Ex post unbalanced tournaments

• Author: Lorens Imhof,Matthias Kräkel
• DOI: http://doi.org/10.1111/1756-2171.12119
• Published date: 01 February 2016
• Date: 01 February 2016

RAND Journal of Economics

Vol.47, No. 1, Spring 2016

pp. 73–98

Ex post unbalanced tournaments

Lorens Imhof∗

and

Matthias Kr¨

akel∗

Tournaments create strong incentives under the assumption that the competition between the

agents is balanced. If, at the outset, one agent is stronger than the other, the tournament is ex

ante unbalanced and incentives break down. Handicaps can in this case restore incentives. In

practice, competing agents are often overall equally strong but have different sorts of strengths.

Then, competition will typically be unbalanced ex post and incentives break down, but handicaps

cannot be used. We show how a simple means, introducing the possibility of a tie, can often

resolve the problem. We examine under what conditions incentives are maximized by ties.

1. Introduction

In tournaments, players compete for given prizes that are distributed according to relative

performance.1The best performing player gets the highest prize, the second-best the second

highest prize, and so on. There are many examples of tournaments in real life: in sports contests,

individual athletes or teams try to win against other individuals or teams, respectively. Many

firms use tournaments to determine bonus payments for workers and managers. In job-promotion

tournaments, workers compete for a more attractiveposition at the next hierarchy level. Politicians

participate in election contests to run for higher-level offices. The main reasons for the widespread

use of tournaments are that they induce high efforts, are simple to implement, and that they can

be applied even when other incentive schemes do not work (e.g., under subjective or unverifiable

performance measurement; Malcomson, 1984, 1986).

A key requirement is that the competition in tournaments must be balanced to avoid dis-

couragement of all players (Lazear and Rosen, 1981; Nalebuff and Stiglitz, 1983; O’Keeffe,

Viscusi, and Zeckhauser, 1984). Tournament competition can be unbalanced in two ways. On

the one hand, individual players may have a competitive advantage known at the beginning of

the tournament (e.g., a higher productivity). Such ex ante unbalanced competition can be easily

∗University of Bonn; limhof@uni-bonn.de, m.kraekel@uni-bonn.de.

We thank two anonymous referees and the Editor, David Martimort, for many helpful comments. Financial support by

the Deutsche Forschungsgemeinschaft (DFG), grant no. SFB/TR 15, is gratefully acknowledged.

1The theory of tournaments and contests builds on the seminal articles by Tullock(1980), Lazear and Rosen (1981),

and Nalebuff and Stiglitz (1983). More recent work comprises Marino and Z´

abojn´

ık (2004), Feess, Muehlheusser, and

Walzl (2008), Gershkov, Li, and Schweinzer(2009), Parreiras and Rubinchik (2010), and Suzuki (2012).

C2016, The RAND Corporation. 73

74 / THE RAND JOURNAL OF ECONOMICS

cleared by an appropriate seeding of players or the introduction of handicaps (Lazear and Rosen,

1981; Nalebuff and Stiglitz, 1983). On the other hand, it is also possible that players start in

a fair tournament; but, when a winner is selected, it turns out that one player has obtained an

unpredictable competitive advantage, so that we can speak of ex post unbalanced competition.

In that case, the aforementioned solutions—seeding or handicaps—do not work, because the

advantaged player is not known ex ante.

There exist many tournament situations with ex post unbalanced competition. For example,

workers with comparable qualifications will usuallyhave different strengths and weaknesses. One

of the strengths may turn out to be decisive, but ex ante neither the firm nor the workers know

whose strengths will be decisive ex post. More generally, individuals typically differ in many

respects, but often it is not clear in advance which specific difference will yield a competitive

advantage when completing complex or innovative tasks or selling to customers with unknown

preferences. As another example, imagine a sales contest with two competing agents and suppose

that there is one big client (that buys a lot of product) and many small clients. The difference in

sales between the twoagents is likely to be disproportionately influenced by the event which agent

will serve the big client. If this event is mainly determined by luck (e.g., which sales agent arrives

first), one of the agents will be advantaged ex post,butex ante it is unclear which agent will be

the lucky one. Finally, imagine a situation with subjective performance evaluationby a supervisor

and agents that differ in at least one dimension (e.g., age, appearance, gender, or nationality).

If the agents expect that the supervisor has some prejudice regarding the dimension(s) in which

they differ, again agents might start in a fair tournament but end up in an ex post unbalanced

competition. In all these examples, no one knows ex ante which specific agent will be favored,

but if it is sufficiently likely that the winner is mainly determined by a competitive advantage

ex post, the incentives of all agents will be harmed.

In such situations, it seems useful to introduce an additional hurdle for the winner to leave

the tournament outcome open. In this article, we analyze a two-agent tournament and show under

which conditions a minimum distance by which the best-performing agent must outperform his

opponent is an effective hurdle. If neither agent has won by this minimum distance, a tie will

occur. In that case, either the sum of the tournament prizes is equally shared among the agents or

a coin toss decides which player obtains the winner prize and which one the loser prize.2Either

alternative ensures Malcomson’s (1984, 1986) self-commitment proper ty of a tournament, which

is important to prevent opportunistic behavior of the tournament organizer—the principal—in

case of unverifiable performance measures.3Contracts of this type retain much of the simplicity

of the standard tournament contract without ties, and we show that the possibility of a tie is often

beneficial for the principal as a measure against ex post unbalanced competition. In this case, the

principal voluntarily introduces the possibility of a tie, even though the monitoring technology

identifies a clear tournament winner. Our results can answer questions such as: Should a worker

obtain a large winner bonus even if his performance is only slightly better than that of his

opponent? In a song contest, should the best singer win by a minimum lead of 1 or 10 merit

points?

Our findings show that, if the problem of ex post unbalanced competition is sufficiently

severe and dominates the influence of idiosyncratic noise, the principal will prefer the possibility

of a tie. Ex post unbalanced competition will be a relevant problem if either agent might have a

2Besides the seniority criterion, random selection of candidates in case of a tie is widely used as part of the

official selection and promotion rules in the US civil services; see, for example, the “Rules of the Springfield Civil

Service Commission,” “The City of Watauga Firefighters’ and Police Officers’ Civil Service Rules and Regulations,”

“Rules of the Civil Service Commission of the City of Cleveland.” Similar tie-breaking rules can also be found in private

corporations (e.g., Kim, 1995). In addition, we observe random election of politicians and judges in case of a tie; see, for

example, Koppel (2012) and Schwarz (2014).

3If the sum of paid tournament prizes is not the same under any event, a rational principal will ex post always

claim the tournament outcome that minimizes his wage costs. As this opportunistic behavior is anticipated by the agents,

incentives would be erased.

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