 10 bidders. For instance, Laffont, Ossard and Vuong (1995)

estimate the number of bidders in their eggplant auctions to be 18. Porter (1995) nds that the number of bidders in

OCS wildcat tract auctions for the right to drill oil and gas is between 10 and 18 for 180 of their 2,510 auctions.

In structural estimation of auction models, researchers attempt to learn an agent’s private information by assuming

that an agent optimizes and from observing the empirical distribution of bids. In the Bayes-Nash equilibrium model,

the rst order condition can be rewritten as:

c = b ¡ 1 ¡ F (b)

f(b)(N ¡ 1)(19)

An implication of equation (19), rst exploited by Elyakime, Laffont, Loisel and Vuong (1994), is that if we can

estimate the empirical distribution of bids, then we can learn an agent’s private information ci from observing her bid

bi.

13

From the discussion above, it is clear that a large set of bids are approximately prot maximizing. If we use the

rst order condition (19) to recover an agent’s unobserved private information c from observing b and agents make

small mistakes as in the logit equilibrium model, our estimates of c will not be very reliable with a large number of

bidders.

3.3 Discussion.

The results above demonstrate that the distance between the Bayes-Nash equilibrium and the logit equilibrium of the

rst price auction can be large even for empirically plausible small amounts of error. The essential insight is that the

probability of winning for a given percentile of the cost distribution F (c) falls exponentially as the number of bidders

increases. This property will hold in many other auction models besides the rst price auction. In most auction

models, the equilibrium bid function is monotonic in an agent’s private information and all agents are symmetric.

Following arguments analogous to the above section, this implies that in any auction model having these properties,

the prot for a given percentile of the type distribution will fall exponentially in the number of bidders.

The Bayes-Nash equilibrium will fail to be robust to many other types of error prone decision making other than

the logit equilibrium. For instance, if we computed all of the " equilibria to our auction game, the set would become

large as N increased. This once again follows from the fact that the prot function becomes at as the number of

bidders become large.

Many other games have the property that the objective function is at near the maximum. In a median voter model

for instance, the probability that any given voter is pivotal is very small. Yet, the models assume that all voters act

as if they are pivotal. Recent experience in Florida suggests, many Nader voters preferred Gore for president but

did not act in accordance with the median voter theory because they believed that the probability that they would be

marginal was small. Ex ante that is correct, however, as it turns out ex post in the 2000 U.S. Presidential election they

were wrong. The problem of at objective functions near the maximum is also present in contribution games where

everyone decides on a donation level and the public project is implemented if the total contributions sum to at least

some xed amount.

As a practical matter, it is important to consider whether our policy recommendations or parameter estimates are

robust to empirically plausible amounts of error prone decision making. One way to do this is to run an experiment.

In designing the spectrum auctions and in electricity deregulation, experiments were used to simulate how new auction

rules would perform in the eld.

14

For many economic problems, however, conducting an experiment may not be feasible because the economist does

not have convenient access to a lab. An alternative procedure is to simulate a logit equilibrium model and compare

the results to the Bayes-Nash equilibrium model for a range of values of ¸. At a minimum, it is important to plot the

objective functions of the agents at the equilibrium values. Common sense would lead an economist to be skeptical

of any policy recommendation that hinged on agents nding the exact maximum to an objective function that is very

at as in Figure 1.

4 Submission Error Model.

In the previous section, we developed a model of bidding when rms make errors in measuring their expected prot.

In this section, we consider the consequences of a different type of error: rms occasionally submit an unintended bid.

Such errors occur in real world procurements due to clerical errors and simple lapses in attention. See Bartholomew

(1998), Chapter 11 for a discussion. In this section, we demonstrate that failure to account for errors in submitting

bids will typically result in an upward bias in estimated markups.

In the submission error model, with probability 1 ¡ " the rms bid according to their Bayes-Nash equilibrium

strategy. However, with probability " > 0 the bid is drawn uniformly from the bidder’s strategy space [c; c]. The

Bayes-Nash equilibrium is a limiting case of the submission error model when " = 0. It would be desirable for

structural parameter estimates to be robust to small, empirically plausible values of ".

We shall rst conduct a computational experiment where we estimate an auction model where a single outlier is

placed in the data set using the Bayesian techniques proposed by Bajari (1997). Only parameter values that give

all of the bids positive probability are in the support of the posterior distribution. As we shall demonstrate, placing

a single outlier into the data set will cause us to dramatically change our parameter estimates. Second, we shall

prove a theorem that demonstrates that as the number of observations generated from the submission error model

becomes sufciently large, the only way to rationalize the observed data is to assume that the cost distribution F (c) is

concentrated near c. This implies that any estimation procedure that gives positive probability to all of the observed

bids will tend in the limit to overestimate markups.

4.1 A Computational Experiment.

In this section, we conduct a computational experiment to explore the sensitivity of econometric estimates to outliers

generated by errors in submitting the bids. In our computational experiment, there are N = 4 rms who have a

15

truncated normal cost distributions with support on the interval [0; 8]. In our experiment, we generalize the model of

Section 2 and allow the distribution of costs to be rm specic. Let Fi(cijµi) denote the cumulative distribution for

the costs of rm i and let fi(cijµi) denote the probability density function. We shall let µi = (¹i; ¾i) denote the rm

specic parameters. The pdf fi(cijµi) can be written as:

fi(ci; µi) /(

1

(2¼¾2i )

12

exp³

¡ (ci¡¹i)2

2¾2i

´if 0 < ci < 8

0 otherwise(20)

The computational experiment will have two parts. We begin by computing the equilibrium bid function using the

methods described in Bajari (2001). We then simulate the model and generate 100 random bids for each rm. Let

t = 1; :::; 100 index each simulation of the model and letBIDi;t denote the randomly generated bid. We then assume

that the rst bid in our data set was “unintended”. Next, we estimate the model parameters using both the Bayes-Nash

equilibrium model and the logit equilibrium model.

4.2 Estimation Procedure.

In this subsection, we shall describe the econometric procedure used to estimate the model parameters assuming the

data was generated by the Bayes-Nash equilibrium model. As is well known, the asymptotics of maximum likelihood

are problematic in auction models because of the shifting support problem. To circumvent this problem, we use

the Bayesian procedure proposed by Bajari (1997) who applied the Hastings-Metropolis algorithm to simulate the

posterior distribution of model parameters.

Let Ái(bijµi) be the inverse of the bid function for rm i and let Á0i(bijzt; µ) be its derivative. Bajari (2001) describes

computational algorithms that can be used to efciently calculate these values. The probability density function for

rm i’s bid will be denoted by pi(bijµ) and by standard change of variable arguments must satisfy:

pi(bijµ) = fi(Ái(bijµ)jµ)Á0i(bijµ) (21)

Let t = 1; :::; 100 index all of the auctions, let BIDT = (BID1;1; :::; BID4;T ) be the vector of all bids submitted in

all auctions and let µ = (µ1; µ2; µ3; µ4). The likelihood function for BIDT can therefore be written as:

p(BIDTjµ) =100Yt=1

4Yi=1

pi(BIDi;tjµi) (22)

Let ¼(µ) be the econometrician’s prior beliefs about the model parameters. By Bayes’ Theorem, the posterior distri-

bution of model parameters, which we denote as p(µjBIDT); is proportional to the prior times the likelihood function,

16

that is,

p(µjBIDT) / ¼(µ) ¤ p(BIDTjµ) (23)

For simplicity, ¼(µ) will be a constant function on a compact support. If µ satises the following equalities, ¼(µ) is a

constant, otherwise it is zero:

¼(µ) =

½constant if 0 < ¹1;i < 10 and 0 < ¾ < 5

0 otherwise(24)

As a practical matter, the specication of the prior will have relatively little impact on the estimation procedure. The

magnitude of the prior is relatively small when compared to the likelihood in the posterior. Standard methods can be

used to examine the sensitivity of our results to changes in the prior, see for instance Geweke and Petrella (1998).

A large body of recent work in Bayesian statistics and econometrics has developed numerical methods for simu-

lating the posterior distribution of model parameters. Posterior simulators generate a sequence of S random vectors

µ1; µ2; µ3; :::µS from a Markov Chain with an invariant distribution equal to the posterior p(µjBIDT). The posterior

simulators can be used to estimate the posterior expected value of some function of the model parameters, g(µ): Under

weak regularity conditions stated in Geweke (1996), it can be shown that:

limS!1

PSi=1 g(µi)

S!

Zg(µ)p(µjBIDT) (25)

That is, the sample mean of g(µ) evaluated at µ1; µ2; µ3; :::µS can be used to consistently estimate the posterior ex-

pected value of g(µ):

This research will use a variant of the Hastings-Metropolis algorithm called a normal random walk described

in Tierney (1994) to generate the random sequence µ1; µ2; µ3; :::µS: The rst step in the algorithm is to nd a

rough approximation to the mode of the posterior p(µjBIDT) which we will refer to as µ0: Let H be a numerical

approximation to the Hessian of p(µjZT; BIDT) at µ0:

The algorithm starts with the rst element in the sequence as µ0 and subsequent draws are made according to the

following steps:

1. Given µn draw a candidate value eµ from a normal distribution with mean µn and variance-covariance matrixproportional to ¡H¡1:

2. Let

17

® = minf p(eµjBIDT)

p(µnjBIDT); 1g:

3.Set µn+1 = eµ with probability ®: Otherwise, let µn+1 = µn

4.Return to 1.

Simulating the posterior distribution when we assume that the data is generated from the logit equilibrium model

is done in a similar fashion. We compute the equilibrium to the model and generate a likelihood function for the

observed bids conditional on the parameter values. We use the prior distribution (24) when simulating the posterior

distribution in the logit equilibrium model. We place a dogmatic prior on ¸ and assume that it is equal to 25 with

probability one. This is equivalent to making the standard deviation of "ij equal to 0.05. We set ¸ equal to 25 in

order to compare the fully rational Bayes-Nash equilibrium model with a model of error prone decision making where

the variance of the errors is relatively small.

4.3 Results.

We begin by estimating our Bayes-Nash equilibrium model on fake data generated by simulating the Bayes-Nash

equilibrium 100 times. We simulate the posterior distribution of the model parameters using the Hastings-Metropolis

algorithm. We set the number of simulation draws S equal to 10,000. To understand the effect of an outlier generated

by the error submission model on our results, we letBID1;1 = 2:4: The lowest bid that has positive probability under

the true parameters is approximately 2.6.

Table 2: Posterior Simulations For Bayes-Nash With An OutlierParameter True Value Posterior Mean Posterior Std. Dev.Firm 1’s Mean 4.0 3.6908 0.1904Firm 2’s Mean 4.3 3.8550 0.1992Firm 3’s Mean 4.5 4.0600 0.1447Firm 4’s Mean 4.7 4.1715 0.2130Std. Deviation 1.5 2.1530 0.1069

Table 2 shows how the inclusion of a single outlier in the data set has biased all of the estimates of rm specic

means downward and the value of the standard deviation upwards by a consider amount. The bid that was generated

by a mistake, BID1;1 = 2:4 does not have positive probability when the likelihood function is evaluated at the true

parameters. In order give the bid BID1;1 = 2:4 positive probability, the model produces a downward bias in the

estimates of the rm’s costs. This is consistent with economic intuition: when the costs of all of the rms are lower,

the lowest bid submitted in equilibrium will also fall.

18

The second computational experiment that we perform is to estimate the logit equilibrium model when we place

an outlier in the data set. The computational experiment with the logit equilibrium model is the same as above. We

generate data from the Bayes-Nash equilibrium model and we place an outlier in the data set by lettingBID1;1 = 2:2

(an even lower bid than in the example above). We then estimate the parameter values for the logit equilibrium model

using the Hastings-Metropolis algorithm.

Table 3: Posterior Simulations For Logit Equilibrium Model With An OutlierParameter True Value Posterior Mean Posterior Std. Dev.Firm 1’s Mean 4.0 3.9582 0.1588Firm 2’s Mean 4.3 4.1064 0.1623Firm 3’s Mean 4.5 4.4328 0.1759Firm 4’s Mean 4.7 4.7358 0.1884Std. Deviation 1.5 1.3373 0.1082

The logit equilibrium model does not exhibit the sensitivity to outliers that Bayes-Nash model does. The logit equilib-

rium model gives all actions positive probability and predicts that “very low” bids should sometimes be observed in

the data.

In practice, there are two other approaches for dealing with outliers in the data. Therst is trimming the distribution

of bids as in Guerre, Perrigne and Vuong (2000). The second is generalizing the Bayes-Nash equilibrium model to

allow for bids to be generated by an error prone process. In Bajari (2001), we allow for bids to be generated from a

purely random distribution with some small but positive probability.

4.4 Why Structurally Estimated Auction Models Tend to Generate Markups That Are TooLarge.

In this section, we shall prove a theorem that demonstrates if the true data generating process is the error submission

model, estimates of markups consistent with the Bayes-Nash equilibrium model will tend to innity in the limit. For

the purposes of proving this theorem, we consider the following thought experiment. Assume once again that all

rms are ex ante identical, that is, fi(cijµ) = f (cijµ) and N rms bid in the procurement auction. With probability

1 ¡ ", rms bid according to their Bayes-Nash equilibrium strategy. However, with probability " > 0 due to errors in

submission the bid is drawn uniformly from the interval [c + ±; c] where ± > 0 is a small, positive constant: There are

t = 1; :::; T replications of this experiment and the randomly drawn bids are labeledBIDi;t.

For simplicity, it is assumed that the f (cijµ) are distributed truncated normal with mean ¹ and standard deviation

¾ (although this is not critical for our results). Theorem 2 states that as ± ! 0 and T ! 1; the only parameter

values for which the likelihood function generated by the Bayes-Nash equilibrium model has positive probability is

19

¹ = c and ¾ = 0. That is, the econometrician infers from the data that the rms have the lowest possible cost with

probability near one.

Theorem 2. Let S(T ) be the set of parameter values that give positive probability to all of the bids observed inauctions t = 1; :::; T . Then with probability one

lim±!0

sup limT!1

sup Lf(¹; ¾) 2 S(T ) \ ¹ > c \ ¾ > ° > 0g = 0 (26)

where Lf:g is the Lebesgue measure of the set argument.Proof. Suppose that equation 7.6 is false. By letting T tend to innity and ± tend to zero then with probability one it

will be possible to nd a bid arbitrarily close to c that has positive probability. In equilibrium, rms never bid less than

cost, therefore, it must be the case that rms with cost arbitrarily close to c must bid arbitrarily close to c. However,

this is only possible if in equilibrium rms with cost c anticipate near zero prot. This can only be true if, in the limit,

the probability of winning with the lowest possible cost draw c is zero. This in turn is only possible if all rms in the

limit have costs of c with probability one.

Theorem 2 shows that if the bidder makes mistakes as we have specied, then we will estimate markups to be

innite (if c = 0). The problem occurs because the only way that bids generated by “mistakes” which are less than

b(c) can receive positive probability is to put additional mass on cost draws near c than occurs under the “true” model

parameters.

4.5 Discussion.

Bids that appear to violate the support conditions of structural auction models are a problem in empirical applications.

In Bajari and Ye (2001), using a procedure similar to Guerre, Perrigne and Vuong (2000), we nd that 3 percent of

the bids in our data set of bidding by construction rms cannot be rationalized by positive cost draws. In Hortacsu

(2001), we nd that a small, but non-trivial fraction of the bids in a T-Bill auction are only rationalized by demands

that are orders of magnitude larger than prices in the secondary market.

In addition to errors in submitting bids, unobserved, auction specic heterogeneity can generate bids that appear

to be inconsistent with support conditions. Alternatively, simple errors in coding or recoding the data will cause us to

sometimes insert a bid that is very low into the data set. Finally, if the data is generated from a model such as the logit

equilibrium model, then we will see very low bids with positive probability.

As a practical matter, several steps can be taken to examine the sensitivity of parameter estimates to very low bids.

20

The rst is to use standard techniques to detect outliers, such as plotting the data and searching for observations with

very large residuals when bids are regressed on cost controls. The second is to trim the data, as suggested by the

non-parametric estimation procedures employed by Guerre, Perrigne and Vuong (2000). It is worth mentioning that

the results of Theorem 2 will clearly hold in the limit even if a trimming procedure is performed.

In our experience, it can be hard to discover in empirical data sets which bids are “too low”. An alternative

strategy is to develop a model where bids are submitted with error with some small positive probability. In Bajari

(2001), we estimate an auction model where the likelihood function is a mixture of the Bayes-Nash equilibrium and a

model where bids are generated uniformly out of a distribution with full support with a small prior probability. As the

analysis of this section suggests, the parameter estimates change substantially compared to just using the Bayes-Nash

equilibrium model.

First price auction models are not the only models where the support of the equilibrium actions depends on the pa-

rameter values. Search and stochastic production frontier have similar properties and will display a similar sensitivity

to outliers. See Hong (1997) for a discussion.

5 An Experimental Case Study

In the previous sections, we examined how slight errors in decision making by bidders can confound empirical pre-

dictions based on an equilibrium concept that does not allow such errors. In this section we supplement our ndings

using actual bidding data. In particular, we will show that small errors in decision-making can lead to relatively large

deviations from equilibrium behavior, and thus can confound empirical estimates of structural parameters that depend

on equilibrium assumptions.

The data set we use was provided to us by John Kagel, and contains a series of IPV rst-price auction experiments

conducted by Dyer, Kagel, and Levin (1989). In these experiments, bidders were assigned private valuations drawn

i.i.d. from a uniform distribution on [$0; $30], and were paid their prots at the end of the experimental run. A novel

feature of this experiment was that the bidders were faced with two possibilities as to how many competitors they

were going to face: with probability 1=2, the market would have N = 3, and with probability 1=2, N = 6. However,

instead of a single bid, the bidders were asked to submit two different bids: one bid that would become effective only

if N = 3, another bid that would become effective only if N = 6. That is, if bidder A, with valuation $25, bid $20

contingent on there being 3 bidders, and $18 on the contingency that there would be 6 bidders, and if it turned out

that the coin ip favored N = 3, bidder A would pay $20 upon winning. After each auction, bids and corresponding

21

private values were posted on a blackboard for the bidders to see.6

Since the type of auction used in the experiments is a rst-price auction, as opposed to a lowest-bid procurement

auction, we will have to modify the notation slightly; however, the discussion in Section 2.1 applies in its entirety to the

game analyzed here. In particular, since the game being played is a rst-price IPV auction with uniformly distributed

valuations, the equilibrium bidding strategies in theN person game can be computed to be:7

bN(v) =N ¡ 1

Nv

5.1 Deviation From Equilibrium Behavior.

One of the main predictions of Section 3 was that if the payoff function was sufciently at, then a large set of bids

is consistent with approximate maximization on the part of the bidders. In this section, we will quantify deviations

from equilibrium behavior in terms of how far were players from their maximal payoffs and how far were the actual

bids away from the equilibrium bids.

To get at the expected payoff, we have to calculate the probability that the observed bid will win the auction. One

could either use the distribution predicted by Nash equilibrium, U [0; 30 N¡1N ], or use the observed distribution of bids

in the data set. We use the latter; we t a uniform distribution to the bid distribution we observe to get F (b), the cdf of

the bid distribution, and calculate F N¡1(b), the probability that a bid of b will win the auction.

We nd that the average difference between the expected payoff of the bidder from his observed action and his

Nash equilibrium action is $0.36 ($0.45) for N = 3, and $0.05 ($0.10) for N = 6. This means that deviations from

equilibrium actions have much smaller consequences in terms of payoffs – or, to put it another way, small errors in

assessing payoffs can lead to large deviations in observed actions. Dyer, Kagel, and Levin (1989) report that the bids

they observe are in general in accordance with the overbidding phenomenon reported extensively in the experimental

literature on rst-price auctions.8 In fact, for the 3 player game, the mean (absolute) deviation from Nash equilibrium

bids was $2.55 with a standard deviation of $ 1.64. In the 6 player game, the mean deviation is $0.94 with a standard

deviation of $0.78.

6 Another novel feature of the experiment which we do not utilize here is that after a few rounds of this contingent bidding, thebidders were asked to submit an additional, non-contingent bid after submitting their contingent bids. That is, if bidder Asubmitted $17 as his non-contingent bid, this would be his payment if he won the auction, regardless of whetherN = 3 orN = 6. The payoff ofbidder A was calculated separately for the contingent and non-contingent bidding procedures – i.e. the payoff of bidder Awith the above contingent and noncontingent bids would be $5 + $8 = $13 if turned out thatN = 3, and bidder A won theN = 3 auction.7 This equilibrium was rst calculated by Vickrey (1961).8 See, for example, Cox, Smith, and Walker (1988), Harrison (1989).

22

Table 4: Deviation From Nash Behavior.N = 3 N = 6

Observed bid - Nash bid $2.55 (1.64) $0.94 (0.78)Exp. payoff from bid - Exp. Nash payoff $0.36 (0.45) $0.05 (0.10)

Onemight wonder whether the Nash equilibrium actions are indeed the best-responses of the bidders when reacting

to a distribution of bids generated by “error-prone” agents like themselves. To assess this, we calculated the best-

response of the bidders given the empirical distribution estimated above. We found that Nash equilibrium predictions

are almost identical to the “empirical” best responses.

As we pointed out in Section 3, the atness of the payoff functions can explain observed deviations from equilib-

rium predictions. Figure 4 plots the expected foregone earnings of a bidder from deviating from the Nash equilibrium

bid for three different valuations, v = 3; 12;and 24 for the three bidder case (the corresponding Nash equilibrium bids

are b¤ = 2; 8, and 16 respectively). As can be observed from this gure, the expected cost of deviating from the Nash

equilibrium strategy is quite small for even large deviations: for a bidder with v = 24, the cost of over-bidding by $4

is about $1. Especially for a bidder with small value, say v = 3, there is almost negligible cost to submitting any bid

that is below the value draw.

Figure 4: Cost of Deviating From Nash Behavior.

-6 -4 -2 0 2 4 60

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Bid - Nash bid

Nas

h Pa

yoff

- Bid

Pay

off

v=3 v=12v=24

23

One intriguing feature of the bidding data is that the deviations from Nash equilibrium seem to be smaller in the

N = 6 case. In the following gure, we plot the foregone payoff for deviations from Nash equilibrium for a bidder

with v = 24. From this gure, we see that it is more costly to bid above the Nash strategy when N = 6 than when

N = 3; on the other hand, it is more costly to bid below the Nash strategy whenN = 6. If costs of deviation regulate

errors in decision-making, this gure suggests that we should observe less overbidding when N = 6 as opposed to

N = 3, but more underbidding. However, what we see in the data is that the degree of underbidding is also less for

theN = 6 case, which can not be accounted for by a straightforward “foregone payoff” argument.9

Figure 5: Cost of Deviating From Nash Behavior, 6 Bidder Case Verus 3 Bidder Case.

-6 -4 -2 0 2 4 60

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Bid-Nash bid

Nas

h pa

yoff

- Bid

Pay

off

N=3, v=24N=6, v=24

5.2 Structural Estimation of Bayes-Nash Equilibrium Model.

Next, we assess how well a structural estimation procedure performs that attempts to estimate bidder’s private infor-

mation under the assumption that bidders play a Bayes-Nash equilibrium. As we have seen in the previous section,

bidders are close to maximizing prots, as the Bayes-Nash equilibrium assumes, however, there is a sizeable distance

between their equilibrium bids and their actual bids.

9 Nevertheless, Dyer, Kagel and Levin (1989) argue that adding risk aversion to the equilibrium model might explain this anomaly.

24

We use the estimator in equation (19) in Section 2 to “back-out” the valuations corresponding to each bid. In Figure

6, we plot the point estimates of the valuations against the true valuations of the bidders. The estimation procedure

we apply is similar to the non-parametric approach advocated by Guerre, Perrigne and Vuong (2000).

Figure 6: Estimated Versus True Valuations.

0 5 10 15 20 25 300

10

20

30

40

True valuation

Estim

ated

val

uatio

n

N=3

0 5 10 15 20 25 300

5

10

15

20

25

30

35N=6

Estim

ated

val

uatio

n

True valuation

We see that the estimated valuations of the bidders overstate their true valuations (they lie above the 45 degree

line), a bias that naturally arises from the overbidding phenomenon. The amount of upward bias is more pronounced

for bidders with higher valuations, as presumably they have more room to make mistakes than bidders with lower

valuations.

In Table 6, we report the actual and estimated average bid-shading undertaken by the bidders. We see that in both

cases, the valuation estimates given by equation (X) lead to bid-shading (or in the procurement context, mark-up)

estimates that are too high. The bias is actually quite stark in the N = 3 case, where an average 20% of shading is

estimated as 40%.

25

Table 6: Estimated and Actual Bid Shading.N = 3 N = 6

Actual % bid shading 0.20 (0.14) 0.14 (0.13)Est. % bid shading 0.40 (0.07) 0.17 (0.02)

From Table 6 we conclude that, even though bidders are close to maximizing expected prot, the empirical esti-

mates generated by a structural model of bidding are quite poor in the N=3 case. The markup is over-estimated by a

factor of two. The estimated markups are, however, are one average much closer to the true markups for the N=6 case.

From this exercise we learn that just because bidders are approximately maximizing, we need to carefully examine the

robustness of the estimates and policy recommendations based on a structural model to error prone decision making.

The estimates in this section indicate that sometimes, both the estimates and policy recommendations based upon them

may be misleading.

5.3 Estimation of the Logit EquilibriumModel.

In this section, we will attempt to assess the performance of the logit equilibrium model. Modifying equation (8) to

the “highest-bid” vs. least cost case, we see that the probability of seeing a bidder with valuation v bid b is given by:

¾(bjv) =exp(¸¼(b; v))P

b02B exp(¸¼(b0; v))(27)

Since we can estimate the expected payoff term ¼(b; v), and since we also observe the empirical distribution of

bids in the data set, the only free parameter in equation (27) is ¸, the precision of the i.i.d. decision error. Equation (8)

allows us to form a likelihood function for the observed bids in the data set, which we can maximize to estimate ¸.10

The results of this estimation are reported in Table 7. According to these estimates, the standard deviation of the logit

error term in the 3-bidder case is about $0.62, and in the 6-player case it is about $0.08. Observe that these gures are

similar in magnitude to the payoff-deviations reported in Table 4 above.

Table 7: ML estimates of ¸ (standard errors in parentheses)N = 3 N = 6

¸ 1.17 (0.10) 9.41 (0.78)log L -2126 -1960

Using the estimates of ¸, we can compare the bid distribution predicted by logit equilibrium to the actual distribu-

tion of bids. To do this, we xed three valuations, v = 22; 24 and 26, and plotted ¾(bjv) given by the logit equation.

We then pooled the bids submitted by bidders with valuations between 22 and 26 (which yielded 62 bids), and made

a histogram of the bids. As we can see from Figure 7, in the N = 3 case, the logit distribution is much atter than10 We discretized the action space of each bidder into a set B of 300 discrete points from [$0; $30], spaced by 10 cent increments.

26

the actual distribution, and predicts a lot more probability mass on the lower tail. In theN = 6 case, the observed bid

distribution matches the prediction of the logit equilibrium model for v = 6 quite well; however, since we are pooling

bids, once again we see that the logit equilibrium model predicts a distribution that is centered to the left of the actual

distribution.

Figure 7: Logit Bid Distribution Versus Actual Bid Distribution.

0 5 10 15 20 25 300

0.02

0.04

0.06

0.08

0.1

0.12

Bid

Bid

prob

abilit

y

N=3

v=22v=24v=26

0 5 10 15 20 25 300

0.02

0.04

0.06

0.08

0.1

0.12N=6

Bid

Bid

prob

abilit

y

v=22v=24v=26

5.4 Discussion

In the preceding section, we found that experimental data does not conform entirely with the predictions of the Bayes-

Nash equilibrium for the risk-neutral IPV rst-price auction game, and that it is quite plausible to think that deviations

from predicted equilibrium behavior can be explained by errors in bidders’ decision-making. In particular, we found

that the magnitude of the error in decisions is several times greater than the magnitude of the error in optimization.

This amplication of the decision error is consistent with the results in Section 3, and as we discussed in Section

4, this fact can have important implications for structural estimation and policy decisions based on such estimation

exercises.

However, our analysis also revealed that the logit equilibrium model, although a formal model of error-prone

27

decision-making, has important deciencies when it is taken to the data. Some of the main deciencies are:

1. Although the logit equilibrium model assigns positive probability to every action, we do not see bidders playingdominated strategies (nobody bids above their value). This, unfortunately, violates the full support assumptionregarding the distribution of the error term.

2. The full support assumption also leads to the following pathological outcome as the number of actions, B, thebidder has to choose from increases (which would be the case if we discretized a continuous game very nely):the payoff function of a bidder becomes with probability one everywhere discontinuous in his actions.11 Thisdiscontinuity is an undesirable property especially in the case where the action space is continuous and the logitequilibrium model is applied to a discretization of the continuous game.

3. The estimated magnitude of ¸, variance of the logit error term, changes with the number of bidders in the game.In the basic logit equilibrium model, there is no accounting for how different types of games lead to differenttypes of errors. Even if we let ¸ be a (monotonic) function of the expected payoff (that is, the bidder tries harderto resolve his payoff uncertainty if there is more at stake), this would predict that decision errors should be largerin theN = 6 game. We found that this is not the case. However, we nd that instead of bidding anything, biddersessentially bid their valuation given that they know they have a very low probability of winning. Therefore, thepredictions of the equilibrium model actually become more consistent with the theory as the number of biddersincreases. This is the opposite of what the logit equilibrium model suggests.

We believe that our analysis suggests that while error prone behavior is important in understanding bidding by

experimental subjects, the set of errors made by experimental subjects, while consistent with some but not all of the

predictions of the logit equilibrium model. One approach to this problem is weakening the logit assumption, as has

been done by McKelvey and Palfrey (1995). A second important direction for future research is to develop richer

models of error prone behavior.

6 Conclusion.

This research has studied the implications of error prone decision making in a simple model of the rst price auction

with independent private values. We studied two types of errors. First, the logit equilibrium model where rms

11

(a) To see why this comes about, think about the difference between the bidders chosen bid, bk; and thebid that is immediately larger, bk+1 = bk + ¢, where¢ is the discretization increment. Let "k be theQRE error term associated with bk, and ¼(bk; "k) be the payoff associated with bk and "k (which isthe maximum payoff realization among the alternatives). Also, let bM be the bid associated with themaximum QRE error draw among the B draws "1; :::; "B. Then:

E¼(bk; "k) ¡ E¼(bk+1; "k+1) ¸ E¼(bM ; "M) ¡ E¼(bk+1; "k+1)

since bk; "k is the payoff maximizing pair. But E¼(bM ; "M) ¡ E¼(bk+1; "k+1) goes to innity asB ! 1, since "M is the maximum among B random draws and " has full support.

28

measure their expected prot with an i.i.d. error term and second, the submission error model where occasionally a

bidder incorrectly submits her bid.

When the number of bidders is sufciently large, the distribution of bids in the logit equilibrium model can differ

greatly from the distribution of bids in the Bayes-Nash equilibriummodel even if the variance in making errors is small.

This difference comes from the fact that the prot function becomes “atter” as the number of bidders increases and

that a large number of bids are consistent with maximizing prot with only a small error. The sensitivity of the

Bayes-Nash equilibrium model to small errors in maximizing prot calls into question parameter estimates based on

the Bayes-Nash model and policy recommendations based on the model when the prot function is at due to a large

number of bidders.

If rms implement their bidding strategies imperfectly, as in the submission error model, structural estimation of

auction models tend to overestimate bidders’ markup. This is because in a Bayes-Nash equilibrium, not all bids can

occur with positive probability. In equilibrium, bids that are unusually small can only occur with positive probability

when the cost distribution has sufcient mass on the left tail. The presence of even a single, unusually small bid,

generated for instance when a rm submits an unintended bid, will force the econometrician to underestimate rms’

costs if all bids are to be given positive probability. Therefore, rms’ markups will be overstated.

The experimental data we examine qualitatively supports our theoretical predictions. First, while bidders maximize

prots to within a few cents, the deviations of the actual bids from the equilibrium bids is many times larger. Second,

we nd that the average markup in one of our experiments is estimated to be 40 percent while the actual markup is

20 percent. The markups tend to be overestimated as we discussed in Section 4. However, the overestimate in the

markup is also consistent with the overbidding phenomenon.

While the experimental data demonstrated that the Bayes-Nash equilibrium has deciencies, we also found that

the logit equilibrium model was decient in explaining the data along several dimensions. First, the logit equilibrium

model predicts that some players should bid over their valuation. This was not seen in the data. Second, the prot

function in the logit equilibrium model will become discontinuous as the strategy space is discretized. Third, the

estimate of the standard deviation of the error term increased by a factor of 8 from the 6 bidder auction to the 3 bidder

auction. There is no underlying theory of why this should occur.

We conclude that while bidders are close to maximizing prots, the parameter estimates in structural models must

be examined critically to determine how robust they are to at least small errors in decision making. Also, policy

recommendations based on such models should be subjected to the same robustness test, since there is often a large set

29

of outcomes that are consistent with approximate maximization. Finally, while models of error pone decision making

such as the logit equilibrium model do capture important qualitative features of the data, there is more work to be done

in developing a theory of errors in auction models and in games more generally. Our results suggest that developing

a richer theory of error prone decision making is crucial in getting reliable estimates of structural parameters and in

making policy recommendations based on equilibrium auction models.

30

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