Overconﬁdence in Economic Contests∗
Munetomo Ando†
National Graduate Institute for Policy Studies
First Draft: September 10, 2003
This Version: August 25, 2004

Abstract
This paper studies an economic contest with two participants, who
are overconﬁdent in their own relative abilities. We examine two different sources of overconﬁdence, overestimation of one’s own ability
and underestimation of the rival’s ability, and compare the behavioral
consequences of each situation with the correctly estimated case. The
main result is that the former always induces the participants’ aggressive behavior, whereas the latter sometimes brings about less aggressive
behavior of one or both participants.
JEL classiﬁcation numbers: D44, D82.
Keywords: Overconﬁdence, economic contest, relative ability, all-pay
auction, distribution of abilities.

∗

The author is grateful to Michihiro Kandori, Hajime Kobayashi, Akihiko Matsui,
Toshihiro Matsumura, Hitoshi Matsushima, Daisuke Oyama, Masatoshi Tsumagari,
Noriyuki Yanagawa, and seminar participants at the Contract Theory Workshop (CTW),
Hosei University, Kyoto Game Theory Workshop, Kyoto University, Osaka University,and the University of Tokyo for their comments and suggestions. This paper was alsopresented in the 2004 Spring Meeting of the Japanese Economic Association, the 2004 Far
Eastern Meeting of the Econometric Society, and the Second World Congress of the Game
Theory Society.
†
Address: 2-2 Wakamatsu-cho, Shinjuku-ku Tokyo, 162-8677 Japan, TEL. +81-3-33410748, FAX. +81-3-3341-0220, E-mail: m.ando@grips.ac.jp (M. Ando).

1

1

Introduction

Overconﬁdence is one of the famous stylized facts about human behavior.
Several studies in psychology and experimental economics have shown thathumans are overconﬁdent in their own (relative) abilities. For example,
Svenson (1981) reported that almost all drivers in Texas believe that theirown driving skills are above average, and he concluded that a considerable
number of subjects were overconﬁdent in their own relative driving skills.
In addition, these studies usually indicate that overconﬁdence induces
aggressive behavior. For example, Camerer and Lovallo (1999) used anexperimental approach and showed that overconﬁdence commonly induces
excess entry into markets and business failure.1
The aim of this paper is to study the behavioral consequences of participants’ overconﬁdence in economic contests with two participants. Well
known applications of economic contests are job promotion in internal labor markets, political elections, rent seeking activities, and R&D rivalry. In
this paper, we examine the situations in which each participant is overconﬁdent in his/her relative ability. Note that there are some possible situations
where a participant is overconﬁdent in his/her relative ability. Overconﬁdence in relative ability comes from overestimating one’s own ability and/or
underestimating the rival’s ability.
To simplify the analysis, we examine two extreme situations, where eachparticipant has overestimated his/her own ability or underestimated therival’s ability, and compare the behavioral consequences of each situation
with the correctly estimated case. By this comparison, we will show that thediﬀerent sources of overconﬁdence have diﬀerent behavioral consequences.
More precisely, we will show that in the former case overconﬁdence always
induces aggressive behavior by the participants, whereas in the latter case it
sometimes brings about less aggressive behavior by one or both participants.
In this paper, we study the following contest game. The principal hires
two risk neutral agents for a speciﬁc period and assigns a task to each ofthem. For exposition, we suppose that the principal is female and the agentsare male. Each agent outlays his eﬀort for winning. Agents diﬀer in ability.
Higher ability agents can obtain higher returns when they win the contest.
We refer to agents’ abilities as their types, so each type is equal to themonetary value of their winning the contest. Examples of such situations
are the aforementioned promotion contests in ﬁrms and political elections.
The types are independently and identically distributed.
An agent has prior beliefs about his own type and the distribution oftypes. In a standard model of economic contest called the “all-pay auction”
with incomplete information, the prior (or subjective) beliefs are correct. 2
1

See Camerer (1997) for further references.
The diﬀerence between the “ﬁrst price sealed bid auction” and the “all-pay auction”
is the following. In the ﬁrst price sealed bid auction, the winner is the only person who22

Moreover, it is assumed that the setting of the game is common knowledge.
However, as we already know, overconﬁdence is frequently observed in thereal world. So, in this paper, we will consider a model of economic contests
that includes participants’ overconﬁdence.
We consider situations where an agent’s beliefs about his own type andthe distribution of types may or may not be correct. More precisely, theagent has incorrect information about either his own type or the distributionof types.3 Here, we assume that the agents do not know that they and theirrival are overconﬁdent in their own types ex ante.4 Each agent’s eﬀort ischosen to maximize his expected proﬁt on the basis of prior (or subjective)
information. The eﬀort level of each agent is observable by all players at theend when the agents have already chosen their eﬀort levels. For the principal,
larger eﬀorts are proﬁtable, so that she welcomes a larger expected eﬀort
per agent.
Our results are as follows. First, we derive the symmetric Bayesian Nashequilibrium strategy in the contest game without the agents’ overconﬁdence
as the benchmark. Because the contest game considered here is the standard
all-pay auction, we can identify the symmetric equilibrium eﬀort strategy
and derive the expected eﬀort level per agent by using the standard method
used in auction literature. The equilibrium eﬀort levels are strictly increasing
with respect to types, so that the probability of winning in the contest is
equivalent to the probability that one’s own type is no lower than the rival’s
type.
Second, we show that overestimation of one’s own type always increases
one’s eﬀort and, therefore, it is proﬁtable for the principal. The reason
overestimation of one’s own type increases his eﬀort is as follows. In thesituation of overestimation of one’s own type, the agent behaves as if he hasa higher type. Since the eﬀort strategy is increasing in types, he chooses an
eﬀort that is higher than that in the benchmark case.5
Third, we show that underestimation of the type distribution6 increases
has to pay his own bid. However, in the all-pay auction, all bidders have to pay theirown bid. Note that promotion contests in ﬁrms and political elections can be regarded as
the all-pay auctions because all bidders (participants) have to pay their own bid (eﬀort),
irrespective of winning or losing.
3
Under an auction setting called the all-pay auction under incomplete information,
we formulate the situation where a participant underestimates the rival’s ability by adominance relationship between the true distribution of types and the subjective belief
about the type distribution (the deﬁnition of a dominance relationship is described in
Section 3.3). This means that the participant believes that there are more lower ability
types than would occur in the true distribution of types.
4
An agent may ﬁnd his and/or his rival’s overconﬁdence by the realized value of winningthe contest and the realized eﬀort levels ex post.
5
Note that each player believes that his strategy is an “equilibrium” strategy; the eﬀort
strategies chosen by the players do not constitute an equilibrium, however.
6
We use the terms “underestimation of the type distribution” instead of “underestimation of the rival’s ability” here. See footnote 3.

3the eﬀort of an agent of low ability, whereas it decreases the eﬀort of anagent of high ability. Therefore, it may or may not be proﬁtable for the
principal. Additionally, we show that underestimation increases an agent’s
expected eﬀorts in some cases, but decreases it in other cases. Thus, theaggregate eﬀect of underestimation is ambiguous. The fundamental reason
why underestimation of the type distribution changes each type’s eﬀort isthat underestimation changes each type’s subjective probability of winning.
There are two strands of theoretical works related to the present paper.
The ﬁrst is the analysis of economic contests as a variation of the all-pay
auction. Examples are Amann and Leininger (1996), Krishna and Morgan(1997), and Che and Gale (2000).7 Amann and Leininger (1996) analyzed
an asymmetric all-pay auction and Krishna and Morgan (1997) analyzed an
all-pay auction with aﬃliated types. Che and Gale (2000) studied the probability of winning function in economic contests carefully and they studied the
all-pay auction as a special case in their contests. Recently, contest design
problems from the viewpoint of the contest designer have attracted muchattention. Examples include Singh and Wittman (2001) and Ando (2004).8
These papers and the present paper focus on the fact that changes in eachtype’s probability of winning yield the changes in each type’s equilibrium
behavior. However, these studies, other than the present paper, assumed
that the prior beliefs about one’s own type and the type distribution are
correct. Hence, the present paper is very diﬀerent from these studies.
The second strand is the study of overconﬁdence in the ﬁeld of behavioral economics.9 Most studies of overconﬁdence focused on situations of one
person’s decision-making. Moreover, these studies usually conclude that
overconﬁdence brings about aggressive behavior by an agent. For example, Dubra (2003) studied a search model with an optimistic individual. In
contrast to these studies, the present paper studies an auction setting incorporating overconﬁdence and we show that overconﬁdence sometimes yields
less aggressive behavior by an agent.
The rest of this paper is as follows. Section 2 describes the environmentof our model. Section 3 examines in turn the benchmark case, the behavioral consequences of overestimation of one’s own type, and the behavioral
consequences of underestimation of the type distribution. Section 4 contains
our conclusions.
7

Noussair and Silver (2003) is an experimental study of all-pay auctions under incomplete information. We will refer again to Noussair and Silver (2003) in Section 4.
8
Singh and Wittman (2001) studied the contest in which each agent’s output exhibits
non-increasing returns to his eﬀort. Additionally, it is assumed that the participant withthe highest output need not win. They described the properties of optimal contests andshow that, for an open interval of types, an optimal contest uses the rule that the agent
with the highest output wins. Ando (2004) studied an economic contest with identicalprizes and described the eﬀects of symmetric division of the contest on the agents’ behavior.
9
Camerer (2003) is a useful reference on behavioral game theory. Itoh (2004) is a study
toward the behavioral contract theory.

4

2

Model

We consider an economic contest with two risk neutral agents, agents 1 and2. They compete to win the contest. Each agent i decides his eﬀort ei .10
Eﬀorts are outlaid simultaneously and independently. The agents’ eﬀort
levels are observable by the principal and the agents at the end when theagents have already chosen their eﬀort levels.
Each agent has a diﬀerent type, which represents the monetary value of
his win. The true type of agent i is denoted by θi and we assume that theexact value can be perceived only after winning. Moreover, we assume thatno one knows the exact value of θi ex ante. We assume that agent i hasa prior (or subjective) belief about his type, ai , instead of the knowledge
about his true type θi , and ai is his private information. The true typeof each agent is drawn independently from an interval [0, 1] according to
the distribution function F that has a continuous and everywhere strictly
positive density function f .11 However, both agents may not know the shapeof the distribution function F .
In the standard model of economic contests based on all-pay auctions,each agent’s prior belief about his own type is exactly the same as the true
type and the shape of the distribution function is assumed to be common
knowledge for the principal and both agents. However, in this paper werelax these assumptions. Agent i believes that his type is ai . However, it
may not be equal to the true type θi from the viewpoint of us (i.e., theanalysts of the model). Moreover, the agent believes that the rival j’s priorbelief about j’s type (i.e., aj ) drawn from an interval [0, 1] according to
the distribution function Gi .12 However, Gi may not be equal to the true
distribution F from our viewpoint. Additionally, the agent i believes thatthe rival j’s prior belief about the type distribution is also Gi , and that theabove-mentioned belief structures are common knowledge. Here, we assume
that each agent does not know the fact that both agents may have incorrect
information ex ante. Note that we do not assume that both agents havethe same wrong belief each other. Our assumptions mean that an agent
believes that his prior belief is the common prior belief among the players.
This setting enables us to use the technique of comparative statics.
In this contest, the real payoﬀ of agent i is, θi − ei if he wins, and −ei
if he does not. However, each agent chooses his eﬀort in order to maximize
his expected payoﬀ on the basis of his prior beliefs about his own type ai
and the type distribution Gi . Therefore the eﬀort strategies chosen by the10

For exposition, we suppose that the principal is female and the agents are male.
We assume that the type space is [0, 1]. This restriction is only for analytical convenience. We can preserve all our results in any non-negative type with bounded support
cases.
12
We assume that Gi has a continuous and everywhere strictly positive density function
gi .
11

5

agents may not constitute an equilibrium, whereas each agent believes that
his strategy is an equilibrium strategy. For the principal, larger eﬀorts are
proﬁtable, so that she welcomes a larger expected eﬀort per agent.

3

Analysis

3.1

The benchmark caseIn this subsection, we consider the standard all-pay auction as the benchmark case. That is, we consider the situation where ai = θi and Gi = F ,i = 1, 2, from the viewpoint of the analysts of the model.
In Proposition 1, we show the symmetric equilibrium eﬀort strategy inthis contest game. The eﬀort strategy in the benchmark case is denoted byβ(θ, F ) instead of β(a, Gi | a = θ and Gi = F ) to simplify the notation.
Proposition 1. In a symmetric equilibrium, the eﬀort strategy is∫ θ
yf (y)dy,β(θ, F ) =(1)0

and the symmetric equilibrium is unique.
Proof. The equilibrium strategy and its uniqueness can be easily derived
with the standard method in the auction literature. Hence, the proof isomitted.
In the symmetric equilibrium, since β(·, F ) is strictly increasing, agent
i’s probability of winning (hereafter p(θi , F )) is equivalent to the probability
that his type is no lower than the rival’s type; that is, p(θi , F ) = F (θi ).
Note that, for all θ ∈ [0, 1], the gradient of the probability of winning
function reﬂects on the gradient of the equilibrium eﬀort strategy, since∂β(θ, F )/∂θ = θf (θ) = θ∂p(θ, F )/∂θ. This is an important characteristic of
the equilibrium strategy. We will use this later.
In this contest, the expected eﬀort per agent is∫ 1
E(e, F ) =β(θ, F )f (θ)dθ.
(2)0

By using integration by parts with expressions (1) and (2), we obtain thefollowing.
Proposition 2. The expected eﬀort per agent is∫ 1
E(e, F ) =(1 − F (θ))θf (θ)dθ.
0

We provide an example.
Example 3.1. If F (θ) = θ, β(θ, F ) = θ2 /2 and E(e, F ) = 1/6.
6

(3)

3.2

Overestimation of one’s own typeIn this subsection, we examine the consequences of overestimation of one’s
own type in economic contests. We assume that agent i’s prior knowledge
about the type distribution is correct (i.e., Gi = F ). He believes that his
type is ai ; however, it is incorrect information from the viewpoint of theanalysts of the model. Now, we deﬁne the following.
Deﬁnition 1. Agent i overestimates his own type if, for given θi ∈ [0, 1),
ai > θ i .
In this situation, agent i derives the eﬀort strategy β by Gi (= F ). Then,he chooses his action by calculation with β(·, F ) and ai . From the above
facts, we obtain the following.
Proposition 3. Compared with the benchmark case, overestimation of one’s
own type increases his eﬀort (i.e., β(ai , F ) > β(θi , F ) for all θi ∈ [0, 1) andfor all ai > θi ).
Proof. The eﬀort strategy β is constructed by F and, therefore, β(a, F ) =∫θ∫a∫a
0 yf (y)dy. Since yf (y) is positive for all y ∈ (0, 1], 0 yf (y)dy > 0 yf (y)dy
for all a > θ. Thus, we obtain β(ai , F ) > β(θi , F ).
This proposition implies that, in the situations of overestimation of one’s
own type, an agent who has the true type θi behaves as if he has a higher
type, ai . Since the equilibrium strategy is increasing in types, he chooses ahigher eﬀort compared with the benchmark case. Thus, overconﬁdence fromoverestimation of one’s own type is always proﬁtable for the principal.
We provide an example. In this example, an agent with the true typeθ ∈ (0, 1) has a √type overestimation that is constructed by the following
simple rule, a = θ.13 This formulation permits us to directly calculate theexpected eﬀort level.
Example 3.2. Suppose that agent√ i’s prior information is (ai , Gi ) that satisﬁes Gi (θ) = F (θ) = θ and ai = θi . In this situation, his eﬀort strategy isβ(ai , F ) = θi /2 and the expected eﬀort level is E(β(ai , F ), F ) = 1/4. If agent
i knows his true type, he should follow the eﬀort strategy β(θi , F ) = θi 2 /2and the expected eﬀort level is E(β(θi , F ), F ) = 1/6.
The eﬀort strategies in the above example are depicted in Figure 1. The
horizontal axis is the true types and the vertical axis is the eﬀort levels.

13

Note that we know this relationship between one’s prior information about type andtrue type, whereas the players in this game do not know it.

7eβ(a, F )12β(θ, F )1θFigure 1:

3.3

Underestimation of the type distribution

In this subsection, we examine the consequences of underestimation of thetype distribution. We assume that agent i’s prior information about his type
is correct; that is, ai = θi . However, he has incorrect information about thetype distribution (i.e., Gi ̸= F ). To simplify the notation, we remove thesubscript i from Gi hereafter.
The next proposition shows agent i’s eﬀort strategy under incorrect information about the type distribution.
Proposition 4. If agent i has incorrect information about the type distribution, he chooses his eﬀort according to the following strategy:∫ θβ(θ, G) =yg(y)dy.

(4)0

Proof. The derivation of the strategy is the same as that in Proposition 1.
In this situation, the expected eﬀort is∫ 1
E(e, G) =β(θ, G)f (θ)dθ.

(5)0Note that the expected eﬀort is derived from the following facts. Each type’s
eﬀort level is calculated by β(·, G) and θi . However, the expectation is basedon the true distribution of types, F .
By using integration by parts with expressions (4) and (5), we obtainthe following.

8

Proposition 5. If agent i has incorrect information about the type distribution, his expected eﬀort level is∫ 1
E(e, G) =(1 − F (θ))θg(θ)dθ.
(6)0Now, we turn our attention to the characteristics of incorrect information
about the type distribution. We deﬁne the following.
Deﬁnition 2. For G ̸= F , agent i underestimates the type distribution if F
ﬁrst order stochastically dominates G; that is,∀θ ∈ [0, 1], G(θ) ≥ F (θ).
An intuitive explanation of underestimation is that agent i believes that
there are more lower types than would occur in the true distribution oftypes.
Next, we describe how underestimation changes agents’ behavior.
Proposition 6. Compared with the benchmark case, underestimation of thetype distribution changes agent i’s behavior as follows. For G ̸= F ,1. there exists θ̂ ∈ (0, 1) such that, for all θ ∈ (θ̂, 1], β(θ, G) < β(θ, F ),and2. there exists an interval of types (θa , θb ) such that for all θ ∈ (θa , θb ),β(θ, G) > β(θ, F ).
Proof. The proof of the former statement is as follows. β(θ, G) − β(θ, F ) =∫θ∫θyg(y)dy−
0 yf (y)dy. By∫using integration by parts we obtain β(θ, G) −0
θβ(θ, F ) = θ(G(θ) − F (θ)) − 0 (G(y) − F (y))dy. This is strictly negative at∫θθ = 1, since θ(G(θ) − F (θ)) is zero and 0 (G(y) − F (y))dy is strictly positiveat θ = 1 by deﬁnition. From the fact that β is increasing and continuous,
we can conclude that there exists θ̂ ∈ (0, 1) such that, for all θ ∈ (θ̂, 1],β(θ, G) < β(θ, F ).
The proof of the latter statement is as follows. We deﬁne θc = inf{θ |G(θ) > F (θ)}. For any θ ∈ [0, θc ), β(θ, G) = β(θ, F ), since for any θ ∈∫θ∫θ[0, θc ), g(θ) = f (θ), and β(θ, G) = 0 yg(y)dy and β(θ, F ) = 0 yf (y)dy. For∫ θ +εsuﬃciently small ε > 0, β(θc + ε, G) − β(θc + ε, F ) = θcc y(g(y) − f (y))dy
and this is strictly positive, since g(θc + ε) > f (θc + ε) by deﬁnition of
θc . From the fact that β is increasing and continuous, we can conclude
that there exists an interval of types (θa , θb ) such that for all θ ∈ (θa , θb ),β(θ, G) > β(θ, F ).
The implication of the above proposition is that, if an agent underestimates the type distribution, some types over work and other types under9

work compared with the benchmark case. Note that there are cases that thesign of β(θ, G) − β(θ, F ) changes more than once. However, to simplify the
exposition, in the rest of this paper we restrict our attention to situations
where β(θ, G) and β(θ, F ) are single-crossing in the interval (0, 1) (that is,θa = 0 and θb = θ̂).
We can describe a suﬃcient condition for single-crossing of β(θ, G) andβ(θ, F ).
Proposition 7. β(θ, G) and β(θ, F ) are single-crossing in the interval (0, 1),
if there exists θ̄ ∈ (0, 1), such that ∀θ < θ̄, f (θ) < g(θ), and ∀θ > θ̄,f (θ) > g(θ).
∫θ
Proof. β(θ, G) − β(θ, F ) = 0 y(g(y) − f (y))dy. This is positive for the typeclose to zero, since g(y) > f (y) at y = 0. Moreover, g(y) and f (y) arecrossing only once at a certain y ∈ (0, 1) and at the ∫left hand side, g(y) >1f (y), and at the right hand side, g(y) < f (y). Since 0 (g(y) − f (y))dy = 0∫1
and y is strictly increasing function, we obtain 0 y(g(y)−f (y))dy < 0. Thisimplies β(θ, G) < β(θ, F ) at θ = 1.
Next, we show that the magnitude relationship between the gradientof β(θ, G) and that of β(θ, F ) changes only once in (0, 1). More precisely,∃θ́ ∈ (0, 1), ∀θ < θ́, ∂β(θ, G)/∂θ > ∂β(θ, F )/∂θ and ∀θ > θ́, ∂β(θ, G)/∂θ <∂β(θ, F )/∂θ, since ∂β(θ, G)/∂θ = θg(θ) and ∂β(θ, F )/∂θ = θf (θ).
From the above and the fact that β is increasing and continuous, we canconclude that β(θ, G) and β(θ, F ) are single-crossing in the interval (0, 1).
This proposition tells us that, if the density functions g and f are singlecrossing, β(θ, G) and β(θ, F ) are also single-crossing in the interval (0, 1).
We brieﬂy describe the reason that underestimation of the type distribution changes an agent’s eﬀort under the single-crossing situations. Whenan agent with type θ underestimates the type distribution, his subjective
probability of winning the contest is changed from F (θ) to G(θ). In this underestimated situation, since the probability of winning at the left tail (notethat the probability of winning for an agent with the lowest ability is alwayszero) increases faster compared with the benchmark case, underestimationinduces more aggressive eﬀort at the left tail. For the higher types, underestimation decreases the gradient of the probability of winning function. This
reﬂects on decreases in the gradient of the eﬀort strategy function for thehigher types.
A more intuitive explanation is as follows. Since underestimation of
the type distribution implies that the (subjective) probability of winningis higher for every type, an agent with a lower type has a non-negligible
probability of winning and therefore he increases his eﬀort. On the other
hand, an agent with a higher type (almost) certainly wins and therefore hecan slightly decrease his eﬀort.
We provide an example.
10

Example 3.3. Suppose that agent i’s prior information is (ai , G) that satisﬁes ai = θi and G(θ) = 2θ − θi 2 , and that F (θ) = θ. In this situation, his
eﬀort strategy is β(θ, G) = θ2 − 2θ3 /3. If he knows the true distribution oftypes, he should follow the strategy β(θ, F ) = θ2 /2.
The above example is depicted in Figure 2. The horizontal axis is thetypes and the vertical axis is the eﬀort levels.
e12β(θ, G)β(θ, F )1θFigure 2:
Finally, we describe the aggregate eﬀect of underestimation of the type
distribution on the expected eﬀort level. In the following proposition, weshow that the necessary and suﬃcient condition for E(e, G) > E(e, F ), thatis, underestimation of the type distribution is proﬁtable for the principal.
Otherwise, underestimation decreases the agents’ expected eﬀorts and is not
proﬁtable for the principal.
Proposition 8. In the situations∫of underestimation of the type distribution,
1
E(e, G) > E(e, F ) if and only if 0 (1 − F (θ))θ(g(θ) − f (θ))dθ > 0.
Proof. This statement follows straightforwardly from Propositions 2 and5.
We provide two examples. The former shows that underestimation may
increase an agent’s expected eﬀort and the latter shows that underestimation
may decrease an agent’s expected eﬀort.
Example 3.4. Suppose that agent i’s prior information is (ai , G), whereai = θi , and that F (θ) = θ. In the correctly estimated case, E(e, F ) = 1/6.
• If G(θ) = (3θ − θ3 )/2, E(e, G) = 7/40 (> 1/6).
• If G(θ) = 3θ − 2θ3/2 , E(e, G) = 11/70 (< 1/6).
11

4

Concluding Remarks

In this paper, we have examined two diﬀerent sources of overconﬁdence
and have compared the behavioral consequences of each situation with thebenchmark case. The main result is that overestimation of one’s own ability
always induces aggressive behavior by the participants, whereas underestimation of the rival’s ability (or the type distribution) sometimes brings
about less aggressive behavior by one or both participants. This means that
overconﬁdence does not always induce aggressive behavior. This is quite
diﬀerent from the situations of one person’s decision-making.
To conclude the paper, three remarks are in order. First, for an agent,both overestimation of his own type and underestimation of the type distribution increase his subjective probability of winning the contest. For example, consider an agent with true type 1/2 in the situation where F (θ) = θ.
In the correctly estimated case, his probability of winning√is 1/2. However,in the situation of overestimationof his own type as ai = θi , his subjective√
probability of winning is 1/2, which is strictly larger than 1/2. In thesituation where he underestimates the type distribution as G(θ) = 2θ − θ2 ,his subjective probability of winning is 3/4, which is also strictly larger than
1/2. In both situations, he has overestimated the probability of winning thecontest. However, these two situations may yield diﬀerent consequences.
The former is always proﬁtable for the principal but the latter may not be.
Second, experiments on economic contests under a situation where participants are overconﬁdent may be interesting. However, they involve somediﬃculties. In the all-pay auction experiments without overconﬁdence, Noussair
and Silver (2003) observed over-bidding behavior by the subjects.14 So, if we
observe over-bidding (or under-bidding) in experiments with overconﬁdence,
we cannot conclude immediately that overconﬁdence yields over-bidding (orunder-bidding). Consequently, we should design experiments carefully toreach a proper conclusion.
Third, we have shown that diﬀerent sources of overconﬁdence have diﬀerent behavioral consequences. This type of conclusion is not only in situationsof contests or auctions. Consider Bertrand competition in a diﬀerentiatedduopoly. Since strategic complementarities exist, lower marginal costs of
one’s own ﬁrm (i.e., higher own relative ability) yield his aggressive behavior, whereas higher marginal costs of the rival ﬁrm (i.e., lower rival’s relative
ability) yield his less aggressive behavior.

14
Dorsey and Razzolini (2002) attempted to describe the source of over-bidding behaviorof bidders in ﬁrst-price auctions by auction experiments and compared choices under aﬁrst-price auction and an incentive-wise identical lottery. They concluded that the subjects
cannot calculate the probability of winning in auctions appropriately, and so over-bidding
occurs.

12References[1] Amann, E. and W. Leininger, 1996, “Asymmetric All-pay Auctions
with Incomplete Information: The Two-player Case,” Games and Economic Behavior, 14, 1-18.
[2] Ando, M., 2004, “Division of a Contest with Identical Prizes,” Journalof the Japanese and International Economies, 18, 282-297.
[3] Camerer, C., 1997, “Progress in Behavioral Game Theory,” Journal of
Economic Perspective, 11, 167-188.
[4] Camerer, C., 2003, Behavioral Game Theory: Experiments in Strategic
Interaction, Princeton University Press.
[5] Camerer, C. and D. Lovallo, 1999, “Overconﬁdence and Excess Entry:
An Experimental Approach,” American Economic Review, 89, 306-318.
[6] Che, Y.-K. and I. Gale, 2000, “Diﬀerence-form Contests and the Robustness of All-pay Auctions,” Games and Economic Behavior, 30, 2243.
[7] Dorsey, R. and L. Razzolini, 2002, “Explaining Overbidding in First
Price Auctions Using Controlled Lotteries,” Experimental Economics,
6, 123-140.
[8] Dubra, J., 2004, “Optimism and Overconﬁdence in Search,” Review of
Economic Dynamics, 7, 198-218.
[9] Itoh, H., 2004, “Moral Hazard and Other-regarding Preferences,”
Japanese Economic Review, 55, 18-45.
[10] Krishna, V. and J. Morgan, 1997, “An Analysis of the War of Attrition
and the All-pay Auction,” Journal of Economic Theory, 72, 343-362.
[11] Noussair, C. and J. Silver, 2003, “Behavior in All-pay Auctions with
Incomplete Information,” mimeograph, Emory University.
[12] Singh, N. and D. Wittman, 2001, “Contests where There is Variationin the Marginal Productivity of Eﬀort,” Economic Theory, 18, 711-744.
[13] Svenson, O., 1981, “Are We All Less Risky and More Skillful than Our
Fellow Drivers?” Acta Psychologica, 47, 143-148.

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