Winner-Take-All and Proportional-Prize Contests:
Theory and Experimental Results
Roman M. Sheremeta
a
, William A. Masters
b
, and Timothy N. Cason
c
a
Argyros School of Business and Economics, Chapman University,
One University Drive, Orange, CA 92866, U.S.A.
b
Department of Food and Nutrition Policy, Tufts University
150 Harrison Avenue, Boston, MA 02111, U.S.A.
c
Department of Economics, Krannert School of Management, Purdue University,
403 W. State St., West Lafayette, IN 47906-2056, U.S.A.
February 27, 2012
Abstract
This study provides a unified theoretical and experimental framework in which to
compare three canonical types of competition: winner-take-all contests won by the best
performer, winner-take-all lotteries where probability of success is proportional to performance,
and proportional-prize contests in which rewards are shared in proportion to performance. We
introduce random noise to reflect imperfect information, and collect independent measures of
risk aversion, other-regarding preferences, and the utility of winning a contest. The main finding
is that efforts are consistently higher with winner-take-all contests. The lottery contests have the
same Nash equilibrium as proportional prizes, but induce contestants to choose higher efforts and
receive lower, more unequal payoffs. This result may explain why contest designers who seek
only to elicit effort offer lump-sum prizes, even though contestants would be better off with
proportional rewards.
JEL Classifications: C72, D72, D74, J33
Keywords: contests, rent-seeking, lotteries, incentives in experiments, risk aversion
Corresponding author: Roman M. Sheremeta; E-mail: sheremet@chapman.edu
For helpful comments we thank Marco Faravelli, Lise Vesterlund, and seminar participants at Purdue University,
Universities of Pittsburgh, Hawaii and Monash, as well as participants at the 2010 International Economic
Association Conference and the 2010 Southern Economic Association Conference. Any errors are our responsibility.
1
1. Introduction
A wide variety of competitions arise in economic life, and new ones are regularly
introduced to attract effort and reward achievement. Such competitions are commonly modeled
as contests, in which agents compete for prize funds by expending costly resources. Although
there are many possible contest designs, most theoretical models and most artificially-designed
competitions use predetermined and exogenous lump-sum prizes (Konrad, 2009), even when
payments could be made proportional to relative performance (Cason et al., 2010). This paper
provides a unified theoretical and experimental framework in which to compare contest designs
and tests how contestants respond to lump-sum as opposed to proportional incentives.
The simplest contest model in the literature is a winner-take-all competition in which the
highest performing contestant wins the prize (Hillman and Riley, 1989). In some versions, such
as the rank order tournament of Lazear and Rosen (1981), performance is stochastically related
to effort, perhaps due to noise in the observation of effort or in the process by which effort is
translated into performance. Even with noise, incentives in such contests follow a step function,
offering high-powered incentives for the winner relative to the next-best performer, and then
lower incentives for all other contestants. As a result, some contestants may be discouraged from
entering (Cason et al., 2010) or from performing well (Brown, 2011) by the presence of a high-
skill competitor.
A closely-related form of competition is the winner-take-all lottery contest of Tullock
(1980), in which the exogenously fixed prize is allocated probabilistically in proportion to
observable efforts. This contest format has been most widely used to model naturally-occurring
competitions for a lump-sum reward such as political lobbying (Krueger, 1974; Tullock, 1980;
Snyder, 1989) or patent races (Fudenberg et al., 1983; Harris and Vickers, 1985, 1987).
2
An extensive experimental literature investigates various forms of winner-take-all
contests; for a review see Sheremeta et al. (2012). Almost without exception, experimental
studies find that contestants incur expenditures that exceed Nash equilibrium levels. Although
sometimes desirable (Morgan and Sefton, 2000; Sheremeta, 2010, 2011), over-expenditures
typically reduce individual payoffs and decrease economic welfare (Sheremeta and Zhang, 2010;
Cason et al., 2011). Moreover, the stark win-or-lose structure of payoffs results in a highly
inequitable distribution of economic welfare (Frank and Cook, 1996).
An alternative to winner-take-all competition that might generate more efficient and more
equitable outcomes would be to divide the prize in proportion to observable effort (Cason et al.,
2010; Schmidt et al., 2011; Eisenkopf and Teyssier, 2012).
1
In a proportional-prize contest, the
fixed prize is shared among contestants according to their performance. The resulting incentives
would be similar to a lottery contest, but with lower risks and greater equality of payoffs among
contestants. Proportional prizes arise naturally in economic situations such as shared rents (Long
and Vousden, 1987), profit sharing and labor productivity (Weitzman and Kruse, 1990), and
labor contracts (Zheng and Vikuna, 2007). Contest designers could choose to divide rewards in
this way, but typically prefer to make lump-sum awards (McKinsey & Company, 2009).
This paper offers a unifying model in which the three contest types are special cases of a
common theoretical structure. For each case we derive the Nash equilibrium for risk-neutral and
self-interested competitors, and implement that contest in a laboratory experiment. A novel
feature of the model and the experiment is to vary the random noise that affects the mapping
1
There are only a few experimental studies comparing different contest structures. Davis and Reilly (1998) and
Potters et al. (1998) compare behavior in all-pay auctions to lottery contests. Both studies find that, as predicted by
theory, perfectly discriminating all-pay auctions generate higher efforts than probabilistic lottery contests, and that
in both contests subjects’ efforts are higher than Nash equilibrium predictions. However, both of these studies
compare only winner-take-all contests and efforts are perfectly observable to the contest designer, so the prize is
always allocated to the contestant with the highest effort (an in the all-pay auction). In contrast, our study examines
both winner-take-all and proportional-prize contests. Moreover, we introduce noise so that efforts are not perfectly
observable. What’s observed is only performance, which is a function of noise and effort.
3
between a contestant’s effort and their observed performance. This exogenous noise represents
the effect of imperfect information, for contestants who may not know how well their efforts will
produce results, and for employers or contest judges who may not be able to observe results
directly. We also collect independent measures of subjects’ risk aversion, other-regarding
preferences, and utility of winning a contest, and use these factors to help understand their
choices in the various contests.
Our central finding is that the simple winner-take-all contest generates the highest efforts
and consequently the lowest net payoff to participants, which is consistent with the predicted
Nash equilibria. The lottery and the proportional-prize contest have the same, lower Nash
equilibrium level of effort. Actual competitors in both contests typically over-contribute and
hence receive lower payoffs than the Nash equilibrium, but sharing the prize reduces the amount
of wasted effort. Sharing the prize also makes effort levels less sensitive to random noise or the
subject’s measured risk aversion and utility of winning. This direct comparison of the three
contest types helps reveal how winner-take-all awards, whether paid deterministically or by
lottery, can induce excess contributions and be preferred by contest designers, even though
competitors would be better off if prizes were shared proportionally. Contest designers are likely
to prefer proportional prizes only if they wish to reduce excess effort, make payoffs more
equitable, or make efforts more consistent in the face of variation in noise and contestants’
individual preferences.
The rest of the paper is organized as follows: Section 2 presents the theoretical model;
Section 3 describes the experimental design, procedures and hypotheses; Section 4 reports the
results of the experimental sessions; and Section 5 concludes.
4
2. The Theoretical Model
Our unified model is a contest in which two risk-neutral players and compete for a
prize . Both players expend individual efforts
and
. Every player who exerts effort has to
bear cost , where
,
0. The performance
of player is determined by a production
function
|
, (1)
where
is a random variable which is drawn from the distribution on the interval 0,∞.
This multiplicative production function (1) has been used by O’Keefe et al. (1984), Hirshleifer
and Riley (1992), and Gerchak and He (2003). The random component,
, can be thought of as
production luck, imperfect information about performance, or measurement error. It can also be
easily interpreted as an unknown ability
(Rosen, 1986).
The share of the prize received by player depends on the relative individual
performance:
,
|
,
/
. (2)
The share of the prize (2) can also be interpreted as the contest success function (CSF), i.e. the
probability of winning the contest (Skaperdas, 1996).
2
Given (1) and (2), the expected payoff for
player can be written as:
. (3)
A deterministic winner-take-all contest similar to the rank-order tournament of Lazear
and Rosen (1981) is obtained using the restriction ∞. A simple all-pay auction of Hillman
and Riley (1989) can be obtained by further restriction of the random component, i.e.
1.
2
The production function (1), with multiplicative noise, implies that the CSF (2) satisfies the axioms introduced by
Skaperdas (1996). In particular, the CSF satisfies the conditions of a probability distribution:
∑
,
,
1
and
,
,
0, for all
and
. Multiplicative noise also guarantees that the contest success function is
homogeneous, i.e.,
,
,
,
,
for all 0.
