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Games and Economic Behavior 68 (2010) 731–747

Contents lists available at ScienceDirect

Games and Economic Behavior

www.elsevier.com/locate/geb

Experimental comparison of multi-stage and one-stage contests

Roman M. Sheremeta1

The George L. Argyros School of Business and Economics, Chapman University, 1 University Drive, Orange, CA 92866, USA

a r t i c l ei n f o a b s t r a c t

Article history:

Received 12 December 2008

Available online 11 August 2009

JEL classification:

C72

C91

D72

Keywords:

Rent-seeking

Contest

Contest design

Experiments

Risk aversion

Over-dissipation

This article experimentally studies a two-stage elimination contest and compares its

performance with a one-stage contest. Contrary to the theory, the two-stage contest

generates higher revenue than the equivalent one-stage contest. There is significant over-

dissipation in both stages of the two-stage contest and experience diminishes over-

dissipation in the first stage but not in the second stage. Our experiment provides evidence

that winning is a component in a subject’s utility. A simple behavioral model that accounts

for a non-monetary utility of winning can explain significant over-dissipation in both

contests. It can also explain why the two-stage contest generates higher revenue than the

equivalent one-stage contest.

© 2009 Elsevier Inc. All rights reserved.

1. Introduction

Contests are economic, political, or social interactions in which agents expend resources to receive a certain prize. Exam-

ples include marketing and advertising by firms, patent races, and rent-seeking activities. All these contests differ from one

another on multiple dimensions including group size, number of prizes, number of inter-related stages, and rules that reg-

ulate interactions. The most popular theories investigating different aspects of contests are based on the seminal model of

rent-seeking introduced by Tullock (1980). The main focus of rent-seeking literature is the relationship between the extent

of rent dissipation and underlying contest characteristics (Nitzan, 1994).

The majority of rent-seeking studies are based on the assumption that contests consist of only one stage. Many contests

in practice, however, consist of multiple stages. In each stage contestants expend costly efforts in order to advance to the

final stage and win the prize. Two major purposes of our study are to compare the performance of a one-stage contest

versus a two-stage elimination contest and to examine whether over-dissipation is observed in both stages of the two-stage

contest. The experiment is also designed to elicit non-monetary utility of winning from subjects in order to explain potential

over-dissipation in contests.

We find that, contrary to the theory, the two-stage contest generates higher revenue and higher dissipation rates than the

equivalent one-stage contest. Over-dissipation is observed in both stages of the two-stage contest and experience diminishes

over-dissipation in the first stage but not in the second stage. Our experiment also provides evidence that winning is a

E-mail address: sheremet@chapman.edu.

1I am particularly grateful to Tim Cason for excellent guidance and support. I also want to thank two anonymous referees and an associate editor, Jack

Barron, Subhasish Modak Chowdhury, Kai Konrad, Dan Kovenock, and Jingjing Zhang, as well as seminar participants at Purdue University and participants

at the June 2008 Economic Science Association meeting for helpful comments. This research has been supported by National Science Foundation Grant

(SES-0751081). Any remaining errors are mine.

0899-8256/$ – see front matter © 2009 Elsevier Inc. All rights reserved.

doi:10.1016/j.geb.2009.08.001

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R.M. Sheremeta / Games and Economic Behavior 68 (2010) 731–747

component in a subject’s utility. A simple behavioral model that accounts for a non-monetary utility of winning can explain

significant over-dissipation in both contests. It can also explain why the two-stage contest generates higher revenue than

the equivalent one-stage contest.

Recent theoretical models of multi-stage elimination contests reveal interesting dynamic aspects. Gradstein and Konrad

(1999) consider a multi-stage elimination contest in which a number of parallel contests take place at each stage and only

winners are promoted to the next stage. The authors show that, depending on the contest success function, a multi-stage

contest may induce higher effort by the participants than a one-stage contest. Under a lottery contest success function,

however, the two structures are equivalent. In the same line of research, Baik and Lee (2000) study a two-stage elimination

contest with effort carryovers. In this contest, players in two groups compete non-cooperatively to win a prize. In the first

stage, each group selects a finalist who competes for the prize in the second stage. First-stage efforts are partially (or fully)

carried over to the second stage. Baik and Lee (2000) demonstrate that, in the case of player-specific carryovers, the rent-

dissipation rate (defined as the ratio of the expended total effort to the value of the prize) increases in the carryover rate

and the rent is fully dissipated with full carryover. Other theoretical studies of multi-stage elimination contests have been

conducted by Rosen (1986), Clark and Riis (1996), Gradstein (1998), Amegashie (1999), Stein and Rapoport (2005), Fu and

Lu (forthcoming), and Groh et al. (forthcoming).2All these studies investigate different aspects of multi-stage contests such

as elimination procedures, interdependency between the stages, asymmetry between contestants, and resource constraints.

Since rent-seeking behavior in the field is difficult to measure, researchers have turned to experimental testing of the

theory, with almost all studies focused on one-stage contests (Millner and Pratt, 1989, 1991; Shogren and Baik, 1991;

Davis and Reilly, 1998; Potters et al., 1998; Anderson and Stafford, 2003).3Despite considerable differences in experimental

design among these studies, most share the major finding that aggregate rent-seeking behavior exceeds the equilibrium pre-

dictions.4Several researchers have offered explanations for such behavior based on non-monetary utility of winning (Parco

et al., 2005), misperception of probabilities (Baharad and Nitzan, 2008), quantal response equilibrium, and heterogeneous

risk preferences (Goeree et al., 2002; Sheremeta, forthcoming).

There are currently only a few experimental studies that investigate the performance of multi-stage contests.5Schmitt

et al. (2004) develop and experimentally test a model in which rent-seeking expenditures in the current stage affect the

probability of winning a contest in both current and future stages. Two other experimental studies are based on a two-

stage rent-seeking model developed by Stein and Rapoport (2005). In this model all players have budget constraints. In

the first stage, players compete within their own groups by expending efforts, and the winner of each group proceeds to

the second stage. In the second stage, players compete with one another to win a prize by expending additional efforts

subject to budget constraints. The experimental studies of Parco et al. (2005) and Amaldoss and Rapoport (2009) reject the

equilibrium model of Stein and Rapoport (2005) because of significant over-dissipation in the first stage. Both experimental

studies conjecture that the non-monetary utility of winning plays a crucial role in explaining excessive over-dissipation in

the first stage. Our experimental design is based on Gradstein’s and Konrad’s (1999) theoretical model, which compares the

performance of a one-stage contest versus a multi-stage elimination contest.

2. Theoretical model

In a simple one-stage contest N identical players compete for a prize of value V . Each risk-neutral player i chooses his

effort level, ei, to win the prize. The probability that a contestant i wins the prize is given by a lottery contest success

function:

ei

?N

The contestant’s probability of winning increases monotonically in own effort and decreases in the opponents’ efforts.

The expected payoff for risk-neutral player i is given by

pi(ei,e−i) =

j=1ej

.

(1)

E(πi) = pi(ei,e−i)V − ei.

That is, the probability of winning the prize, pi(ei,e−i), times the value of the prize, V , minus the effort expended, ei.

Differentiating (2) with respect to eiand accounting for the symmetric Nash equilibrium leads to a classical solution (Tul-

lock, 1980)

(2)

e∗=(N − 1)

N2

V.

(3)

2Another type of multi-stage contests is the multi-battle contests. In a multi-battle contest, players compete in a sequence of simultaneous move contests

to win a prize and the player whose number of victories reaches some given minimum number wins the prize. Such contests have been studied by Harris

and Vickers (1985, 1987), Klumpp and Polborn (2006), and Konrad and Kovenock (2009).

3For empirical results on multi-stage elimination tournaments in sports see Ehrenberg and Bognanno (1990) and Bognanno (2001).

4Shogren and Baik (1991) do not find excessive expenditure.

5Exception is a study by Amegashie et al. (2007) on multi-stage all-pay auction.

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Table 3.1

Experimental design and equilibrium effort levels.

TreatmentOSTS

Value of the prize, V

Number of players, N

Number of groups, K

120

4

1

120

4

2

Effort in stage 1, e1

Effort in stage 2, e2

Total revenue

Dissipation rate

22.5

–

90

0.75

7.5

30

90

0.75

The simple model considered above is the building block of contest theory. Gradstein and Konrad (1999) extended this

model to study a multi-stage elimination contest. In their contest, N players expend irreversible efforts in an attempt to

advance to the final stage. In the first stage, all players are divided into several groups. The winner of each group proceeds

to the second stage, where contestants again are divided into competing groups, etc. The winner of the final stage receives

a prize of value V . For our analysis, assume that there are only two stages. In the first stage, all players are divided into

K equal groups (N/K players per each group), with the winner of each group proceeding to the final stage. To analyze the

two-stage contest, we apply backward induction. According to (3), in the second stage each finalist will expend effort of

e∗

2=(K − 1)

The resulting expected payoff in the second stage is E(π∗

each group compete as if the value of the prize was E(π∗

given by

K2

V.

(4)

2) = V/K2. Knowing this, in the first stage N/K players within

2). Therefore, according to (3), the first stage equilibrium effort is

e∗

1=(N − K)

N2K

It is straightforward to show that, under the equilibrium strategy, the second order conditions hold and the resulting

expected payoff is non-negative.6Formulas (4) and (5) demonstrate how the first and second stage equilibrium efforts of

each player depend on the prize value and the number of contestants in each stage.

V.

(5)

3. Experimental design and procedures

3.1. Experimental design

Our experiment consists of two different contests. The outline of the experimental design and theoretical predictions for

each contest are shown in Table 3.1. In each contest there are 4 players and the prize value is 120 experimental francs. In a

baseline treatment, all 4 contestants compete with each other for the prize in a one-stage (OS) contest. In equilibrium the

revenue collected in this contest is 90. The resulting dissipation rate, defined as the total efforts divided by the value of the

prize, is 0.75.

The second treatment is a two-stage (TS) contest which consists of 4 players divided between 2 equal groups. The first

stage winner of each group proceeds to the second stage and the winner of the second stage receives the prize. This contest

resembles many real life situations. For instance, swimming or track tournaments often place competitors in different groups

called “heats” with the winner of each “heat” proceeding to the finale. The major competition in TS arises between the two

players in the second stage (see Table 3.1). Therefore, the revenue collected from the second stage is substantially higher

than the revenue collected from the first stage. The total revenue collected from both stages in the TS treatment is 90,

which is equivalent to the revenue collected in the OS treatment. This equivalence was proved by Gradstein and Konrad

(1999) for a more general multi-stage contest under lottery contest success function.

3.2. Experimental procedures

The experiment was conducted at the Vernon Smith Experimental Economics Laboratory. A total of 84 subjects par-

ticipated in seven sessions (12 subjects per session). All subjects were Purdue University undergraduate students who

participated in only one session of this study. Some students had participated in other economics experiments that were

unrelated to this research.

The computerized experimental sessions were run using z-Tree (Fischbacher, 2007). Each experimental session proceeded

in four parts. Subjects were given the instructions, available in Appendix A, at the beginning of each part and the exper-

imenter read the instructions aloud. Before the actual experiment, subjects completed the quiz on the computer to verify

6For a more detailed derivations, see Amegashie (1999), Gradstein and Konrad (1999), and Baik and Lee (2000).

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Table 4.1

Average statistics.

Treatment OS TS

Equilibrium ActualEquilibrium Actual

Effort in stage 1

Effort in stage 2

Net payoff

Total revenue

Dissipation rate

22.5

–

7.5

90

0.75

34.1 (0.7)

–

−4.1 (1.1)

136

1.14

7.5

30

7.5

90

0.75

18.9 (0.6)

47.2 (0.9)

−12.5 (1.2)

170

1.42

Note. Standard error of the mean in parentheses.

their understanding of the instructions. The experiment started only after all subjects had answered all quiz questions. In

the first part subjects made 15 choices in simple lotteries, similar to Holt and Laury (2002).7This method was used to elicit

subjects’ risk preferences. The second and the third parts corresponded to OS and TS treatments ran in different orders. In

three sessions we ran the OS treatment first and in three other sessions we ran the TS treatment first. Each subject played

30 periods in the OS treatment and 30 periods in the TS treatment.

In each period, subjects were randomly and anonymously placed into a group of 4 players and designated as participant

1, 2, 3, or 4. Subjects were randomly re-grouped after each period. In the first stage of the TS treatment, participant 1

was paired with participant 2 and participant 3 was paired with participant 4. In the OS treatment, all 4 participants were

paired against each other. At the beginning of each period, each subject received an endowment of 120 experimental francs.

Subjects could use their endowments to expend efforts (make bids). After all subjects submitted their efforts, the computer

chose the winner by implementing a simple lottery rule. In the TS treatment, the two finalists — one from each pair — again

made their effort choices in the second stage. At the end of the second stage the computer chose the winner of the prize

and displayed the following information to all subjects: the opponent’s effort in the first stage, the other opponent’s effort

in the second stage, the result of the random draw in the first and second stage, and personal period earnings. Subjects

who did not proceed to the second stage in the TS treatment did not receive any information about the decisions made in

the second stage. All subjects were informed that by increasing their efforts, they would increase their chance of winning

and that, regardless of who wins the prize, all subjects would have to pay for their efforts. The instructions explained the

structure of the game in detail.

In the final fourth part of the experiment, subjects were given an endowment of 120 francs and were asked to expend

efforts in a one-stage contest in order to be a winner. The procedure followed closely to the OS treatment. The only differ-

ence was that the prize value was 0 francs. Subjects were told that they would be informed whether they won the contest

or not. We used this procedure to receive an indication of how important it is for subjects to win when winning is costly

and there is no monetary reward for winning.

At the end of the experiment, 1 out of 15 decisions subjects made in part one was randomly selected for payment.

Subjects were also paid for 5 out of 30 periods in part two, for 5 out of 30 periods in part three, and for the 1 decision they

made in part four. The earnings were converted into US dollars at the rate of 60 francs to $1. On average, subjects earned

$25 each which was paid in cash. Each experimental session lasted about 90 minutes.

4. Results

4.1. General results

Table 4.1 summarizes average efforts, average net payoffs, and average dissipation rates over the treatments. The first

striking feature of the data is that, on average, net payoffs in both OS and TS treatments are negative and the actual

dissipation rates are significantly greater than predicted.8Similar findings are also reported in Davis and Reilly (1998) and

Gneezy and Smorodinsky (2006). In both studies, revenues collected repeatedly exceeded the prize and subjects earned, on

average, negative payoffs.

Result 1. There is significant over-dissipation in one-stage and two-stage contests.

There are several possible explanations for significant over-dissipation. First, it is possible that subjects expend signif-

icantly higher efforts because each period they receive a “free” endowment of 120 francs.9Note that this endowment is

7Subjects were asked to state whether they preferred safe option A or risky option B. Option A yielded $1 payoff with certainty, while option B yielded

a payoff of either $3 or $0. The probability of receiving $3 or $0 varied across all 15 lotteries. The first lottery offered a 5% chance of winning $3 and a 95%

chance of winning $0, while the last lottery offered a 70% chance of winning $3 and a 30% chance of winning $0.

8Separately for each treatment, we estimated a random effects model, with individual subject effects, where the dependent variable is effort and the

independent variables are a constant and a period trend. A standard Wald test, conducted on estimates of a model, clearly rejects the hypothesis that the

constant coefficients are equal to the predicted theoretical values as in Table 4.1 (p-value < 0.01).

9The endowment was chosen for several reasons. First, the endowment was chosen to be equal to the prize value to be consistent with other studies

(Anderson and Stafford, 2003; Herrmann and Orzen, 2008). Second, the endowment of 120 francs was also chosen to be substantially higher than the Nash

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735

Fig. 4.1. Average efforts by treatments.

substantially higher than the Nash equilibrium predictions. While the endowment itself has no theoretical impact, it cer-

tainly may have a behavioral impact, causing subjects to over-dissipate. The second explanation, related to the endowment

size effect, is that subjects are likely to make “errors.” Sheremeta (forthcoming) shows how the quantal response equilib-

rium developed by McKelvey and Palfrey (1995), which accounts for errors made by individual subjects, can explain some

over-dissipation in lottery contests. Finally, and probably most importantly, subjects may have a non-monetary utility of

winning. If that is the case, then in addition to the monetary value of 120 francs, subjects also compete to be winners. In

Section 4.3 we provide evidence consistent with subjects having a non-monetary utility of winning which may explain why

there is persistent over-dissipation in both treatments.

It is important to emphasize that the over-dissipation in the TS treatment takes place in both stages of the competition.

In the first stage of TS treatment, subjects expend an average effort of 18.9 which is more than double the equilibrium effort

of 7.5 (Table 4.1). In the second stage, instead of the equilibrium effort of 30, subjects expend an average effort of 47.2. The

first and the second stage efforts in TS treatment are higher than theoretical values in all periods of the experiment (Fig. 4.1).

Result 2. In the two-stage contest, significant over-dissipation is observed in both stages.

This result is very different from previous experimental findings. In a related study, Parco et al. (2005) find significant

over-dissipation only in the first stage of a two-stage contest. Given the first stage over-dissipation, and the fact that subjects

are budget constrained, there is significant under-dissipation in the second stage. Our study shows that, after eliminating

the budget constraints, over-dissipation in a two-stage contest occurs in both stages.

It is often argued that subjects need to get some experience in order to learn how to play the equilibrium. For that

reason, Fig. 4.1 displays the average effort over all 30 periods of the experiment. As players become more experienced, the

average efforts made in the first stage of OS and TS treatments decrease. A simple regression of the first stage effort on a

period trend shows a significant and negative relationship (p-value < 0.01). Although this is true for the first stage, it is not

the case for the subjects’ behavior in the second stage.

Result 3. Experience diminishes over-dissipation in the first stage but not in the second stage.

One possible reason for this finding is that at the beginning of the TS treatment, subjects apply similar strategies to both

stages of the competition. This may occur because the decisions are cognitively difficult, which causes subjects to apply

similar heuristics or “rules of thumb” to both stages (Gigerenzer and Goldstein, 1996). But with the repetition, subjects

learn the strategic aspect of the two-stage contest and correctly redistribute their efforts between the first stage and the

second stage. Note that in the second half of the experiment the magnitude of relative to the equilibrium over-dissipation

in the first stage is very similar to the magnitude of relative over-dissipation in the second stage (efforts are approximately

one and a half times higher than the equilibrium predictions).10

equilibrium predictions in order to make sure that in the two-stage contest subjects are not budget constrained (otherwise, we would have to provide

additional endowment in the second stage of a two-stage contest which would cause substantial differences in earnings between two treatments).

10We estimated a convergence model as in Noussair et al. (1995) and found that the first stage effort in OS and TS treatments does not converge to the

predicted level of 22.5 and 7.5 (p-value < 0.01 for both treatments) and the second stage effort in TS treatment does not converge to the predicted level of

30 (p-value < 0.01).