Discussion Paper No. 747

AUCTIONS WITH
ENDOGENOUS PRICE CEILING:
THEORETICAL AND EXPERIMENTAL
RESULTS
Róbert F. Veszteg, Kenju Akai,
Tatsuyoshi Saijo and Shigehiro Serizawa

July 2009

The Institute of Social and Economic Research
Osaka University
6-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan

Auctions with endogenous price ceiling:
Theoretical and experimental results∗Róbert F. Veszteg†, Kenju Akai‡, Tatsuyoshi Saijo§ and Shigehiro Serizawa¶
July 6, 2009

Abstract
This paper analyzes an auction mechanism that excludes overoptimistic bidders inspiredby the rules of the procurement auctions adopted by several Japanese local governments.
Our theoretical and experimental results suggest that the endogenous exclusion rule reduces
the probability of suffering a monetary loss induced by winning the auction, and also mitigates the problem of the winner’s curse in the laboratory. However, this protection comesat the price of a lower revenue for the seller.
Keywords: common-value auctions, experiments, winner’s curse
JEL Classification Numbers: C91, D44, D82∗

Veszteg gratefully acknowledges the financial support from project SEJ2006-10087 of the Spanish Ministryof Education and Sciences, and from the Plan de Investigación de la Universidad de Navarra (PIUNA).
†
Universidad Carlos III de Madrid, Departamento de Economı́a, c/Madrid 126, 28903 Getafe (Madrid), Spain;
e-mail: rveszteg@gmail.com‡
Japan Society for the Promotion of Science, 1-6 Chiyoda-Ku, Tokyo 1028471, Japan; e-mail: kakai@iser.osaka-u.ac.jp§
Research Institute for Sustainability Science, Osaka University, 6-1 Mihogaoka, Ibaraki, Osaka 5670047,
Japan; and CASSEL, University of California at Los Angeles, Los Angeles, CA90095, United States of America;
e-mail: saijo@iser.osaka-u.ac.jp¶
Institute of Social and Economic Research, Osaka University, 6-1 Mihogaoka, Ibaraki, Osaka 5670047, Japan;
e-mail: serizawa@iser.osaka-u.ac.jp1

1

Introduction

Several Japanese local governments have adopted a special auction format for their procurement
procedures. There has been no official justification for the change, nevertheless we believe that
local governments aimed at increasing the success ratio of the public projects assigned withthese auctions by reducing the probability of a monetary loss by winning the auction and thewinner’s curse. The winner’s curse is a severe empirical problem that has received attention bothfrom theoretical and experimental economists. In case of procurement auctions, the winner’s
curse may arise due to an overly optimistic estimation of the total cost of carrying out the
project and/or to the fierce competition among bidders. Although the winner’s curse representsa systematic overestimation of the true value of the auctioned good, and therefore does not occur
in a theoretical equilibrium, it represents a serious empirical problem.1
We believe that negative payoffs that do appear, although average out, in the theoreticalequilibrium also endanger the fulfillment of public contracts. Even rational bidders who win theauction will often find that the price they have paid is higher than the real value of the project.
As a consequence, paralyzed construction works and unexploited natural resources—among
other problems—require the government to repeat the costly auction procedure. The lack of orlate production may also reduce consumers surplus, therefore represents an efficiency loss forthe society. Ganuza (2007) presents the empirical impact of the negative winning payoffs bylisting European and US horror stories about procurement projects with long delays and hugecost overruns.2
1

The number of cases in which companies claim they fell prey of the winner’s curse is very large and present
a large variety as for the market it occurred in. The early examples of oil companies appear in Capen, Clapp andCampbell (1971). There has been systematic overpayments in book publication rights auctions, on the market ofprofessional sport players, also in numerous situation where the state sold out telecommunication rights.
2
In his model, it is the sponsor, i.e. the seller’s equilibrium strategy that makes these cost overruns likely due

2The auction rule adopted by the Japanese local governments excludes some of the most optimistic bidders from the competition, i.e. those whose bid is among the highest price-bids or
the lowest cost-bids. The exclusion is based on an endogenous price ceiling that is computed as
the average of the most optimistic bids that is often corrected with a fixed scalar. For example,
Nagano prefecture excludes those who send a cost-bid lower than the 80% of the five lowest
cost-bids. With unchanged bidding behavior this measure looks promising as a means for reducing the probability of a financial loss. Nevertheless if participants are rational, we expect
them to update their bidding behavior and adapt to the new rules. The question whether the
new auction format is able to decrease the probability of negative payoffs is non-trivial. Basedon numerical results that tackle the otherwise intractable mathematical problem, we show that
excluding the overoptimistic can indeed reduce the probability of suffering a loss by winning
the auction. However, this positive effect comes at the price of a smaller revenue for the seller.3
The idea of eliminating the overoptimistic is not a new one even it has not received enoughattention in the academic literature. The European Commission, for example, identifies the socalled abnormally low tenders that “in the light of the client’s preliminary estimate and of all
the tenders submitted, it seems to be abnormally low by not providing a margin for a normal
level of profit”.4 The Commission points out the potential risk behind these tenders and calls fortheir prevention, detection and elimination. Calveras et al. (2004) studies these abnormally low
tenders from a theoretical point of view and concentrates on surety bonds as a possible solution.
to an underinvestment in design specifications. In ours, it is due to a similar uncertainty over the real cost of the
project.
3
Some authors, e.g. Ishii (2006) who uses empirical data, suggest that the Japanese procurement auctions sufferfrom collusion and bidding rings. Although it is an important question, with our one-stage common-value model
we cannot address whether the specific auction rules have any effect on the possibility of collusion. Our paper
focuses on how optimal bidding strategies and revenues change if the auction rules exclude some of the highest
bids (and bidders) from the competition.
4
Refer to http://ec.europa.eu/enterprise/construction/alo/altfin.htm for more details.

3

In this paper, we take a closer look at an other tool to cope with the problem that is based onlarge price deviations, i.e. when a submitted bid is considered as outlier when compared to someprice (bid) average. Cox et al. (1994) consider a similar problem of cost overruns in procurement
contracting with a model that includes both adverse selection and moral hazard and uses firstand second-price auctions to award contracts. Rather than studying the auction schemes, they
compare fixed-price and cost-sharing contracts and find that although cost-sharing reduces theseller’s expenses, it is inefficient as it induces cost overruns, i.e. negative payoffs. Abbink et al.
(2006) study experimentally the so-called Spanish auction format used by the Bank of Spain tosell government bonds. The speciality of that auction is that for winning bids above the averagewinning bid, bidders are charged the average winning bid. They compare the Spanish auction
in a multi-unit common-value environment to the discriminatory and the uniform auctions, andfind that the Spanish auction, along with the uniform auction, raise higher revenue. Note that,
although it is similar in its spirit to the Japanese format, the Spanish auction does not exclude
overly optimistic bidders, it reduces the price they have to pay when winning the auction.
Our benchmark model is a first-price sealed-bid auction (FPSB). We compare its empiricalproperties to an auction with endogenous price ceiling (EPC) that mimics the procurement
auction applied by the Japanese local governments. In both cases participants are required tosimultaneously place a single bid for an object whose value is fixed, but unknown. Participantsreceive noisy signals about this value. In the first-price sealed-bid auction, the bidder with thehighest bid receives the object in exchange for which she is required to pay her bid. In theauction with endogenous price ceiling, bids above the average of the highest three bids are
excluded, and the winner is the bidder with the highest bid among the remaining bids.
In addition to the theoretical results, we also analyze data from laboratory experiments4where we implemented four common-value auctions: two first-price sealed-bid auctions andtwo endogenous price-ceiling auctions with 5 and 10 participants.5 Our data confirm the theoretical results that predict a lower probability of suffering a loss by winning the endogenous
price-ceiling auction and a smaller revenue. As it is commonly observed in the experimentallaboratory, subjects often make systematic mistakes and fall prey of the winner’s curse.6 Our
treatments suggest that by introducing an endogenous price-ceiling the seller can also reduce
the problem (probability) of the winner’s curse.7
The article is organized as follows. Section 2 presents the theoretical results, while sections
3 and 4 the experimental ones. Section 5 concludes. The main theoretical results are statedin numbered predictions, while the experimental ones as numbered observations in the text.
Tables appear in the text, while figures are in the appendix.

2

Theoretical results

The theoretical results detailed by Milgrom and Weber (1982) and Wilson (1977) offer a generalsolution to the bidding problem in common-value auctions (without price ceilings). Nevertheless, numerical examples with computed equilibrium strategies are rare due to the complexity of
the calculations involved. In what follows, we present theoretical and experimental results for5

Apart from the our goal to study the effect of the endogenous price ceiling on the winner’s curse, the use of
the experimental method is due to the mathematical complexity of the theoretical model whose solution presented
in this paper is based on approximation.
6
While rational bidders apply a sufficiently high discount when observing the private signal and never sufferof this curse, it is a commonly observed problem in the experimental laboratory. Check for example Dyer et al.
(1989) and Kagel and Levin (1986).
7
The experimental literature has studied price ceilings (and floors) in a competitive environment induced by
the double auction, e.g. refer to Isaac and Plott (1981), and Smith and Williams (1981). Differently from our case,they typically focus on non-binding price controls and find that they affect the competitive equilibrium in the labeven if in theory they should not. Gode and Sunder (2004) offer a simple dynamic model with “zero-intelligence”
traders to explain the discrepancy.

5the endogenous price-ceiling auction in a family of situations that are frequently implemented
and studied in the experimental laboratory. Readers who are not interested in the technical details of obtaining the theoretical results may want to skip the next two subsections and follow
our discussion from subsection 2.3.
We follow the common-value paradigm and assume that the value of the auctioned object,v, is not observable by bidders. In our information structure, it is assumed to be drawn from theuniform distribution over the [0; 1] interval, and the signals, s, privately received by participants
are independently drawn from the uniform distribution over [0; 2v]. The corresponding density
function are g for the value of the object and f for the signal. This way, signals are unbiased,since their expected value coincides with the true value of the object, i.e. E(s|v) = v. However,2

their variance increases with the value of the object, since V ar(s|v) = v3 , i.e. they are lessinformative for larger values.8 These pieces of information, except for the value of the object,v, are assumed to be common knowledge in the auction as is the number of bidders, n.

2.1

Bidding behavior

In order to describe the formal model, we first introduce some additional pieces of notation
following Wilson (1977). The equilibrium bidding function in the symmetric Bayes-Nash equilibrium of the auction with n bidders is denoted by bn (s), while its inverse by σn (b).9 Given
the true value of the object, v, let Qn (s|v) be the conditional probability of winning the auction
with signal s and by bidding bn (s). The support of Qn (s|v) is denoted by
8

P

S(v), and it is theThis is a special case ( = v) of the general [v − ; v + ] specification. The literature, e.g. the seminal work2
by Kagel and Levin (1986), often chooses a fixed value for  that implies a constant precision, V ar(s|v) = 3 . We
have explored both information structures and haven’t found a qualitative difference. This is why we only present
here the case that we believe to lie closer to reality.
9
Since we are looking for a symmetric equilibrium, in which all bidders use the same bidding function, we do
not have to include a subindex for bidders in our notation.

6[0; 2v] interval for our information structure.
Bidders are assumed to be risk-neutral and to maximize their expected payoffs after receiving the signal.10 If a bidder is using the bidding function b∗ (s) while everybody else is bidding
according to bn (s), her ex-ante expected utility can be written formally as

Z 1Z0[v − b∗ (s)] · Qn (σn (b∗ (s))|v) · f (s|v) · g(v)dsdv,

P

(1)

S(v)

1
where in our uniform model f (s|v) = 2v
whenever s ∈ [0; 2v], and g(v) = 1 if v ∈ [0; 1].

Otherwise, both functions are equal to zero. The first term of the integrand is the net surplus
that the winner of the auction enjoys (the value of the object minus her bid), while the others
represent the ex-ante probability of winning the auction.11 The interim expected utility, i.e. theexpected utility for an observed and therefore fixed signal s isZP[v − b∗ (s)] · Qn (σn (b∗ (s))|v) · g(v|s)dv,

(2)

S(v)

where Bayes’ rule gives g(v|s) = RPf (s|v)g(v)S(v) f (s|w)g(w)dw. Bidders are assumed to maximize thisfunction by choosing the bidding function b∗ (s). The first order condition of the bidders’ interim
10

In the symmetric theoretical Bayes-Nash equilibrium of any auction, bidders are assumed to maximize their
expected utility by using the same bidding strategy. The solution of this maximization problem, i.e. the equilibrium
bidding function, is a fixed point, since given that all the others’ are using that particular bidding function, the best
an expected-profit maximizer agent can do is to opt for the same bidding function.
11
Qn (s|v) is the conditional probability of winning the auction with signal s, while f (s|v) is the conditional
probability of receiving a signal s if the value of the object is v. Finally, g(v) represents the probability of the valueof the object being equal to v.

7expected utility maximization problem yields the following equation:

Z

∗

P

[v − b (s)] · Q0n (σn (b∗ (s))|v) · σn0 (b∗ (s)) · g(v|s)dv =
S(v)

Z
P

Qn (σn (b∗ (s))|v) · g(v|s)dv.
S(v)

(3)
Since σn (bn (s)) = s, and σn0 (bn (s)) = b0 1(s) , and in a symmetric Bayes-Nash equilibriumnb∗ = bn , we can write the above condition as a differential equation.12ZZ

b0n (s)· P

S(v)Qn (s|v)·g(v|s)dv+bn (s)· P

Q0n (s|v)·g(v|s)dv =

Z
P

S(v)

Q0n (s|v)·v·g(v|s)dv
S(v)

(5)
We assume that participants bid zero (the smallest possible bid) if they observe the smallest
possible signal, i.e. bn (0) = 0.13 Now the solution to the differential equation that represents
the equilibrium bidding function can be computed asbn (s) = e

F (s)

R

Z s
·

e

−F (t)0PS(t)Q0n (t|v) · v · g(v|t)dv

R
P

S(t)

Qn (t|v) · g(v|t)dv

dt,

(6)

with

Z t RP
Q0 (x|v) · g(v|x)dv
S(x) nR
F (t) = −dx.
P
Q (x|v) · g(v|x)dv0
S(x) n(7)

Since the conditional probability of winning, Qn (s|v), itself depends on the equilibrium bid12

For more details refer to Wilson (1977). For our information structure, this equation takes the form ofb0n (s) ·Z 1

1
Qn (s|v) dv + bn (s) ·
s
2v
2

Z 1
s
2

13

1
1
Q0n (s|v) dv =
2v

2

Z 1
s
2

Q0n (s|v)dv.

(4)

Although a zero signal is possible for any value of the object in our information setup, this normalizing
assumption seems plausible. Given the risk neutrality of the bidders it is not particularly restrictive and it allows
for a unique solution.

8

ding function, bn (s), we are facing a difficult mathematical problem: bn (s) and Qn (s|v) need
to be determined simultaneously. In a standard auction model, such as the first-price sealed-bid
auction, this difficulty can be avoided by considering a strictly increasing equilibrium bidding
function. If the equilibrium bidding function, bn (s), of the first-price sealed-bid auction isincreasing, then independently of its exact form, Qn (s|v) will be simply the conditional probability that the bidder’s signal is the highest, i.e. Qn (s|v) = (s/2v)n−1 .14 Then, the equilibrium
bidding function can be computed according to equations 6 and 7. Given that the resulting bidding function is strictly increasing for Qn (s|v) = (s/2v)n−1 , the previous technical assumption
is typically justified.
In the endogenous price-ceiling auction, we cannot find the equilibrium bn (s) and Qn (s|v)
in the same way. If bn (s) were linear, then Qn (s|v) would be the conditional probability thatthe bidder’s signal is the highest among those that are not higher than the average of the threehighest signals.15 However, the optimal bidding function of the endogenous price-ceiling auction computed according to equations 6 and 7 is not linear even in our information setup wherethe value of the auctioned good, v, and the private signal, s, are uniformly distributed. Consequently, the probability of winning, Qn (s|v), in the endogenous price-ceiling auction cannot be
traced back to the probability of receiving the highest signal. Moreover, even if the equilibrium
bidding function were linear, the probability of winning the endogenous price-ceiling auction
could not be derived in a straightfoward way (due to the existence of the endogenous priceceiling). It constitutes a real mathematical challenge for which we have not found any exact
solution. We approximate bn (s) and Qn (s|v) in the following way.
14

Note that if bn (s) is increasing, then s is the highest signals if and only if the corresponding bn (s) is thehighest bids.
15
Note that if bn (s) = c · s + d, c > 0, then s ≤ (s1 + s2 + s3 )/3 ⇔ bn (s) ≤ (bn (s1 ) + bn (s2 ) + bn (s3 ))/3,
where s1 , s2 , s3 are the three highest signals.

9First, although bn (s) is not linear, we interpret Qn (s|v) as the conditional probability ofreceiving the winning signal, i.e. the highest signal among those that are not higher than the
average of the three highest signals.16 Second, instead of deriving the exact functional formof Qn (s|v), we approximate it by a Beta distribution with parameters α and β. Third, we
compute bn (s) using that Beta distribution as a proxy for Qn (s|v). Four, in order to evaluate thegoodness of the approximation, we study again the empirical form of Qn (s|v) computed fromthe bn (s) by simulation and approximate it by Beta distribution. Finally, we compare the α andβ parameters of the Beta distribution estimated in the second and the fourth steps. We consider
that our method is justified if this comparison renders small differences between the estimatedparameters. In what follows we present the detailed procedure to derive bn (s) and Qn (s|v).
Our choice of the Beta distribution is based on two arguments. On one hand, the Beta
distribution with its two parameters is flexible enough to approximate asymmetric distributions.
On the other hand, the kth order statistics (i.e., the kth largest of n − 1 draws) of the uniform
distribution follows the Beta distribution with parameters α = k and β = n − k. Therefore,
while the Beta distribution offers a good approximation in the endogenous price-ceiling auction,it also represents the exact mathematical solution for auctions without price ceiling or with anexogenous price ceiling (e.g. the first-price sealed-bid auction). Since the Beta distribution iss. When weusually defined over the [0; 1] interval, we use normalized signals defined as 2v

say that the winning signal follows a Beta distribution given the value v, we refer to these
normalized signals.
As for our particular endogenous price-ceiling auction, the use of the Beta distribution as aproxy is justified by a Monte Carlo experiment. Our Monte Carlo experiment involved 1 million
16

This simplifying assumption is less severe for situations with a large number of bidders. Our numerical results
show that as n increases, the equilibrium bidding function loses curvature.

10

independent cases. In each of these cases, given the number of bidders, n, we generated n
independent draws from the uniform distribution over the [0; 1] interval and selected the numberthat was not larger than the average of the three largest draws. We used the empirical distributionof these numbers to estimate the two parameters of the Beta distribution.17 The upper part oftable 1 contains the parameters of the Beta distribution that describes the distribution of thewinning signal for 5, 10 and 20 bidders.18 Although the Beta distribution does not produce a
perfect fit, it closely follows the empirical distribution of the winning signal and captures its
skewed shape.
Once the equilibrium bidding function is computed for the endogenous price-ceiling auction, it can be used to check the goodness of the above described approximation. Let us anticipate the equilibrium results and check here the precision of our approximation. We simulated
the distribution of the winning bid for 10 possible values of the object, v, (between 0.1 and 1with a step-length of 0.1) for the three studied group sizes. Then we re-estimated the parametersof the Beta distribution and report a summary of the results in the lower part of table 1. Since
the estimates vary little, we only include the maximum and the minimum estimates in the table.
The results suggest that the chosen approximation is adequate, especially for lower values ofthe object and for larger number of bidders.
Given the above simplifications in the endogenous prince-ceiling auction model, the equi, α, β)librium bidding functions can be computed. In other words, we use Qn (s|v) = B( s−(v−)2
in equations 6 and 7, where B is the cumulative distribution function of the Beta distribution.
17

The two parameters of the Beta distribution were estimated using the method of moments, i.e. the following
formula: α̂ = x̄( x̄(1−x̄)− 1) and β̂ = (1 − x̄)( x̄(1−x̄)− 1), where x̄ is the sample mean and v is the samplevvvariance.
18
In order to check the performance of the applied Monte Carlo algorithm we computed the empirical distribution of the winning bid for the other three auction formats, too, and compared them to the theoretical results. Our
simulation results match the theoretical ones with a precision of at least 1 decimal.

11

Table 1: Estimated and re-estimated parameters, (α, β), of the Beta distribution that describe
the distribution of the normalized winning signal given the value of the object. FPSB: first-pricesealed-bid auction; EPC: endogenous price-ceiling auction. *Exact results. **Results based onMonte Carlo experiments.
number of bidders
5

10

20

parameter

FPSB*
EPC**
minimum
maximum

α
β
α
β
α
β
parameters for equilibrium bids
4.0 1.0 9.0 1.0 19.0 1.0
2.4 2.9 7.5 2.8 17.8 2.8
re-estimated parameters (EPC**)
2.2 2.9 7.4 2.8 17.7 2.8
2.4 3.0 7.5 3.0 17.9 3.0

We present the theoretical results in the form of tabulated values based on numerical integrals in table 2. Since answering our main question related to the probability of suffering amonetary loss by winning the auction makes computer simulation and numerical methods unavoidable, we believe that this choice does not limit the practical interpretations of our results.
In all computations we used the tabulated equilibrium bidding functions and applied linear interpolation. The equilibrium bidding functions gain slope and lose curvature as the number ofbidders increases. The opposite pattern appears as the government changes the auction format
and introduces protection by an endogenous price ceiling. For low signals, more protection implies more aggressive bidding behavior, but the pattern is the opposite for larger signals. Since
the equilibrium bidding functions are strictly increasing, a higher signal will always induce
more optimism, i.e. a higher bid. While the first-price sealed-bid auction assigns the object tothe most optimistic bidder, the endogenous price-ceiling auction assigns it to the second or third
most optimistic one.

12

Table 2: Tabulated equilibrium bidding functions. FPSB: first-price sealed-bid auction; EPC:
endogenous price-ceiling auction.
auction
FPSB
number of bidders
510
signal0.1
0.05 0.05
0.2
0.11 0.10
0.3
0.16 0.15
0.4
0.21 0.20
0.5
0.26 0.25
0.6
0.32 0.30
0.7
0.37 0.35
0.8
0.42 0.40
0.9
0.46 0.46
1.0
0.51 0.51

2.2

EPC
20

5
bid
0.05 0.08
0.10 0.15
0.15 0.21
0.20 0.27
0.25 0.31
0.30 0.36
0.35 0.39
0.40 0.42
0.45 0.45
0.50 0.47

10
0.05
0.11
0.16
0.21
0.27
0.32
0.37
0.43
0.48
0.52

FPSB
20
0.05
0.10
0.15
0.20
0.25
0.30
0.36
0.41
0.46
0.51

signal
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2.0

5

10

0.55
0.60
0.64
0.68
0.71
0.75
0.78
0.81
0.84
0.87

0.56
0.61
0.65
0.70
0.75
0.79
0.83
0.87
0.90
0.93

EPC
20

5
bid
0.55 0.49
0.60 0.51
0.65 0.52
0.70 0.53
0.75 0.54
0.80 0.55
0.85 0.55
0.89 0.55
0.93 0.55
0.97 0.55

10

20

0.57
0.61
0.65
0.68
0.71
0.73
0.75
0.76
0.77
0.77

0.56
0.61
0.66
0.71
0.75
0.79
0.83
0.86
0.87
0.88

Endogenous price-ceiling auctions and the Japanese rules

Although the rules of our endogenous price-ceiling auction are inspired by the rules of the
procurement auctions used by several Japanese local governments, they do not exactly coincide.
The local governments usually exclude bidders who bid a cost below some k times the averageof some of the highest bids, where k is typically less than 1. In our setup, given that we consider
prices instead of costs, this rule can be translated into eliminating all those who bid above c times
the average of the three highest bids, with c ≥ 1. If c is large enough, the endogenous priceceiling loses its importance and the distribution coincides with the distribution of the winningsignal in the first-price sealed-bid auction. Although the changes we observe in the empirical
distributions seem to be smooth, with the increase of c the densities tend to lose their bell
shapes that makes the Beta approximation less accurate. The intuition behind the double peaks
that appear in the histograms is that very large bids increase considerably the average of the13three largest bids. When it is multiplied by a fixed c, for sufficiently large bids the endogenous
price-ceiling stops being a binding constraint as it grows above the highest bid. Therefore thecorresponding graphs are some combinations of two distributions: the one that belongs to ourendogenous price-ceiling auction with c = 1, and the one that corresponds to the first-pricesealed-bid auction. The complexity of this case made us leave the auction formats with c > 1
for further research and present the simplest, yet novel and interesting c = 1 case here.

2.3

Negative payoffs and the winner’s curse

Since bidders only receive a noisy signal about the value of the auctioned object, in someoccasions the winner, who is always the most optimistic (admitted) bidder, has to pay morethan the real value of the object. Rational players apply a discount to take into consideration thisinformational problem. Nevertheless, negative payoffs occur even if players play according to
the Bayes-Nash equilibrium strategies, as for a given signal extremely low values (for the valueof the auctioned object) do appear with a positive, although relatively small, probability. The
winner’s curse occurs if players fail to correct their bids correctly, and bid above the expectedvalue of the objects conditional on that the signal they hold is the winning one. Although
the winner’s curse does not occur in equilibrium, both analyzed auction formats are prone tonegative winning payoffs.
Prediction 1. The probability of negative payoff from winning the auction decreases with theintroduction of a price ceiling, and with the number of bidders.
Table 3 shows the probability of a negative winning payoff computed by a Monte Carloexperiment with 1 million draws. Although negative winning payoffs occur in all cases, their14

probability decreases as the number of bidders increases. More importantly, the results showthat the endogenous price ceiling in fact reduces the probability of negative profits.
Table 3: The probability of obtaining a loss by winning the auction, Pr(loss), ex-ante expectedutility, E(utility), and ex-ante expected revenue E(revenue). FPSB: first-price sealed-bid auction; EPC: endogenous price-ceiling auction.
auction
FPSB

EPC

number of bidders
5
10
20
5
10
20
Pr(loss)
11.9% 7.3%
4.1%
8.3%
1.9%
0.3%
E(utility)
0.0175 0.0044 0.0012 0.0427 0.0119 0.0032
E(revenue) 0.4130 0.4535 0.4762 0.2876 0.3810 0.4361

2.4

Expected payoffs and revenue

Prediction 2. The expected payoff of bidders increases with the introduction of a price ceiling,
and decreases with the number of bidders.
One expects that reducing the probability of suffering a loss by winning the auction increases
bidders’ expected payoff. We used equations 1 and 2 to confirm this hypothesis and tabulated
the ex-ante expected payoffs in table 3. The numbers show that more competition reduces
expected utility, while the implicit protection makes it to increase.
Prediction 3. The expected revenue of the seller decreases with the introduction of a priceceiling, and increases with the number of bidders.
Similarly to expected payoffs, we computed ex ante and interim expected revenues for theseller. Table 3 contains intuitive numerical results according to which more competition in-15

creases revenues, while more protection decreases it. Similar results hold for interim expected
payoffs and interim expected revenue.

3

Experimental design

We conducted four experimental sessions to check the empirical properties of the analyzed
auction formats. For each session we invited 20 subjects to a computer laboratory at the Instituteof Social and Economic Research at Osaka University who participated in a 90-minute longexperiment. We decided to adopt the instructions used by Kagel and Levin (1986) to make ourresult comparable with others from the literature. In order to avoid fluctuation in the number ofparticipants, players received a monetary endowment in each period before learning their private
signals and posing their bids in the auction. This feature made bankruptcy and a reduction in thenumber of active bidders impossible. This is in line with our theoretical model that describes a
one-shot interaction between people, excluding any type of dynamic strategic considerations.
We used computers and zTree (Fischbacher, 2007) in all sessions. Table 4 offers a session
summary showing the most important characteristics of the experimental design. In all cases thevalue of the object was generated by the uniform distribution over the [0; 1000] interval, making
experimental monetary units (EMU) close to real monetary units (Japanese Yen). Subjects
received random draw—rounded to the closest integer—from the uniform distribution over the[0; 2v] interval as private signals and were asked to bid. After the bidding we showed them all
the bids, identified the winner and computed the payoffs. We used the same payment scheme(833 JPY show-up fee, and the 10 EMU = 1 JPY conversion rule) in all treatments.

16

Table 4: Experimental session summary. Number of practice rounds between parenthesis.

4

sessionauctionroundsgroup size

participants

1
2
3
4

FPSB
EPC
FPSB
EPC

(3) + 20(3) + 20(3) + 20(3) + 20

5
5
10
10

20
20
20
20

net earnings (JPY)
min. average max.
1938
2019
2096
1979
2038
2136
1797
2021
1987
1937
2054
1999

Empirical results

Theory predicts that the bidding functions are strictly increasing in the private signal, therefore
the winner of a first-price sealed-bid auction should be the person with the most optimistic
estimate. According to our data, this is the case in 58% of the experimental markets in our
FPSB session with 5 bidders, and 50% with 10.19 Table 5 shows the distribution of the rank of
the winner’s signal. In the session with groups of 5 bidders, the rules of the endogenous priceceiling auction managed to shift the distribution and to make that the person with the second
highest bid become winner in 40% of the cases. Curiously, the empirical distributions in thesessions with 10 bidders are statistically identical for the two auction formats.20
Observation 1. The expected payoff of bidders increases with the introduction of a price ceiling,
and decreases with the number of bidders.
Participants’ net profits are the best numbers to give a quick overview of the empiricalperformance of the auctions. With 5 bidders, in the FPSB treatment their sum amounted to 953
EMU per group after the 20th period, while in the EPC treatment this number was equal to 1950
EMU. With 10 bidders, the fiercer competition reduced these numbers considerably and turned
19This results is comparable to the 42%-89% interval that includes the same proportion in the treatments reported
by Kagel and Levin (1986).
20
The statistical comparisons described in this section are based on one-sided parametric tests. The usual qualifier refers to a significance level of at most 15%.

17

Table 5: Empirical distribution of the auction winners according to their signal. Number ofbidders between parenthesis.
auction
FPSB

EPC

number of bidders
5
winning signal’s rank
1
2
3
4
5

58%
20%
13%
10%
0%

10
5
% of cases
50% 25%
28% 40%15% 23%
3% 4%
5% 9%

10
50%
30%
10%
3%
8%

the average net profit into average net loss. The sum of net profit was -1314 EMU per group on
average in the FPSB session, and -86 EMU per group in the EPC session.
Observation 2. The probability of a negative winning payoff does not increase with the introduction of a price ceiling, and it increases with the number of bidders.
As for the number of winners who achieved a net loss instead of a net gain, figure 2 plots
the accumulated number of winners over the session who bid above the real value of the object.
We estimate the probability of suffering a monetary loss by winning the auction using the cumulative data, and make a statistical comparison between the two auction formats. We find thatthe difference is either insignificant or shows that the endogenous price-ceiling auction implies
a smaller probability independently from the number of bidders.
If we take all 20 round into account, the final point estimate for the probability of sufferinga monetary loss by winning the auction with 5 bidders is 30% for both the first-price sealed-bid
and endogenous price-ceiling auctions. With 10 bidders these estimates are much higher: 63%
for the first-price sealed-bid and 48% for the endogenous price-ceiling auction. It is impor-18

tant to point out that all these numbers are higher than the expected probabilities for all usualsignificance levels. Moreover, it seems that the number of bidders increases the probability of
suffering a loss in the laboratory instead of reducing it as suggested by the theoretical results.
Observation 3. The probability of the winner’s curse decreases with the introduction of a priceceiling, and it increases with the number of bidders.
Systematic overbidding, i.e. winner’s curse, and negative payoffs did occur in many different
cases, however both problems typically appeared for small signals. Figure 1 shows the theempirical and the theoretical bidding functions and also includes the expected value of the object
given a specific winning signal.21 We can talk about the winner’s curse when bids shoot overthis expected value. According to our data this systematic mistake appears for signals between
0 and 300 in the first-price sealed-bid auction, and for signals below 100 in the endogenous
price-ceiling auction if there are 5 bidders. More competition makes the situation worse forwinners as the area of the winner’s curse expands until signals around 600 in the first-pricesealed-bid, and below 200 in the endogenous price-ceiling auction.22 In the FPSB sessions theempirical bidding function practically coincides with the theoretical one for signals around 500
and above. In case of the endogenous price-ceiling auction, participants bid more aggressively
than expected for especially small signals (below 300) and for especially large ones (above1100). Although the winner’s curse does not constitute a problem in theory, our theoreticalresults plotted in figure 1 suggest that participants in the EPC auction are less prone to sufferfrom it. The separation between the two grey curves, i.e. the equilibrium bidding function and
21

In order to present a relatively smooth empirical bidding function we created 21 grid point over the [0; 200]
interval for signals using 100 EMU as distance and computed the average bid for the rounded signals.
22
Although the estimated bidding function appears to stay above the expected value of the object for larger
signals, too, the region reported in the text is characterized by statistically significant differences.

19the expected value of the object (given that the observed signal is the winning one), is larger incase of the EPC auction than in case of the FPSB auction for a fixed number of bidders.
Theoretically, bidders’ expected payoff is an increasing non-negative function of their signal. Our experimental data is partially in line with this result. It is the winner’s curse that makes
expected payoffs and the winner’s payoff tend to be negative for small signals.
Observation 4. The average revenue of the seller does not increase with the introduction of aprice ceiling, and it does not decrease with the number of bidders.
Both the theoretical and the empirical results suggest that the endogenous price-ceiling auction offers more protection and therefore a better situation for bidders for any given number ofcompetitors. However this protection comes at a price as predicted by the drop in the revenue
raised by the auction. Figure 3 shows the average revenue given the value of the auctioned object. The relatively small number of observations makes the picture incomplete and statisticalcomparison difficult. Nevertheless, our data confirms that the endogenous price-ceiling auction
never achieves a higher revenue that the first-price sealed-bid auction for a given value of theobject.23 Similarly, the seller’s revenue tend to increase with the competition.

5

Conclusions

We have presented theoretical and empirical results related to novel auction formats whose
main purpose is to protect bidders from a financial loss by excluding the overoptimistic ones.
Ours is the first step to systematically analyze and understand how this exclusion technique canreduce the probability of the winner’s curse and/or suffering a monetary loss by winning the23

The statistical tests, where possible to complete, show either insignificant differences or the superiority of thefirst-price sealed-bid auction in raising revenue.

20

auction. Our laboratory data confirms that bidders indeed tend to be too optimistic and makesystematic mistakes (as compared to theoretical results). Therefore the use of the proposed
auction mechanisms by several authorities, like the local governments in Japan or the European
Commission, is recommended to cope with the problem. Sellers, on the other hand, should
be aware of the cost of this protection, since excluding the overoptimistic bidders reduces therevenue raised by the auction both in theory and in the experimental laboratory.

References[1] Abbink, K., Brandts, J., Penazis-Christou, P. (2006) Auctions for governments securities:
A laboratory comparison of uniform, discriminatory and Spanish designs, Journal of Economic Behavior and Organization, 61: 284-303.
[2] Calveras, A., Ganuza, J.J., Hauk, E. (2004) Wild bids. Gambling for resurrection in procurement contracts, Journal of Regulatory Economics, 26(1): 41-68.
[3] Capen, E.C., Clapp, R.V., Campbell, W.M. (1971) Competitive bidding in high-risk situations, Journal of Petroleum Technology, 23: 641-653.
[4] Cox, J.C., Isaac, R.M., Cech, P.A., Conn. D. (1996) Moral hazard and adverse selection in
procurement contracting, Games and Economic Behavior, 17: 147-176.
[5] Doornik, J.A. (2006) An object-oriented matrix language - Ox 4, Timberlake Consultants
Press

21

[6] Dyer, D., Kagel, J.H., Levin, D. (1989) A comparison of naive and experienced bidders
in common value offer auctions: A laboratory analysis, The Economic Journal, 99(394):
108-115.
[7] Fischbacher, U. (2007) z-Tree: Zurich toolbox for ready-made economic experiments,
Experimental Economics, 10(2): 171-178.
[8] Ganuza, J.J. (2007) Competition and cost overruns in procurement, The Journal of Industrial Economics, 55(4): 633-660.
[9] Gode, D.K., Sunder, S. (2004) Double auction dynamics: Structural effects of non-binding
price controls, Journal of Economic Dynamics & Control, 28: 1707-1731.
[10] Isaac, E.M., Plott, C.R. (1981) Price controls and the behavior of auction markets: An
experimental examination, The American Economic Review, 71(3): 448-459.
[11] Ishii, R. (2008) Favor exchange in collusion: Empirical study of repeated procurement
auctions in Japan, International Journal of Industrial Organization, 27(2): 137-144.
[12] Kagel, J.H. (1995) Auctions: A survey of experimental research, in: Kagel, J.H., Roth,
A.E. (eds.) The handbook of experimental economics, Princeton University Press[13] Kagel, J.H., Levin, D. (1986) The winner’s curse and public information in common value
auctions, The American Economic Review, 76(5): 894-920.
[14] Milgrom, P.R., Weber, R.J. (1982) A theory of auctions and competitive bidding, Econometrica, 50(5): 1089-1122.

22

[15] Smith, V.L., Williams, A.W. (1981) On nonbinding price controls in a competitive market,
The American Economic Review, 71(3): 467-474.
[16] Wilson, R. (1977) A bidding model of perfect competition, The Review of Economic
Studies, 44(3): 511-518.

FiguresFigure 1: Theoretical and empirical bidding functions, and the expected value of the object
given that the signal is the winning one.
expected bid

expected value of the object

observed bid

expected bid

1000

1000

900

900

800

800

700

700

600

600

500

500

400

400

300

300

200

200

100

100

0

expected value of the object

observed bid

0
0

200

400

600

800 1000 1200 1400 1600 1800 2000

0

200

400

FPSB (5 bidders) - signal
expected bid

expected value of the object

600

800 1000 1200 1400 1600 1800 2000

EPC (5 bidders) - signal
observed bid

expected bid

1000

1000

900

900

800

800

700

700

600

600

500

500

400

400

300

300

200

200

100

100

0

expected value of the object

observed bid

0
0

200

400

600

800 1000 1200 1400 1600 1800 2000

0

FPSB (10 bidders) - signal

200

400

600

800 1000 1200 1400 1600 1800 2000

EPC (10 bidders) - signal

23

Figure 2: Accumulated number of negative winning payoffs over sessions.
FPSB (5 bidders)

EPC (5 bidders)

FPSB (10 bidders)30
accumulated number ofnegative winning payoffs

accumulated number ofnegative winning payoffs

30

EPC (10 bidders)

25
20
15
10
5
0

25
20
15
10
5
0

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

round

round

Figure 3: Revenue raised by the auctions given the value of the object
EPC (5 bidders)

FPSB (10 bidders)
1000

800

800

600

600

revenuerevenueFPSB (5 bidders)
1000

400
200

EPC (10 bidders)

400
200

0

0
0

100 200 300 400 500 600 700 800 900 1000

0

value

100 200 300 400 500 600 700 800 900 1000
value

24

Supplementary materials
INSTRUCTIONS (for EPC)
This is an experiment about market decision making. The instructions are
simple, and if you follow them carefully and make good decisions you may earn a
CONSIDERABLE AMOUNT OF MONEY which will be PAID TO YOU IN CASH at
the end of the experiment.
1. In this experiment we will create a market in which you will act as buyers ofa fictitious commodity in a sequence of trading periods. A single unit of the commodity
will be auctioned off in each trading period. There will be 3 trial periods and 20 paying
periods that determine the amount of money that you will receive by the end of theexperiment.
2. Your task is to submit written bids for the commodity in competition withother buyers. The precise value of the commodity at the time you make your bids will
be unknown to you. Instead, each of you will receive information as to the value of theitem which you should find useful in determining your bid. The process of determining
the value of the commodity and the information you will receive will be described in
Sections 6 and 7 below.
3. The highest bidder among all bidders whose bids are lower than the priceceiling gets the item and makes a profit equal to the difference between the value of thecommodity and the amount they bid. The price ceiling is computed as the average of the
3 highest bids. That is,(VALUE OF ITEM) – (HIGHEST BID AMONG ALL BIDS LOWER THAN THE
PRICE CEILING) = PROFITS
for the highest bidder among all bidders whose bids are lower than the price ceiling. If
this difference is negative, it represents a loss.
If you do not make the highest bid among all bids lower than the price ceilingon the item, you will earn zero profits. In this case, you neither gain nor lose moneyfrom bidding on the item.
4. You will be given 1,000 points of experimental cash at the beginning of eachperiod. Any profit earned by you in each period will be added to these initial points, andany losses incurred will be subtracted from these initial points. The total profit of thesetransactions will be calculated and paid to you in CASH under the conversion rate of 1

25

point equal to 0.1yen at the end of the experiment. Finally, your final earnings will becalculated as follows:
EARNINGS = 0.1× { (TOTAL PROFITS) – (TOTAL LOSSES)+ (20 PERIODS × 1,000 POINTS) }5. At the beginning of each period groups of 5 participants will be formed by
the computer in a random way. It means that you typically play with 4 different
participants in every period. During each trading period you will be bidding in a market
in which all the other participants are also bidding. You are asked in each period is tochoose your bid, write it in the purple cell in the center of the screen, and click OK
within 3 minutes. After all participants click OK, all bids in your group will be
displayed on your screen. Then the computer will display the price ceiling, the highest
value among all bids lower than the price ceiling, the value of the item, and your profit
or loss. Once you reconfirm them, please click OK.
6. The value of the auctioned commodity (V*) will be assigned randomly andwill lie between 0 points and 1,000 points inclusively. For each auction, any value
within this interval has an equally likely chance of being drawn. The value of the item
can never be less than 0 points or more than 1,000 points. The V* values are determined
randomly and independently from auction to auction. As such a high V* in one period
tells you nothing about the likely value in the next period whether it will be high or low.
It doesn't even preclude drawing the same V* value in later periods.
7. Private Information Signals:
Although you do not know the precise value of the item in any particular
trading period, you will receive a private information signal that is related to the V*
value of the item in each auction. This signal is an integer between 0 and twice the V*
value of the object (limits included). Any value within this interval has an equally likely
chance of being drawn and being assigned to one of you as your private information
signal.
For example, suppose that the value of the auctioned item is 550 points. Theneach of you will receive a private information signal which will consist of a randomly
drawn number that will be between 0 points and 2×550 =1,100 points. Any number inthis interval has an equally likely chance of being drawn.
The line diagram below shows what's going on in this example.
0 points

V*=550 points

1,000 points

Signal values may be anywhere in this interval.

26

2V*=1,100 points

The data below show the entire set of signals the computer generated for oursample. (Note we've ordered these signal values from lowest to highest.)
V* = 550 points. Signal values: 964 points,
914 points,
432 points
279 points
22 points
You will note that some signal values were above the value of the auctioned item, andsome were below the value of the item. Over a sufficiently long series of auctions, thedifferences between your private information signal and the value of the item will
average out to zero (or very close to it). For any given auction, however, your private
information signal can be above or below the value of the item. That's the nature of therandom selection process generating the signals.
8. Your signal values are strictly private information and are not to be revealed
to anyone else during the whole session. Finally we will post all of the signal values
drawn along with the bids.
9. No one may bid less than 0 points for the item. Any integral bid isacceptable. In case of ties for the highest bid among all bids lower than the price ceiling,
computer will determine randomly who will earn the item.
10. You are not to reveal your bids, or profits, nor are you to speak to any othersubject while the experiment is in progress.
Let's summarize the main points: (1) Highest bidder among all bidders whosebids are lower than the price ceiling earns the item and earns a profit = value of item
–highest bid price lower than the price ceiling. The price ceiling is computed as theaverage of the 3 highest bids. (2) Profits in each period will be added to your initial
experimental cash of 1,000 points, losses subtracted from it. Your total earnings at theend of experiment will be paid in cash. (3) Your private information signal is randomly
drawn from the interval between 0 points and 2V*. (4) The value of the item will
always be between 0 points and 1,000 points.
Are there any questions?

27

INSTRUCTIONS (for FPSB)This is an experiment about market decision making. The instructions are
simple, and if you follow them carefully and make good decisions you may earn a
CONSIDERABLE AMOUNT OF MONEY which will be PAID TO YOU IN CASH at
the end of the experiment.
1. In this experiment we will create a market in which you will act as buyers ofa fictitious commodity in a sequence of trading periods. A single unit of the commodity
will be auctioned off in each trading period. There will be 3 trial periods and 30 paying
periods that determine the amount of money that you will receive by the end of theexperiment.
2. Your task is to submit written bids for the commodity in competition withother buyers. The precise value of the commodity at the time you make your bids will
be unknown to you. Instead, each of you will receive information as to the value of theitem which you should find useful in determining your bid. The process of determining
the value of the commodity and the information you will receive will be described in
Sections 6 and 7 below.
3. The highest bidder gets the item and makes a profit equal to the difference
between the value of the commodity and the amount they bid. That is,(VALUE OF ITEM) – (HIGHEST BID) = PROFITS
for the highest bidder. If this difference is negative, it represents a loss.
If you do not make the highest bid on the item, you will earn zero profits. In
this case, you neither gain nor lose money from bidding on the item.
4. You will be given a starting capital credit balance of 1,500 yen. Any profit
earned by you in the experiment will be added to this sum, and any losses incurred will
be subtracted from this sum. The net balance of these transactions will be calculated andpaid to you in CASH at the end of the experiment.
The starting capital credit balance, and whatever subsequent profits you earn,
permits you to suffer losses in one auction to be recouped in part or in total in later
auctions. You are permitted to bid in excess of your capital credit balance in any given
period.
5. At the beginning of each period groups of 5 participants will be formed by
the computer in a random way. It means that you typically play with 4 different

28participants in every period. During each trading period you will be bidding in a market
in which all the other participants are also bidding. You are asked in each period is tochoose your bid, write it in the purple cell in the center of the screen, and click OK
within 1 minutes. After all participants click OK, all bids in your group will be
displayed on your screen. Then the computer will display the highest bid, the value ofthe item, and your profit or loss. Once you reconfirm them, please click OK.
6. The value of the auctioned commodity (V*) will be assigned randomly andwill lie between 0 yen and 1,000 yen inclusively. For each auction, any value within
this interval has an equally likely chance of being drawn. The value of the item cannever be less than 0 yen or more than 1,000 yen. The V* values are determined
randomly and independently from auction to auction. As such a high V* in one period
tells you nothing about the likely value in the next period whether it will be high or low.
It doesn't even preclude drawing the same V* value in later periods.
7. Private Information Signals:
Although you do not know the precise value of the item in any particular
trading period, you will receive a private information signal that is related to the V*
value of the item in each auction. This signal is an integer between 0 and twice the V*
value of the object (limits included). Any value within this interval has an equally likely
chance of being drawn and being assigned to one of you as your private information
signal.
For example, suppose that the value of the auctioned item is 550 yen. Theneach of you will receive a private information signal which will consist of a randomly
drawn number that will be between 0 yen and 2×550 =1,100 yen. Any number in thisinterval has an equally likely chance of being drawn.
The line diagram below shows what's going on in this example.
0 yen

V*=550 yen

1,000 yen

2V*=1,100 yen

Signal values may be anywhere in this interval.

The data below show the entire set of signals the computer generated for oursample. (Note we've ordered these signal values from lowest to highest.)
V* = 550 yen. Signal values: 964 yen,
914 yen,
432 yen29

279 yen
22 yen
You will note that some signal values were above the value of the auctioned item, andsome were below the value of the item. Over a sufficiently long series of auctions, thedifferences between your private information signal and the value of the item will
average out to zero (or very close to it). For any given auction, however, your private
information signal can be above or below the value of the item. That's the nature of therandom selection process generating the signals.
8. Your signal values are strictly private information and are not to be revealed
to anyone else during the whole session.
9. No one may bid less than 0 yen for the item. Any integral bid is acceptable.
In case of ties for the highest bid, computer will determine randomly who will earn theitem.
10. You are not to reveal your bids, or profits, nor are you to speak to any othersubject while the experiment is in progress.
11. As promised, everyone will receive 750 yen irrespective of their earnings
for participating in the experiment. Your net balance at the end of the experiment drop
to zero (or less), you will receive only 750 yen for participating in the experiment. You
will not pay your own money to the experimenter if your net balance at the end of theexperiment less than zero.

Let's summarize the main points: (1) Highest bidder earns the item and earns a
profit = value of item –highest bid price. (2) Profits will be added to your startingbalance of 1,500 yen, losses subtracted from it. Your balance at the end of experimentwill be paid in cash. If balance turns negative you're no longer allowed to bid. (3) Yourprivate information signal is randomly drawn from the interval between 0 yen and 2 V*.
(4) The value of the item will always be between 0 yen and 1,000 yen
Are there any questions?

30