Enhancing Effort Supply with Prize-Augmenting Entry Fees:
Theory and Experiments∗
Robert G. Hammond†
Bin Liu‡
Jingfeng Lu§
Yohanes E. Riyanto¶
August 4, 2016
Abstract
Entry fees are widely observed in contests. We study the effect of a prize-augmenting entry
fee on expected total effort in an all-pay auction setting where the contestants’ abilities are
private information. While an entry fee reduces equilibrium entry, it can enhance the entrants’
effort supply. Our theoretical model demonstrates that, at optimum, the entry fee must be
strictly positive and finite. We design a laboratory experiment to empirically test the effect of
entry fees on effort supply. Specifically, we compare an optimally set entry fee to no entry fee
or a high entry fee. Our results provide strong support for the notion that a principal can elicit
higher effort using an appropriately set entry fee to augment the prize purse.
Keywords: All pay auction, Carrot and stick, Effort, Entry Fee, Prize allocation
JEL classification: C91, D44, J41, L23
∗
We acknowledge the Isaac Manasseh Meyer Fellowship, which funded Hammond’s visit to the National University
of Singapore, where part of the work on this paper was completed. We also thank seminar participants at the
International Industrial Organization Conference as well as Pia Weiss for comments and suggestions.
†
Department of Economics, North Carolina State University. Email: robert [email protected].
‡
Department of Economics, National University of Singapore. Email: [email protected].
§
Department of Economics, National University of Singapore. Email: [email protected].
¶
Division of Economics, Nanyang Technological University. Email: [email protected].
1
1
Introduction
Contests are observed in a wide range of environments such as labor markets, investments in
research and development, political campaigns, advertising, lobbying, conflict, and sports, etc. In
many situations, the goal of the contests is to elicit productive effort. For this purpose, the prize
allocation rule has long been recognized as a major instrument that the contest organizer can
use to enhance effort supply. In particular, the principle of “carrot and stick” has been long and
widely followed in practice as a guideline for effective prize and penalty design to provide the right
incentives to agents. In a contest setting, a positive prize can be viewed as a carrot, while a negative
prize can be treated as a stick. In this paper, we study contest design by allowing both positive
and negative prizes. Specifically, we study how a designer can use entry fee as an instrument to
create negative prizes for losers and further enhance the positive prize for the winner, which would
induce higher expected effort.
We consider an all-pay auction with incomplete information, where the bidders’ abilities are
their private information. The organizer can set an entry fee for each participant. The bidders
simultaneously decide whether to participate, and how to bid if they decide to enter without
observing the others’ entry and bidding decisions. Entry fees are added to the original prize purse,
which will be awarded as a single winner prize to the contestant who exerts the highest effort. All
losing participants win nothing but pay the entry fee, which is equivalent to a negative prize ex
post.
Bundling the sum of the revenue collected from awarding negative prizes into the reward paid
as a positive prize has a number of empirical analogs. Sporting contests often require monetary
payments to enter the contest and these entry fees are used to partly fund the prize paid to
the winner. Examples from sports include registration fees for marathons and “buy-ins” in poker
tournaments, both of which are required entry fees in order to compete that are then used as part of
the prize purse. Outside the realm of sports, contests in the creative industries often require prizeaugmenting entry fees, including contests giving awards for best works of writing, photography,
architecture, and design.1
1
Several contests explicitly mention that the entry fees are used to supplement the prize paid to the winner.
Fredriksson (1993) discusses rodeos where the “entrance fees were added to the prize money.” In horse racing,
https://www.toba.org/owner-education/entering-races.aspx says that the “sum of owners’ entry fees” are added to the
prize purse. Concerning design contests, http://www.victoriastrauss.com/advice/contests says that contests “charge
2
The effect of an entry fee on effort supply is two-fold. On one hand, it discourages the low
ability types from participating in the competition, which lowers their effort supply. On the other
hand, entry fees enlarge the winner’s prize, which induces more effort from higher types. The total
effect thus relies on the trade-off of these two effects. Our theoretical study shows that imposing
an entry fee indeed enhances the effort supply of agents if it is set appropriately. We establish that
the effort-maximizing entry fee must be strictly positive and finite. This result helps to explain
why entry fees are widely observed in practice.
We verify our theoretical predictions in a laboratory experiment. Our analysis focuses on the
expected total effort of the contestants, which in our theoretical context is equivalent to the revenue
raised by the contest designer. The experiment uses a 2x3 between-subject design, with treatment
variation in the level of the entry fee and the degree of contestant heterogeneity. The entry fee
E takes one of three values: no entry fee (E = 0), optimal entry fee (E = E ∗ ), and high entry
fee (E = 3E ∗ ). We also vary the bounds of the marginal cost distribution [c, c], which affects the
degree of heterogeneity among contestants. Using the uniform distribution, we consider two cases:
low heterogeneity [c, c] = [1, 2] and high heterogeneity [c, c] = [1, 3].
Our experimental results strongly support our theoretical predictions. We find that revenue is
the highest when the entry fee is set optimally and the prize is augmented with the total entry
fees collected from the entrants. With low contestant heterogeneity, the revenue is 11.56% higher
with the optimal entry fee than with no entry fee and 33.17% higher than with the high entry fee.
With high heterogeneity, the corresponding revenue increases are 10.88% and 50.56%, respectively.
When we consider the efficiency of the contest, with efficiency being defined as contests in which the
lowest marginal cost wins the contest, we find that a higher contestant heterogeneity leads to higher
efficiency rate. Finally, while theory predicts full entry into the contest with no entry fee, there is
less than full entry in these treatments. In the positive entry fee treatments, we also find less than
full entry by the contestants whose marginal cost is below the entry threshold (i.e., the high types)
contrary to the theoretical prediction. Interestingly, in the positive entry fee treatments, we have
excess entry by those contestants whose marginal cost is above the entry threshold (i.e., the low
types). This is in contrast to the theory, which predicts that these contestants would not enter the
a fee to fund the prize.” Finally, in the context of writing contests, https://www.freelancer.sg/feesandcharges says
that entry fees “will be used to increase the contest prize,” while http://thewritelife.com/27-free-writing-contests/
says that entry fees are used as a “way of [growing] the prize purse for each contest.”
3
contest.
We emphasize that entry fees play a different role in revenue-targeting auctions than they do
in effort-eliciting contests. The seller in a revenue-generating auction keeps the entry fees collected
as part of the revenue and the entry fee serves only as an instrument to control entry. Further,
the well-known revenue equivalence theorem says that two mechanisms generate the same expected
revenue given that they implement the same allocation and that the lowest type earns the same
expected payoff. According to the revenue equivalence theorem, if the minimum winning type (i.e.,
the lowest type that wins the auction with positive probability) is higher than the lower bound
of the value distribution, then a continuum of pairs of reserve price and entry fee would generate
the same expected revenue in an all-pay auction. In particular, there is an optimal entry fee that
would maximize revenue if there is no reserve price. However, zero entry fee is optimal if the lowest
winning type equals the lower bound of bidders’ values regardless of the reserve price.
In our paper, we study whether an entry fee would generate higher expected effort. Entry fees
augment the prize purse to induce more effort from the entrants of different types, in addition to
controlling entry. It is thus less clear that a prize-augmenting entry fee would induce higher effort
given that it discourages entry, especially for a type distribution that must entail zero optimal entry
fee in a revenue-generating auction. Nevertheless, we unambiguously establish that, regardless of
the type distribution, a non-zero entry fee must be optimal in an effort-eliciting all-pay contest
with incomplete information.
The paper is organized as follows. Section 2 briefly surveys the related literature. Section 3
presents our theoretical model and its predictions. We then describe our experimental design and
procedures in Section 4 and discuss our experimental results in Section 5. Section 6 concludes.
2
Related Literature
Our paper belongs to the literature on optimal contest design with incomplete information.
Optimal prize allocation has been studied in various all-pay auction frameworks starting from the
seminal work of Moldovanu and Sela (2001). They establish winner-take-all as the optimal prize
allocation rule when contestants’ effort function is linear or concave, while restricting prizes to
be non-negative. They also establish that convex effort cost can invalidate the optimality of the
4
winner-take-all. Minor (2013) maintains the assumption of non-negative prizes and studies the
optimal prize allocation rule when contestants have convex costs of effort or the contest designer
has concave benefit of effort. Moldovanu and Sela (2006) generalize their earlier investigation to
a two-stage all-pay auction framework. Meanwhile, Moldovanu, Sela, and Shi (2007) analyze the
environment where contestants care about their relative status. They further allow for negative
prizes in Moldovanu, Sela, and Shi (2012). In Moldovanu, Sela, and Shi (2012), a negative prize
is costly for the organizer to implement. In our paper, when negative prizes are implemented for
some contestants, the revenue collected, like in Fullerton and McAfee (1999), will be used by the
organizer to reward other contestants.
Thomas and Wang (2013) presents a model in which the highest bidding contestant wins a fixed
prize (V ) and the lowest bidding contestant pays a fine (P ), with P < V . The contest designer
sets the optimal level of fine P . Contestants must decide whether to participate in the contest.
If there is only one bidder who participates in the contest, she will win the prize, but she also
must pay the fine because her bid is both the highest and the lowest. Different from theirs, in our
setup, all contestants must pay an entry fee, regardless of whether their bid is the lowest bid. The
revenues collected from entry fees are then added onto the winning prize. Further, in our setup,
contestants are heterogeneous in their marginal bidding cost. We know from the literature that
greater cost heterogeneity discourages weaker contestants from competing (Schotter and Weigelt,
1992; Gradstein, 1995; Clark and Riis, 1998). This discouragement effect may reduce effort.
Counter to this negative effect, Moldovanu and Sela (2001) show that having multiple prizes
encourages weaker contestants to participate. However, this also comes at a cost of inducing
stronger contestants to lower their effort. Moldovanu and Sela (2001) show that, only when bidding
costs are sufficiently convex, awarding multiple prizes may be optimal. Anderson and Stafford
(2003) test and confirm this prediction in the lab. In their experimental design, contestants have
differing cost of effort and must pay an entry fee to participate in the contest. The show that higher
cost heterogeneity reduces the number of entrants and effort. Other studies have also produced
similar results showing that a single prize generates higher effort than multiple prizes (Vandegrift,
Yavas, and Brown, 2007; Sheremeta, 2011; Stracke, Höchtl, Kerschbamer, and Sunde, 2014; Cason,
Masters, and Sheremeta, 2010).
In our paper, we focus on the optimal design of entry fee in a setting where the total revenues
5
collected from entry fees would be used to boost the winning prize. The entry fee in our setup is
analogous to a punishment and reward scheme, where the punishment is imposed on all contestants
and the augmented reward is given to the winning contestant. Similar to the existing literature,
the presence of cost heterogeneity in our setup reduces weaker contestants’ incentive to compete
and induces stronger contestant to bid less aggressively. The presence of entry fee will reduce
the incentive of weaker contestants further. However, augmenting the prize incentivizes stronger
contestants to exert higher effort. We show that when entry fee is optimally set, the incentive effect
on stronger contestants outweighs the discouragement effect on weaker contestants.
Note that the entry fee in our setup essentially works as an endogenous participation constraint.
Instead of using entry fee to create the participation constraint, Megidish and Sela (2013) use the
minimal effort constraint in which a contestant is only allowed to participate if her effort exceeds the
minimal effort constraint. The minimal effort constraint works in a similar fashion as the entry fee
in our setup. Further, Megidish and Sela (2013) consider two alternative prize allocation schemes.
The first one is the winner-take-all scheme and the second one is the random allocation scheme.
With the random allocation scheme, all contestants who exert efforts that exceed the minimal
effort constraint has an equal chance to win the prize. They show that the random prize allocation
scheme induces greater participation from contestants. The increase in the number of participants
could offset the reduction in effort due to the discouragement effect. They show that the random
prize allocation scheme could potentially be superior than the winner-take-all allocation scheme.
In our paper, we design a lab experiment to test the predictions of our prize-augmenting entry
fee scheme on the total expected effort. The question is whether this scheme creates a strong enough
incentive for stronger contestants to exert high effort, such that it outweighs the loss from weaker
contestants who are less likely to enter the contest because of entry fee. Our paper thus contributes
to the growing experimental literature on contests; see Dechenaux, Kovenock, and Sheremeta (2014)
for the most comprehensive literature survey on experimental research on contests to date.
3
Theoretical Model with Incomplete Information
We consider an all-pay auction in which the player with the highest effort wins a single prize.
There are N (≥ 2) potential bidders whose marginal bidding costs are their private information.
6
Every bidder i’s marginal bidding cost ci is identically and independently distributed on interval
[c, c] with a cumulative distribution function F (·) and a density function f (·). Type c is thus the
most efficient type.
The auction has an initial fixed prize V , and the organizer can set an entry fee E that is collected
from entrants to augment the initial prize. Therefore, when there are n entrants, then the total
prize will be Vn = V + nE, n = 0, 1, 2, ..., N . Entrants make their bid simultaneously without
observing the number of actual bidders. The highest bid among n (≥ 1) entrants then wins the
single prize Vn . If there is no entrant, the organizer keeps the initial prize.
A positive entry fee E induces endogenous symmetric entry, which can be denoted by an entry
threshold ĉ ∈ [c, c]. Only the more efficient types with ci ≤ ĉ would pay the entry fee and participate
in the auction. Given monotonicity of the entrants’ symmetric bidding strategy, an entrant with
the threshold type (i.e., type ĉ) wins if and only if she is the only entrant. In this case, she wins
V1 = V + E with probability [1 − F (ĉ)]N −1 , and must pay the entry fee E. In addition, his
equilibrium bid must be b(ĉ) = 0 in an all pay auction. Therefore, we must have
[1 − F (ĉ)]N −1 [V + E] = E,
which leads to the following lemma.
Lemma 1. With entry fee E, the symmetric entry threshold is
ĉ = F −1 [1 − (
1
E
) N −1 ] ∈ (c, c], ∀E ≥ 0.
V +E
Note that ĉ decreases with E. In particular, when E = 0, we have ĉ = c, i.e., every type participates.
We next derive the monotone equilibrium bidding strategy b(c), ∀c ≤ ĉ. Consider a representative bidder, say bidder 1. Suppose the other bidders i = 2, 3, ..., N adopt the equilibrium entry
threshold ĉ, and hypothetical bidding strategy b(c). If bidder 1 enters and announces a type c0 ≤ ĉ
7
when his true type is c ≤ ĉ, then his expected payoff is
N
−1
X
π1 (c0 , c) =
n=0
N
−1
X
=
n
n
(N −1)−n
CN
[1 −
−1 F (ĉ) [1 − F (ĉ)]
F (c0 ) n
] Vn+1 − b(c0 )c − E
F (ĉ)
n
(N −1)−n
CN
[F (ĉ) − F (c0 )]n Vn+1 − b(c0 )c − E.
−1 [1 − F (ĉ)]
n=0
Together with the first order condition for optimal bid, truth telling (i.e., c0 = c at optimum)
in equilibrium implies
N −1
X
∂π1 (c0 , c)
n
(N −1)−n
0
|
=
−f
(c)
CN
n[F (ĉ) − F (c)]n−1 Vn+1 − b0 (c)c = 0.
c =c
−1 [1 − F (ĉ)]
∂c0
n=1
We thus have
N −1
f (c) X n
CN −1 [1 − F (ĉ)](N −1)−n n[F (ĉ) − F (c)]n−1 Vn+1 ,
b (c) = −
c
0
n=1
with boundary condition b(ĉ) = 0. We thus obtain the solution of b(c), as shown in the following
proposition.
Proposition 1. With entry fee E, the symmetric bidding strategy of entrants is
Z
ĉ(E)
b(c; E) =
c
N −1
f (t) X n
CN −1 [1 − F (ĉ(E))](N −1)−n n[F (ĉ(E)) − F (t)]n−1 Vn+1 dt, ∀c ≤ ĉ(E).
t
n=1
Using the entry equilibrium in Lemma 1 and bidding equilibrium in Proposition 1, we can derive
the equilibrium expected total effort as follows.
Corollary 1. With entry fee E, the equilibrium expected total effort is
Z
T E(E) = N
c
ĉ(E)
N −1
f (t) X n
CN −1 [1 − F (ĉ(E))](N −1)−n n[F (ĉ(E)) − F (t)]n−1 Vn+1 (E)F (t)dt,
t
n=1
1
N −1 ].
where ĉ(E) = F −1 [1 − ( V E
+E )
Proof. See Appendix A.
The focus of this paper is how a prize-augmenting entry fee can enhance effort supply in an all
8
pay auction. With the above results, we are ready to carry out this analysis.
Proposition 2. The optimal entry fee must be positive.
Proof. See Appendix A.
The following proposition further shows that the optimal level of entry fee is finite.
Proposition 3. The optimal entry fee must be finite.
Proof. See Appendix A.
Now we use N = 2 to illustrate the effect of the entry fee on the bidding function and the
expected total effort elicited. Assume that the marginal cost of exerting effort ci follows the
uniform distribution with support [1, 2], so that F (t) = t − 1 and f (t) = 1, t ∈ [1, 2]. For simplicity,
let V = 1. Then according to Proposition 1, the bidding function is
Z
b(c; E) = (1 + 2E)
c
ĉ(E)
1
dt = (1 + 2E)(ln ĉ(E) − ln c), ∀c ≤ ĉ(E),
t
where
ĉ(E) =
2+E
.
1+E
Further, by Corollary 1, the expected total effort is
Z
T E(E) = 2(1 + 2E)
1
ĉ(E)
t−1
dt
t
= 2(1 + 2E)(ĉ(E) − 1 − ln ĉ(E))
= 2(1 + 2E)(
1
2+E
− ln
).
1+E
1+E
Notice that
∂ ∂b(c; E)
2
(
)=− ,
∂E
∂c
c
which implies that the absolute value of the slope of the bidding function is increasing in the level
of entry fee given that c ≤ ĉ(E). This leads to the central trade-off of introducing entry fee: on the
one hand, the introduction of entry fee enlarges the grand prize awarded to the most able entrant,
so that the entrant bids more aggressively when entry fee becomes larger; on the other hand, higher
9
entry fee dampens the entry incentive of the high cost contestants, so that ex ante the expected
number of entrants is smaller. Since the expected total effort depends on both the bidding behavior
and the entry incentive, these two conflicting effects are the trade-off the organizer faces. According
to Proposition 2 and 3, when the entry fee is small enough, the former effect dominates the latter
one; while when the entry fee is large enough, the latter effect dominates. The goal is to balance
these two effects.
Figure 1 illustrates how the bidding functions vary with the magnitude of entry fee.
[INSERT FIGURE 1 HERE]
It draws the bidding function when E = 0, 0.5, and 1, respectively. Notice that the absolute
value of the slope of the bidding function is increasing in the magnitude of entry fee. Though higher
entry fee induces more aggressive bidding from the entrants, it also discourages entry. Figure 2
further illustrates how the total effort elicited varies with the level of entry fee.
[INSERT FIGURE 2 HERE]
As can be seen from the above figure, the total effort first rises when entry fee increases, and
then it descends when the entry fee further increases. The optimal entry fee E ∗ ≈ 0.242.
4
Experimental Design
In this section we outline our experimental design and procedure. The experiment has N = 2
potential contestants. We set different possible parameter values for the bound of the marginal
cost distribution of the contestants [c, c] and the prize value V , then simulated the models to derive
the entry threshold b
c, optimal bid b (b
c), expected total effort T E, and optimal entry fee E ∗ that
maximizes the expected total effort T E. We assumed that the marginal cost of the contestants is
drawn from a uniform distribution over [c, c].
4.1
The Experimental Design
Based on the simulation results, we chose parameter values for our experiment that would be
sufficiently intuitive to implement in our experiment. To this end, we set the prize value V = 100.
10
We interpreted the bound of the marginal cost of the contestants [c, c] as the degree of heterogeneity
of the contestants. The marginal cost is private information of the contestants. We considered two
bounds of [c, c] which we coined as the low and the high contestant heterogeneity. In the former
we set [c, c] = [1, 2] and in the latter we set [c, c] = [1, 3]. Under the low contestant heterogeneity,
the entry threshold b
cL can be derived as
b
cL =
200 + 100E
,
100 + E
while under the high contestant heterogeneity the entry threshold b
cH is
b
cH =
300 + 100E
.
100 + E
Contestants with the marginal cost higher than the entry threshold would not enter the contest.
Given the entry thresholds b
cL and b
cH , we can solve for the equilibrium bid b (b
c) as respectively
b (b
cL ) = (V + 2E) ln
200 + 100E
100 + E
− ln c , ∀c ≤ b
cL ,
300 + 100E
b (b
cH ) = (V + 3E) ln
− ln c , ∀c ≤ b
cH .
100 + E
Next, we derived the expected total effort T E, which would be a function of the entry fee E. The
optimal entry fee E ∗ that maximizes T E can then be calculated. They are respectively
EL∗ = 24.170,
∗
EH
= 40.393.
In addition to varying the degree of contestant heterogeneity, we also varied the level of entry
fee E. Our baseline treatment is a treatment where the optimal entry fee is set optimally at
E = E ∗ . We benchmarked this baseline treatment against the other two treatments with no entry
fee (E = 0) and the high entry fee (E = 3E ∗ ). Relative to E = E ∗ , both E = 0 and E = 3E ∗
should generate a lower T E.
All in all, we have 2x3 treatment variations. The following table summarizes our treatment
11
variations. In each treatment cell, we present the entry threshold (b
c), the optimal bid (b∗ ) and the
expected total effort (T E). It can be seen that, conditional on the bound of the marginal cost, the
entry threshold (b
c) is decreasing in the level of entry fee (E).
[INSERT TABLE 1 HERE]
We employed a between-subject experimental design. In our analysis, we divide these treatments into two comparison groups. Treatments 1 (Zero Entry Fee, Low Heterogeneity), 2 (Optimal
Entry Fee, Low Heterogeneity), and 3 (High Entry Fee, Low Heterogeneity) would form the first
comparison group. These treatments are characterized by the presence of a low degree of contestant heterogeneity. The second comparison group consists of Treatments 4 (Zero Entry Fee, High
Heterogeneity), 5 (Optimal Entry Fee, High Heterogeneity), and 6 (High Entry Fee, High Heterogeneity); all of them have a high degree of contestant heterogeneity. Our main goal is to validate
our theoretical result showing the optimality of augmenting the contest prize with the entry fee
collected from the contestants. As was mentioned previously, augmenting the contest prize with the
entry fee is equivalent to imposing ex-post punishment on the losing contestant and giving ex-post
reward to the winning contestant.
Based on our theoretical model, we should expect that there will be full entry into the contest
when entry fee is absent (E = 0). However, in the presence of entry fee, only contestants with the
marginal cost lower than the entry threshold would participate in the contest. The probability of
participation should be decreasing in the magnitude of entry fee and in our experiment it will be
the lowest when E = 3E ∗ . The contest designer’s goal is to maximize the contest revenue and in
our context, maximizing the contest revenue is equivalent to maximizing the expected total effort
(T E) exerted by the contestants.
4.2
The Experimental Procedure
Our subjects were recruited from the population of undergraduate students at Nanyang Technological University (NTU) in Singapore. They came from various majors such as science, engineering,
business, economics, arts and humanities, and social sciences. The recruitment information was
distributed through the university-level mass email system. We had in total 144 subjects. The
12
number of subjects in each treatment varied. For each treatment, with the exception of Treatment
5, we conducted 1 experimental session. For Treatment 5,we conducted two experimental sessions
because the first session had only 14 subjects. The second session had 22 more subjects. No subjects participated in more than one treatment. Table 2 below gives a summary of the number of
subjects and the gender composition of the subjects in each treatment.
[INSERT TABLE 2 HERE]
Our experiment was programmed using z-Tree (Fischbacher, 2007). At the beginning of each
session, the subjects were given a random six digit numerical user ID as their identifier throughout
the experiment. The subjects were randomly rematched in every period. They went through 30
periods of interactions conducted using the computer interface, but they were not told in advance
the total number of periods they would have to go through to minimize the end-game effect. The
duration of the experiment was approximately 2 hours.
At the beginning of the experiment, written instructions were distributed to the subjects and
were read aloud by the experimenter to ensure that the subjects had common knowledge about the
experiment.2 Subsequently, the subjects were then given 2 trial periods to gain familiarity with
the experimental interface and to ensure that they understood the experimental instructions. They
were told that the payoff from these trial periods would be excluded from the determination of
their final earnings.
At the end of the experiment, 5 periods were randomly selected out of 30 periods and their final
earnings were the sum of the payoffs they earned in these 5 periods. The payoffs were denominated
in experimental currency units (ECUs). To cover the bidding cost, the subjects were provided with
a checkbook of 200 ECU at the beginning of every period. Each subject’s payoff from a period will
be her checkbook amount in that period minus the bidding cost incurred and the winning prize
obtained if she wins the contest in that period. Notice that a subject can earn 200 ECU in every
period (1000 ECU in total). To incentivize subjects to participate in the contest rather than to
sit out and simply earn 1000 ECU, we adopted a two-part payment-conversion scheme. That is,
2
The complete experiment instructions are available in Appendix B.
13
when the final earnings was less than 1000 ECU, the subjects earned S$ 1 for every 150 ECU. For
earnings above 1000 ECU, the subjects earned S$ 1 for every 25 ECU.
After the contest game had been completed, the subjects were required to answer a riskelicitation survey using the multiple price-list procedure of Holt and Laury (2002). On average
the subjects earned between 20 ECU to 30 ECU from the Holt-Laury procedure. Subsequently,
subjects were given 10 quantitative questions to measure their quantitative aptitude. They earned
5 ECU for every correct answer and 0 ECU for every wrong (or blank) answer. On average the
subjects earned 25 ECU from this part. Following the quantitative questions, subjects were asked
to answer a post-experiment questionnaire. The conversion rate for the Holt-Laury elicitation and
the quantitative task was S$ 1 for every 10 ECU.
The subjects were also given a show-up fee of S$ 2. However, when the total earnings of a
subject was less than S$ 8 (around US$ 6), we gave the subject an additional S$ 2 lump-sum fee
at the end of the experiment. We did not, however, tell subjects about this at the beginning of the
experiment to avoid any perverse incentive effect. This additional show-up fee was given to make
the payment to this low earning subject commensurate with the 2 hours spent in our experiment.3
This was done so as not to discourage these subjects from participating in other future experiments
at the lab. After considering all payment components, the average total earnings per subject was
around S$ 21 (around US$ 16).
5
Results from the Lab
Our main concern is the revenue raised by the principal. As the contest structure augments the
prize purse with the entry fees paid by entrants, revenue is equal to the sum of all bids entered.
The analysis first considers aggregate results at the treatment level, then moves to results at the
level of an individual subject.
5.1
Aggregate Results
The first column of Table 3 as well as Figures 3 and 5 provide several looks at the revenue
earned by the principal in each treatment. Table 3 shows treatment-level means, while the figures
3
The payment for 1 hour part-time job in Singapore is around S$ 8.
14
plot CDFs of revenue. Results are shown separately for treatments with low and high contestant
heterogeneity, that is, marginal costs that either vary between one and two or between one and
three. In each set of results, we ask whether the optimal entry fee induces more total effort (i.e.,
higher bids) than either no entry fee or an entry fee that is three times larger than the optimal entry
fee. Given that we expect (and find) substantial overbidding that is ameliorated but not eliminated
by learning, we also repeat the analysis for the final 10 periods out of a total of 30 periods. Panel
B of Table 3 as well as Figures 4 and 6 show the results that account for learning.
[INSERT FIGURE 3 HERE]
[INSERT FIGURE 4 HERE]
[INSERT FIGURE 5 HERE]
[INSERT FIGURE 6 HERE]
[INSERT TABLE 3 HERE]
Overall, the findings strongly support our hypotheses concerning the optimality of a properly
set entry fee. From Column (1) of Table 3’s Panel A, with low contestant heterogeneity, revenue is
11.56% higher with an optimal entry fee than with no entry fee and 33.17% higher than with a high
entry fee. With high contestant heterogeneity, the corresponding increases are 10.88% and 50.56%.
All differences are quantitatively large and statistically meaningful.4 Moving to Panel B, with low
contestant heterogeneity, the increased revenue raised from an optimally set entry fee is smaller
when only the final 10 periods are considered relative to no entry fee but the increase is larger
relative to a high entry fee. The opposite changes are seen with high contestant heterogeneity.
Overall though, the only meaningful difference once subject learning is accounted for is in the
comparison of the optimal entry fee and no entry fee when contestant heterogeneity is low, where
the difference is statistically insignificant when only considering the final 10 periods (z-statistic =
0.57, p-value = 0.57).
However, Figures 3 and 4 jointly suggest that, with low contestant heterogeneity, an optimally
set entry fee has a meaningful revenue advantage over no entry fee, with the primary difference
4
We test for statistically significant differences using a Wilcoxon-Mann-Whitney rank-sum test. The corresponding
test results for the above comparisons are as follows: z-statistics of 2.99, 5.53, 2.53, and 8.41 with p-values of 0.00,
0.00, 0.01 and 0.00, respectively.
15
between the results from all 30 periods and the final 10 periods coming in terms of probability of
entry. Broadly, Figures 3 and 4 show that the optimal entry fee is raising more revenue than the
other treatments, with a substantial portion of contests raising more revenue than the prize purse
itself of 100 points. Note that all pairwise comparisons of the CDFs reject distributional equality
using a Kolmogorov-Smirnov test and a 10% significance level; specifically, in the final 10 periods,
testing the CDF of revenue from the optimal entry fee against the CDF of revenue from zero entry
fee is weakly statistically significant (D-statistic = 0.16, p-value = 0.10). Further, in Figures 5
and 6, we find that the revenue dominance of the optimally set entry fee is larger when contestant
heterogeneity is high than when it is low and the dominance remains large when considering only
the final 10 periods. As with low contestant heterogeneity, pairwise Kolmogorov-Smirnov tests
reject that the revenue distributions are equal.
Moving to the other outcomes of interest in Table 3, Column (2) provides the treatment-level
mean efficiency rate of the contests, where a contest is efficient if the contestant with the lowest
marginal cost wins. There are conflicting patterns regarding efficiency as the entry fee increases
when considering all 30 periods but the differences are much smaller when only the final 10 periods
are used in the analysis. Both with low and high contestant heterogeneity, the optimal entry fee
treatment has the lowest efficiency rate but the differences are small relative to the standard error.
As a result, we do not infer much about how efficiency varies with the entry fee, beyond noting
that high contestant heterogeneity increases efficiency.
Finally, Columns (3) and (4) consider entry into the contests, where Column (3) shows the
mean number of entrants and Column (4) shows the difference between this mean and the number
of entrants predicted by our theoretical model, that is, the number of excess entrants. Consistent
with our intuition, for a given level of contestant heterogeneity, entry decreases as the entry fee
increases. There is too little entry with no entry fee because theory predicts full entry, whereas
our results suggest that a small number of contestants choose to stay out even in these treatments.
Further, there is excess entry in all positive entry fee treatments, a finding that is stronger for a
high entry fee and for the final 10 periods.
16
5.2
Individual-Level Results
Next, we estimate two sets of regressions at the individual level and, as before, repeat the
analysis only considering the final 10 periods to account for subject learning. The first set of
results from this analysis is in Table 4, which presents conditional averages of bids and entry
probabilities for each treatment. The bid estimation is an OLS regression that includes only those
contestants who entered the contest, while the entry estimation is a Probit regression of whether the
contestant entered. Standard errors are clustered at the subject-level to account for dependencies
at the subject level.5 The results shown in Table 4 are conditional averages in the sense that the
regressions control for a host of subject characteristics as well as the treatment in which the subject
participated. We calculate post-estimation conditional means to present the average bid and entry
probability for each treatment, holding observable characteristics constant.
[INSERT TABLE 4 HERE]
Column (1) of Table 4 shows that, conditional on entry, contestants bid meaningfully more when
the entry fee is set optimally than when it is zero or when it is high. This result holds in both low and
high contestant heterogeneity and whether all periods or only the final 10 periods are considered.
All pairwise comparisons are statistically significant and the differences are large. Column (2) of
Table 4 considers whether a contestant enters the contest and provides the intuitive confirmation
that entry falls sharply when the entry fee increases. Again, all differences are large and statistically
significant (each at the 1% significance level, with the exception of the comparison of the optimal
and high entry fees with the final 10 periods, which is significant at the 10% significance level).
Finally, Table 5 presents the full set of individual-level regressions of a contestant’s bid and
entry probability, which are the same regressions that generated the conditional means discussed
above. Here the treatment-level differences are shown relative to the zero entry fee treatment with
low contestant heterogeneity. Further, we show the effects of the contestant’s marginal cost in
that period, the period, and several subject characteristics. We investigate differences in bidding
strategies among the following subject-level variables: probability test score (percent of questions
5
An alternative specification uses subject-level random effects within a panel-data regression model, that is, paneldata random-effects linear regression for bids and panel-data random-effects Probit regression for entry. These results
are shown in Appendix C and are very similar to what is discussed below. See Tables B1 and B2.
17
correct on the post-experiment probability questionnaire), whether the subject has participated in
previous economics experiments, whether the subject has previously taken a game theory course,
whether the subject is risk averse, indicators for the subject’s year of study, indicators for the
subject’s course of study, and whether the subject is female.6 Continuous variables (marginal cost,
period, and probability test score) are each included along with a quadratic term but marginal
effects at the mean are shown. For the estimation of the probability of entry, marginal effects of
dummy variables are shown.
[INSERT TABLE 5 HERE]
Having already discussed the differences across treatments, we turn to the results on contest and
subject characteristics. First, a marginal cost that is one unit higher reduces bids by more than 40
points and reduces entry by around 25 percentage points. Second, the effect of the period variable
indicates that subjects learn to lower their bids to a moderate degree and to enter at a lower rate to a
very small degree. For subject characteristics, demonstrating a better understanding of probability
is correlated with lower bids and less entry, suggesting that some part of the overbidding we observe
is driven by cognitive differences among subjects. Previous experimental experience and knowledge
of game theory both lower entry probabilities but only previous experiments affect bids and only in
the final 10 periods. Next, risk averse subjects bid more and enter at a lower rate, consistent with
results from first-price winner-pays auction experiments. Age does not have a meaningful affect on
bids or entry probabilities. Concerning a subject’s course of study, our results suggest a ranking of
bids from lowest to highest on average as follows: business students, engineering, liberal arts (the
omitted group), and science. Course of study affects entry probabilities as well, where business and
engineering students enter more frequently. Finally, there is strong evidence that female subjects
bid more and weak evidence that they enter at a higher rate. Concerning bidding behavior, several
authors (including Chen, Katušcák, and Ozdenoren (2013) and Schipper (2014)) have found gender
differences in bidding behavior in environments without endogenous entry. Our result on gender
differences in bidding is consistent with the general pattern: female bidders bid in a way that results
in lower profits for female bidders than male bidders.
6
Based on the multiple-price list risk elicitation of Holt and Laury (2002), a subject is considered risk averse who
switches from the safe lottery to the risky lottery later than the fifth of ten rows.
18
6
Conclusions
Contests are an important class of performance evaluation mechanisms. We introduce a new
contest format and consider its theoretical and empirical properties. Contestants whose abilities
are their private information can choose to pay an entry fee to participate in a contest where the
initial prize is augmented with the sum of entrants’ entry fees. Our theoretical analysis shows that
the principal can induce higher effort by setting a positive and finite entry fee. The optimal entry
fee balances the entry-discouraging effect of an entry fee with the incentive effect of a larger prize
(due to the prize-augmenting entry fee). Our experimental results confirm the revenue advantage
of the optimal entry fee relative to no entry fee or a high entry fee. These results are consistent
with observed practice in the sense that entry fees are widely observed in contests in the field.
We conclude that contest designers find entry fees to be an efficient approach for increasing effort
provision in environments such as sporting competitions (e.g., marathons and poker tournaments)
and contests in the creative industries (e.g., writing, photography, architecture, and design).
19
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21
Figure 1: Bidding Functions as a Function of the Entry Fee
Figure 2: Total Effort as a Function of the Entry Fee
22
1
Cumulative Probability
.4
.6
.8
.2
0
0
100
200
300
Sum of Bids
Zero Entry Fee
Optimal Entry Fee
High Entry Fee
0
.2
Cumulative Probability
.4
.6
.8
1
Figure 3: CDFs of Revenue in Low Heterogeneity Treatments, All 30 Periods
0
50
Zero Entry Fee
100
Sum of Bids
150
Optimal Entry Fee
200
High Entry Fee
Figure 4: CDFs of Revenue in Low Heterogeneity Treatments, Final 10 Periods
23
1
Cumulative Probability
.4
.6
.8
.2
0
0
50
100
150
Sum of Bids
Zero Entry Fee
200
Optimal Entry Fee
250
High Entry Fee
0
.2
Cumulative Probability
.4
.6
.8
1
Figure 5: CDFs of Revenue in High Heterogeneity Treatments, All 30 Periods
0
50
Zero Entry Fee
100
Sum of Bids
Optimal Entry Fee
150
200
High Entry Fee
Figure 6: CDFs of Revenue in High Heterogeneity Treatments, Final 10 Periods
24
Table 1: Summary of Experimental Treatments
Entry Fee
Zero Entry Fee
Optimal Entry Fee
(E = 0)
(E = E ∗ )
Treatment 1
Treatment 2
High Entry Fee
(E = 3E ∗ )
Treatment 1
Low Heterogeneity
[c, c] = [1, 2]
E=0
b
c=2
b∗ = 30.6853
T E = 61.3706
Treatment 3
E = 24.1704
b
c = 1.8053
b∗ = 31.8329
T E = 63.6658
Treatment 4
E = 72.5112
b
c = 1.5797
b∗ = 30.0042
T E = 60.0084
Treatment 5
High Heterogeneity
[c, c] = [1, 3]
E=0
b
c=3
b∗ = 22.5347
T E = 45.0694
E = 40.3925
b
c = 2.4246
b∗ = 24.3572
T E = 48.7144
E = 121.1775
b
c = 1.9043
b∗ = 22.2670
T E = 44.5340
Notes: Column (1) denotes the two (2) variations in the range of the marginal biding cost, which
we call low and high cost heterogeneity, respectively. Row (1) denotes the three (3) variations in
the entry fees, namely zero entry fee, optimal entry fee, and high entry fee. In each cell shown in
the table, we provide the magnitude of entry fee (E), the entry threshold (b
c), the equilibrium bid
∗
(b ), and the total effort (T E).
25
26
(1)
22
59%
Notes: Shown are the number of subjects and the proportion of female subjects in each treatment: (1) zero entry fee and low heterogeneity,
(2) optimal entry fee and low heterogeneity, (3) high entry fee and low heterogeneity, (4) zero entry fee and high heterogeneity, (5) optimal
entry fee and high heterogeneity, and (6) high entry fee and high heterogeneity.
No. of Subjects
% of Female
Table 2: Experimental Subjects, Including the Gender Composition
Treatments
Low Heterogeneity
High Heterogeneity
[c, c] = [1, 2]
[c, c] = [1, 3]
(2)
(3)
(3)
(4)
(5)
20
22
22
36
22
35%
36%
55%
50%
59%
Table 3: Treatment-Level Summary Statistics
Panel A: All 30 Periods
Zero Entry Fee, LowHet
Optimal Entry Fee, LowHet
High Entry Fee, LowHet
Zero Entry Fee, HighHet
Optimal Entry Fee, HighHet
High Entry Fee, HighHet
(1)
Sum of Bids
(2)
Efficiency
(3)
Entrants
(4)
Excess Entrants
94.848
(3.166)
107.375
(3.350)
79.479
(3.249)
73.459
(2.309)
83.392
(2.224)
55.900
(2.040)
0.594
(0.027)
0.677
(0.027)
0.670
(0.026)
0.715
(0.025)
0.663
(0.020)
0.779
(0.023)
1.930
(0.015)
1.637
(0.031)
1.282
(0.036)
1.803
(0.023)
1.561
(0.026)
1.412
(0.036)
-0.070
(0.015)
0.017
(0.037)
0.197
(0.043)
-0.197
(0.023)
0.107
(0.033)
0.512
(0.039)
(1)
Sum of Bids
(2)
Efficiency
(3)
Entrants
(4)
Excess Entrants
87.159
(4.908)
90.646
(6.018)
60.533
(5.089)
61.330
(3.346)
79.392
(3.797)
56.472
(3.702)
0.664
(0.045)
0.630
(0.049)
0.645
(0.046)
0.745
(0.042)
0.706
(0.034)
0.736
(0.042)
1.973
(0.016)
1.650
(0.052)
1.327
(0.059)
1.782
(0.040)
1.528
(0.046)
1.409
(0.064)
-0.027
(0.016)
0.100
(0.064)
0.245
(0.074)
-0.218
(0.040)
0.156
(0.058)
0.536
(0.072)
Panel B: Final 10 Periods
Zero Entry Fee, LowHet
Optimal Entry Fee, LowHet
High Entry Fee, LowHet
Zero Entry Fee, HighHet
Optimal Entry Fee, HighHet
High Entry Fee, HighHet
Notes: Column (1) shows the mean revenue raised in the contest, that is, the sum of the bids
entered. Column (2) shows the proportion of contests in which the lowest cost type wins the
contest. Column (3) shows the number of entrants, out of two, to the contest. Column (4) shows
the number of entrants minus the number of contestants who are predicted by theory to enter. For
this and subsequent tables, standard errors are shown in parentheses.
27
Table 4: Conditional Means of Bids and Probability of Entry for Each Treatment
Panel A: All 30 Periods
Zero Entry Fee, LowHet
Optimal Entry Fee, LowHet
High Entry Fee, LowHet
Zero Entry Fee, HighHet
Optimal Entry Fee, HighHet
High Entry Fee, HighHet
(1)
Bid if Enter
(2)
Pr(Entry)
49.136
(5.025)
65.606
(3.432)
62.005
(3.553)
40.742
(3.744)
53.419
(2.616)
39.586
(2.044)
0.965
(0.009)
0.822
(0.028)
0.642
(0.047)
0.898
(0.030)
0.776
(0.031)
0.708
(0.033)
(1)
Bid if Enter
(2)
Pr(Entry)
44.182
(5.699)
54.937
(4.776)
45.607
(5.443)
34.420
(3.722)
51.966
(3.198)
40.077
(2.605)
0.986
(0.007)
0.828
(0.034)
0.662
(0.060)
0.886
(0.040)
0.765
(0.038)
0.708
(0.045)
Panel B: Final 10 Periods
Zero Entry Fee, LowHet
Optimal Entry Fee, LowHet
High Entry Fee, LowHet
Zero Entry Fee, HighHet
Optimal Entry Fee, HighHet
High Entry Fee, HighHet
Notes: For each treatment, column (1) shows the mean bid, conditional on entering the contest, and
column (2) shows the mean probability of entering the contest. These results are post-estimation
conditional means, which present the average bid and entry probability for each treatment, holding
observable characteristics constant. The full regression specifications are shown in Table 5.
28
Table 5: Individual-Level Regression Results for Bids and Entry
All 30 Periods
Optimal Entry Fee, LowHet
High Entry Fee, LowHet
Zero Entry Fee, HighHet
Optimal Entry Fee, HighHet
High Entry Fee, HighHet
Marginal Cost
Period
Probability Test Score
Past Experiments
Had Game Theory
Risk Averse Subject
Subject’s Age
Engineering Student
Science Student
Business Student
Female Subject
Observations
Log Likelihood
Final 10 Periods
(1)
Bid if Enter
(2)
Pr(Entry)
(3)
Bid if Enter
(4)
Pr(Entry)
13.572
(6.047)∗∗
12.056
(5.943)∗∗
2.930
(6.595)
16.582
(5.890)∗∗∗
-0.729
(6.172)
-42.210
(1.855)∗∗∗
-0.572
(0.098)∗∗∗
-17.844
(5.975)∗∗∗
-4.575
(3.053)
-0.971
(2.994)
2.227
(3.394)
-1.093
(1.264)
-2.359
(4.098)
5.781
(3.993)
-10.825
(5.453)∗∗
7.145
(3.328)∗∗
-0.202
(0.043)∗∗∗
-0.360
(0.053)∗∗∗
0.024
(0.032)
-0.073
(0.035)∗∗
-0.080
(0.041)∗
-0.267
(0.019)∗∗∗
-0.000
(0.001)
-0.062
(0.077)
-0.065
(0.032)∗∗
-0.070
(0.030)∗∗
-0.056
(0.030)∗
0.005
(0.011)
0.051
(0.034)
0.001
(0.036)
0.038
(0.042)
0.023
(0.031)
11.001
(7.422)
3.921
(7.616)
5.677
(7.537)
21.696
(6.807)∗∗∗
8.967
(7.472)
-44.471
(2.552)∗∗∗
-0.342
(0.185)∗
-20.718
(7.447)∗∗∗
-9.201
(4.244)∗∗
0.796
(3.503)
-1.441
(3.959)
-0.753
(1.276)
-1.077
(5.105)
7.123
(5.214)
-12.217
(6.195)∗
7.533
(3.859)∗
-0.225
(0.051)∗∗∗
-0.379
(0.074)∗∗∗
-0.031
(0.039)
-0.133
(0.040)∗∗∗
-0.127
(0.049)∗∗∗
-0.248
(0.023)∗∗∗
0.002
(0.003)
0.037
(0.096)
-0.040
(0.040)
-0.074
(0.043)∗
-0.074
(0.036)∗∗
0.011
(0.015)
0.075
(0.043)∗
-0.023
(0.048)
0.056
(0.056)
0.062
(0.039)
3455
-16,234.045
4320
-1,666.874
1154
-5,379.239
1440
-541.047
Notes: Column (1) displays marginal effects from an OLS regression of the contestant’s bid, conditional on entering the contest. Column (2) displays marginal effects from a Probit regression of
whether the contestant entered the contest. ∗, ∗∗, and ∗ ∗ ∗ denote significance at the 10%, 5%,
and 1% level, respectively.
29
Appendix A
Proofs
Proof of Corollary 1. The expected total effort as a function of entry fee E is
T E(E)
Z c
b(c; E)f (c)dc
= N
c
Z
ĉ(E) Z ĉ(E)
= N
c
Z
c
ĉ(E) Z
ĉ(E)
= N
c
= N
f (t)
t
n=1
N
−1
X
n
(N −1)−n
CN
n[F (ĉ(E)) − F (t)]n−1 Vn+1 (E)f (c)dtdc
−1 [1 − F (ĉ(E))]
n=1
N
−1
t
f (t) X n
CN −1 [1 − F (ĉ(E))](N −1)−n n[F (ĉ(E)) − F (t)]n−1 Vn+1 (E)f (c)dcdt
t
c
c
n=1
Z ĉ(E)
N
−1
f (t) X n
CN −1 [1 − F (ĉ(E))](N −1)−n n[F (ĉ(E)) − F (t)]n−1 Vn+1 (E)F (t)dt.
t
c
n=1
Z
= N
c
N −1
f (t) X n
CN −1 [1 − F (ĉ(E))](N −1)−n n[F (ĉ(E)) − F (t)]n−1 Vn+1 (E)dtf (c)dc
t
ĉ(E) Z
Proof of Proposition 2. From T E(E) in Corollary 1, we have
T E 0 (E)
N
Z ĉ(E)
N −1
f (t) X n
CN −1 [1 − F (ĉ(E))](N −1)−n n(n + 1)[F (ĉ(E)) − F (t)]n−1 F (t)dt
=
t
c
n=1
N
−1
X
n f (ĉ(E))
n
(N −1)−n
+
CN
n[F (ĉ(E)) − F (ĉ(E))]n−1 Vn+1 (E)F (ĉ(E))
−1 [1 − F (ĉ(E))]
ĉ(E)
n=1
Z ĉ(E)
N −1
f (t) X n
−
CN −1 (N − 1 − n)[1 − F (ĉ(E))]N −n−2 f (ĉ(E))n[F (ĉ(E)) − F (t)]n−1 Vn+1 (E)F (t)dt
t
c
n=1
Z ĉ(E)
N
−1
o
f (t) X n
+
CN −1 (n − 1)[1 − F (ĉ(E))]N −n−1 f (ĉ(E))n[F (ĉ(E)) − F (t)]n−2 Vn+1 (E)F (t)dt ĉ0 (E).
t
c
n=1
It suffices to show that the limit of the above expression is strictly positive when E → 0. Because
the first integral is strictly positive, showing that that limit of the sum of other terms is positive
30
when E → 0 completes the proof. These terms are
=
β(E)
−1
n f (ĉ(E)) NX
ĉ(E)
Z ĉ(E)
n=1
N
−1
X
f (t)
t
−
c
Z
n
(N −1)−n
CN
n[F (ĉ(E)) − F (ĉ(E))]n−1 Vn+1 (E)F (ĉ(E))
−1 [1 − F (ĉ(E))]
ĉ(E)
f (t)
t
+
c
n=1
N
−1
X
n
N −n−2
CN
f (ĉ(E))n[F (ĉ(E)) − F (t)]n−1 Vn+1 (E)F (t)dt
−1 (N − 1 − n)[1 − F (ĉ(E))]
o
n
N −n−1
n−2
CN
(n
−
1)[1
−
F
(ĉ(E))]
f
(ĉ(E))n[F
(ĉ(E))
−
F
(t)]
V
(E)F
(t)dt
ĉ0 (E).
n+1
−1
n=1
Recall that Vn+1 (E) = V + nE. We thus have
β(E)
N −1
f (ĉ(E)) X n
CN −1 [1 − F (ĉ(E))](N −1)−n n[F (ĉ(E)) − F (ĉ(E))]n−1 [V + (n + 1)E]F (ĉ(E))ĉ0 (E)
=
ĉ(E)
n=1
−1
n Z ĉ(E) f (t) NX
n
N −n−1
+
CN
(ĉ(E))n[F (ĉ(E)) − F (t)]n−2 F (t)dt
−1 (n − 1)[1 − F (ĉ(E))]
t
c
n=1
Z ĉ(E)
N
−1
o
X
f (t)
n
N −n−2
n−1
CN
(N
−
1
−
n)[1
−
F
(ĉ(E))]
f
(ĉ(E))n[F
(ĉ(E))
−
F
(t)]
F
(t)dt
V ĉ0 (E)
−
−1
t
c
n=1
Z ĉ(E)
N
−1
f (t) X n
+
CN −1 (n − 1)[1 − F (ĉ(E))]N −n−1 Eĉ0 (E)f (ĉ(E))n[F (ĉ(E)) − F (t)]n−2 F (t)dt
t
c
n=1
Z ĉ(E)
N −1
f (t) X n
−
{
CN −1 (N − 1 − n)[1 − F (ĉ(E))]N −n−2
t
c
n=1
0
Eĉ (E)f (ĉ(E))n[F (ĉ(E)) − F (t)]n−1 F (t)}dt.
1
N −1 ]. We thus have
Recall that ĉ(E) = F −1 [1 − ( V E
+E )
1
0
ĉ (E) = −
−1
V
N −1
(V E
+E )
(V +E)2
1
N −1 ))
f (F −1 (1 − ( V E
+E )
.
We first show that
lim Eĉ0 (E) = 0.
E→0
31
(1)
This is true because
lim Eĉ0 (E) =
E→0
1
E
1
lim (
) N −1 = 0.
f (c)V E→0 V + E
Note that limE→0 ĉ(E) = c, and limE→0 F (ĉ(E)) = 1. Since (1) is true, we have
lim Eĉ0 (E)[1 − F (ĉ(E))]n = 0, ∀n ≥ 0.
E→0
(2)
We thus have that the limit of the last two terms in β(E) is zero, i.e.,
ĉ(E)
Z
lim
E→0 c
N −1
f (t) X n
CN −1 (n − 1)[1 − F (ĉ(E))]N −n−1 Eĉ0 (E)f (ĉ(E))n[F (ĉ(E)) − F (t)]n−2 F (t)dt = 0,
t
n=1
and
ĉ(E)
Z
lim
E→0 c
N −1
f (t) X n
CN −1 (N −1−n)[1−F (ĉ(E))]N −n−2 Eĉ0 (E)f (ĉ(E))n[F (ĉ(E))−F (t)]n−1 F (t)dt = 0.
t
n=1
We now show that the second term in β(E) is zero, i.e.,
ĉ(E)
nZ
c
Z
−
c
ĉ(E)
N −1
f (t) X n
CN −1 (n − 1)[1 − F (ĉ(E))]N −n−1 f (ĉ(E))n[F (ĉ(E)) − F (t)]n−2 F (t)dt
t
f (t)
t
n=1
N
−1
X
o
n
N −n−2
n−1
CN
(N
−
1
−
n)[1
−
F
(ĉ(E))]
f
(ĉ(E))n[F
(ĉ(E))
−
F
(t)]
F
(t)dt
V ĉ0 (E)
−1
n=1
= 0.
This is true because
N
−1
X
=
n=1
N
−2
X
n
N −n−2
CN
n[F (ĉ(E)) − F (t)]n−1
−1 (N − 1 − n)[1 − F (ĉ(E))]
n
N −n−2
CN
[F (ĉ(E)) − F (t)]n−1
−1 n(N − 1 − n)[1 − F (ĉ(E))]
n=1
= (N − 1)(N − 2)
N
−2
X
n−1
N −n−2
CN
[F (ĉ(E)) − F (t)]n−1
−3 [1 − F (ĉ(E))]
n=1
= (N − 1)(N − 2){[1 − F (ĉ(E))] + [F (ĉ(E)) − F (t)]}N −3
= (N − 1)(N − 2)[1 − F (t)]N −3 ,
32
and
N
−1
X
=
n=1
N
−1
X
n
N −n−1
CN
n[F (ĉ(E)) − F (t)]n−2
−1 (n − 1)[1 − F (ĉ(E))]
n
N −n−1
CN
[F (ĉ(E)) − F (t)]n−2
−1 n(n − 1)[1 − F (ĉ(E))]
n=2
= (N − 1)(N − 2)
N
−1
X
n−2
N −n−1
CN
[F (ĉ(E)) − F (t)]n−2
−3 [1 − F (ĉ(E))]
n=2
= (N − 1)(N − 2)[1 − F (t)]N −3 .
Recall limE→0 F (ĉ(E)) = 1. Based on the above results, we have
lim β(E)
E→0
=
N −1
f (ĉ(E)) X n
CN −1 [1 − F (ĉ(E))](N −1)−n n[F (ĉ(E)) − F (ĉ(E))]n−1 [V + (n + 1)E]F (ĉ(E))ĉ0 (E)
E→0 ĉ(E)
lim
n=1
=
=
=
f (ĉ(E)) 1
CN −1 [1 − F (ĉ(E))]N −2 (V + 2E)F (ĉ(E))ĉ0 (E)
ĉ(E)
1
(N − 1)
E
lim [1 − F (ĉ(E))]N −2 (
) N −1 −1
E→0
c
V +E
N −2
1
E
E
N −1
(N − 1)
lim (
) N −1 (
) N −1 −1 =
> 0.
E→0 V + E
c
V +E
c
lim
E→0
This completes the proof.
33
Proof of Proposition 3. Recall that
Z
ĉ(E)
T E(E) = N
c
N −1
f (t) X n
CN −1 [1 − F (ĉ(E))](N −1)−n n[F (ĉ(E)) − F (t)]n−1 Vn+1 (E)F (t)dt,
t
n=1
1
N −1 ]. The above term is strictly positive. Further notice that
where ĉ(E) = F −1 [1 − ( V E
+E )
N
−1
X
n
(N −1)−n
CN
n[F (ĉ(E)) − F (t)]n−1 Vn+1 (E)
−1 [1 − F (ĉ(E))]
n=1
= (N − 1)
< (N − 1)
N
−1
X
n=1
N
−1
X
n−1
(N −1)−n
CN
[F (ĉ(E)) − F (t)]n−1 Vn+1 (E)
−2 [1 − F (ĉ(E))]
n−1
(N −1)−n
[F (ĉ(E)) − F (t)]n−1 VN (E)
CN
−2 [1 − F (ĉ(E))]
n=1
= (N − 1)[1 − F (t)]N −2 VN (E) ≤ (N − 1)VN (E).
To show that the optimal entry fee is finite, it suffices to show that limE→∞ T E(E) = 0. To this
end, note that
Z
ĉ(E)
0 < T E(E) < N (N − 1)VN (E)
c
<
N (N − 1)VN (E)
c
f (t)
F (t)dt
t
ĉ(E)
Z
f (t)F (t)dt =
c
N (N − 1)VN (E)F 2 (ĉ(E))
.
2c
Thus, we need to show that
lim EF 2 (ĉ(E)) = 0.
E→∞
If we can show that
lim EF (ĉ(E)) < M,
E→∞
where M is some finite number, then (3) holds.
In fact, recall that
F ((ĉ(E)) = 1 − (
34
1
E
) N −1 .
V +E
(3)
Thus,
lim EF (ĉ(E))
E→∞
1
=
=
lim
N −1
1 − (V E
+E )
1
E
1
−1
E
V
1
N −1
(
)
N −1 V +E
(V +E)2
lim
1
E→∞
E2
E→∞
1
V E2
1
E
(
) N −1 −1
E→∞ N − 1 V + E
(V + E)2
V
=
.
N −1
=
lim
This completes the proof.
Appendix B
Experimental Instructions
35
General Instructions Thank you for participating in this study! Please pay attention to the information provided here
and make your decisions carefully. If at any time you have questions to ask, please raise your
hand and an experimenter will come to you.
Please do not communicate with other participants at any point in this study. Failure to adhere
to this rule would force us to stop this experiment and you will be asked to leave the experiment
without pay.
All information collected will be kept strictly confidential and used for the sole purpose of this
study. Specific Stage-by-Stage Instructions We estimate the total duration of this study to be approximately 1.5 hours. All incentives will be
denominated in experimental dollars (expressed as ECU), which will be added to and
subtracted from the balance in your “checkbook” as the experiment progresses. The final
amount of experimental dollars in your checkbook at the end of the experiment’s money-earning
stages will be converted to real dollars using the following conversion rule:
1. For the amount of ECU earned in Stage 2 and Stage 3,
10 ECU = $ 1
2. For the payoff earned in Stage 1:


If your final payoff in Stage 1 1000 ECU:
150 ECU = $ 1
If your final payoff in Stage 1 1000 ECU, it is split into a base part and a bonus
part, where
o Base part = 1000 ECU. The conversion rate for the base part is
150 ECU = $ 1
o Bonus part = your final payoff – base part. The conversion rate for the bonus
part is
25 ECU = $ 1
For example if you earn 1500 ECU in Stage 1, you will receive
1000
150
$6.67
500
25
$20
$26.67
The real-dollar equivalent of your final earning will then be round up to the nearest dollar. It will
be added to your show-up/ completion fee and paid to you in SGD. The exact amount of the
show-up/ completion fee will be determined at the end of the experiment. The maximum fee you
will receive is $5 and the minimum fee you will receive is $2.
You will participate in the following stages: 36
Stage 1 You will interact with other participants anonymously through the computer interface for a
number of periods. Each participant will only be identified by his or her unique numerical ID
displayed on the participant's computer screen throughout the experiment.
In each period, you will be randomly matched with another participant to form a pair. You and
the other participant will compete against each other in a bidding contest to win prize money (V),
which equals 100 ECU.
You and your opponent can decide whether or not to participate in the contest. There is no entry
fee charged for participating in this contest.
----------------------Notes: (not included in the experimental instructions distributed to subjects).
In all treatments with entry fee, the above bold sentence is replaced by
“If you decide to participate in the contest, you must pay entry fee (E). The amount of
entry fee will be indicated in your computer screen. The total amount of entry fees collected
from all participating contestants will be added to the prize money (V).
------------------------If you decide to participate in the contest, you will have to submit your bid. Submitting a bid is
costly, and the cost is equal to
Cost = c * bid
In which c denotes the additional cost you need to incur for an additional dollar of bid. We call it
cost rate. Bid denotes the amount of your bid. To cover your total bidding cost, each of you is
given a checkbook of 200 ECU in every period.
Your cost rate c can take any value from 1 to 2 (up to 2 decimal points).
-----------------Note (not shown in the actual experimental instruction): The range of value is from 1 to 3 in
the high heterogeneity treatments.
------------------The computer will randomly draw the value of your cost rate. All values have an equal chance
of being selected as your cost rate. You will only know your own cost rate c, but not your
opponent's cost rate.
37

If you decide to participate in the contest and submit a non-zero bid, your total cost
incurred will be equal to:
Your Bidding Cost = c*Bid.

If both of you participate in the contest, whoever submit the highest bid wins the
contest. The winning prize will be 100 ECU

If it is only you who participates in the contest, you automatically win the contest. The
winning prize value will be 100 ECU

If you submit the same bid amount as your opponent, the contest ends in a tie, the
computer will randomly select a winner. The winning prize will be 100 ECU

If you decide not to participate in the contest (i.e. submit zero bid), your earnings
from the contest will be 0 ECU.
Five periods out of all periods you participated will be drawn randomly to determine your payoff
from Stage 1. The earning you received in these five drawn periods will be summed and
converted into SGD equivalent and added to your final earning from this experiment when the
experiment is completed.
Stage 2 In this part of the study you will be asked to make a series of choices. How much you receive
will depend partly on chance and partly on the choices you make. The decision problems are not
designed to test you. What we want to know is what choices you would make in them. The only
right answer is what you really would choose.
For each line in the table you will see in this stage, please indicate whether you prefer option A
or option B. There will be a total of 10 lines in the table but just one line will be randomly
selected for payment. You do not know which line will be paid when you make your choices.
Hence you should pay attention to the choice you make in every line.
After you have completed all your choices, the computer will randomly generate a number,
which determines which line is going to be paid out.
Your earnings for the selected line depend on which option you chose: If you chose option A
in that line, you will receive 20 ECU. If you chose option B in that line, you will receive either
60 ECU or 0. To determine your earnings in the case you chose option B, there will be a second
random draw. The computer will randomly determine if your payoff is 0 ECU or 60 ECU, with
the chances set by the computer as they are stated in Option B.
Your earning from Stage 2 will be added to your final earning from this experiment when the
experiment is completed.
38
Stage 3 This stage consists of answering ten (10) quantitative questions. You will be compensated in the
following way. For each correct answer you will receive 5 ECU. If you get a question wrong or
leave a question unanswered (blank) you will receive 0 ECU, so there is no addition or
deduction of points when you make a mistake or leave a question unanswered.
The total amount of time given to you to answer these 10 questions is 8 minutes. Your earning
from Stage 3 will be added to your final earning from this experiment when the experiment is
completed. Please remember to submit your answers before the time allocated runs out,
otherwise your answers will not be recorded.
Stage 4 In this final stage of the experiment, we will ask you to answer some questions about yourself.
Please answer truthfully. When you have finished, we will prepare your final earning, and ask
you to sign a receipt in exchange for the payment you receive.
Thank you again for your participation! If you have any questions, please raise your hand and an
experimenter will come to you. 39
Appendix C
Additional Tables of Robustness Checks
40
Table B1: Conditional Means of Bids and Probability of Entry: Panel-Data Specifications
Panel A: All 30 Periods
Zero Entry Fee, LowHet
Optimal Entry Fee, LowHet
High Entry Fee, LowHet
Zero Entry Fee, HighHet
Optimal Entry Fee, HighHet
High Entry Fee, HighHet
(1)
Bid if Enter
(2)
Pr(Entry)
49.176
(3.479)
64.700
(3.674)
63.512
(3.627)
40.188
(3.495)
51.705
(2.758)
39.154
(3.534)
0.979
(0.010)
0.846
(0.037)
0.654
(0.052)
0.943
(0.020)
0.815
(0.030)
0.722
(0.047)
(1)
Bid if Enter
(2)
Pr(Entry)
44.280
(4.084)
54.837
(4.339)
47.644
(4.315)
34.489
(4.121)
51.812
(3.292)
39.865
(4.193)
0.998
(0.003)
0.864
(0.048)
0.707
(0.065)
0.963
(0.023)
0.805
(0.043)
0.733
(0.066)
Panel B: Final 10 Periods
Zero Entry Fee, LowHet
Optimal Entry Fee, LowHet
High Entry Fee, LowHet
Zero Entry Fee, HighHet
Optimal Entry Fee, HighHet
High Entry Fee, HighHet
Notes: For each treatment, column (1) shows the mean bid, conditional on entering the contest, and
column (2) shows the mean probability of entering the contest. These results are post-estimation
conditional means, which present the average bid and entry probability for each treatment, holding
observable characteristics constant. The full regression specifications are shown in Table 5.
41
Table B2: Individual-Level Regression Results for Bids and Entry: Panel-Data Specifications
All 30 Periods
Optimal Entry Fee, LowHet
High Entry Fee, LowHet
Zero Entry Fee, HighHet
Optimal Entry Fee, HighHet
High Entry Fee, HighHet
Marginal Cost
Period
Probability Test Score
Past Experiments
Had Game Theory
Risk Averse Subject
Subject’s Age
Engineering Student
Science Student
Business Student
Female Subject
Observations
Log Likelihood
Final 10 Periods
(1)
Bid if Enter
(2)
Pr(Entry)
(3)
Bid if Enter
(4)
Pr(Entry)
12.412
(5.340)∗∗
12.994
(5.347)∗∗
3.350
(5.251)
15.248
(4.632)∗∗∗
-0.199
(5.889)
-42.439
(0.959)∗∗∗
-0.565
(0.041)∗∗∗
-17.725
(6.953)∗∗
-5.537
(3.504)
-0.392
(3.423)
0.452
(3.163)
-1.088
(1.032)
-1.902
(3.655)
6.919
(4.001)∗
-9.530
(5.524)∗
5.619
(3.133)∗
-1.153
(0.313)∗∗∗
-1.753
(0.303)∗∗∗
0.469
(0.320)
-0.390
(0.279)
-0.483
(0.335)
-1.867
(0.091)∗∗∗
-0.001
(0.003)
-0.468
(0.383)
-0.371
(0.184)∗∗
-0.441
(0.182)∗∗
-0.339
(0.175)∗
0.036
(0.057)
0.327
(0.203)
-0.099
(0.214)
0.102
(0.298)
0.157
(0.171)
10.411
(6.288)∗
5.527
(6.327)
5.349
(6.191)
20.978
(5.487)∗∗∗
9.394
(6.968)
-42.449
(1.394)∗∗∗
-0.363
(0.179)∗∗
-19.438
(8.415)∗∗
-10.106
(4.144)∗∗
0.554
(4.055)
-3.627
(3.744)
-1.000
(1.216)
-0.684
(4.328)
7.860
(4.741)∗
-13.842
(6.516)∗∗
5.948
(3.721)
-1.873
(0.578)∗∗∗
-2.475
(0.565)∗∗∗
-0.015
(0.591)
-1.188
(0.531)∗∗
-1.154
(0.596)∗
-2.197
(0.188)∗∗∗
0.017
(0.018)
0.266
(0.607)
-0.324
(0.278)
-0.581
(0.278)∗∗
-0.493
(0.272)∗
0.105
(0.094)
0.576
(0.314)∗
-0.245
(0.324)
0.444
(0.485)
0.439
(0.264)∗
3455
-15,626.223
4320
-1,443.891
1154
-5,087.617
1440
-452.829
Notes: Column (1) displays marginal effects from an OLS regression of the contestant’s bid, conditional on entering the contest. Column (2) displays marginal effects from a Probit regression of
whether the contestant entered the contest. ∗, ∗∗, and ∗ ∗ ∗ denote significance at the 10%, 5%,
and 1% level, respectively.
42