Heterogeneity of Players and Aggregate Effort in Contests

• DOI: http://doi.org/10.1111/jems.12028
• Date: 01 December 2013
• Author: Dmitry Ryvkin
• Published date: 01 December 2013

Heterogeneity of Players and Aggregate Effort

in Contests

DMITRY RYVKIN

Department of Economics

Florida State University

TallahasseeFL 32306-2180

dryvkin@fsu.edu

We explore the systematic effects of variation in players’ heterogeneity on aggregate effort in

contests. We show that if costs of effort are convex, a mean-preserving increase in the variation

of players’ abilities can lead to an increase or decrease in aggregate effort, both in contests of

complete and incomplete information, depending on the curvature of the effort cost function.

Specifically, if effort costs are not too steep, aggregate effort increasesin ability variation, whereas

if effort costs are sufficiently steep, aggregate effort decreases in ability variation.

1. Introduction

Contests are situations in which participants compete for a valuable prize by expending

effort or other resources. Examples of such situations include rent seeking (Lockard

and Tullock, 2001; Congleton et al., 2008), conflicts (Garfinkel and Skaperdas, 2006),

R&D competition (Taylor, 1995), sports (Szymanski, 2003), and tournaments in the labor

market (Lazear, 1999).

In many cases contests, similar to auctions, can be viewed as incentive provision

mechanisms whose objective, from the organizer’s perspective, is to extract resources

from participants. For example, in a patent race largerlevels of R&Dinvestment are likely

to lead to faster and more productive innovations that the society (which “organizes”

the race by legislating the patent system) benefits from. In sports, higher effort exerted

by athletes makes competition more attractive to spectators and thus generates more

revenue for the organizers. In the context of the labor market—for example, in promotion

tournaments—higher effort of the employees benefits the firm.

One of the ways in which organizers can manipulate aggregate effort in contests

is by varying players’ abilities, directly or indirectly. For example, professional sports

leagues in the United States have revenue-sharing agreements whereby resources are

transferred from small-market to large-market teams effectively reducing the gap in

winning ability (Fort and Quirk, 1995). Additionally, the leagues can affectthe distribu-

tion of team abilities through draft policies, salary caps, and decisions on the location,

ownership, and the number of teams. Within a firm, managers can sort employees into

groups by their ability levels and run tournaments within groups. Managers can also

use firm’s resources to provide training for less able employees or directly help them

with their assignments, thereby leveling out the playing field.

Other things being equal, more able players are expected to perform better, so

policies directed at increasing average ability would be beneficial, but such policies may

I am grateful to two anonymous referees for their comments.

C2013 Wiley Periodicals, Inc.

Journal of Economics & Management Strategy, Volume22, Number 4, Winter 2013, 728–743

Heterogeneity and Aggregate Effort in Contests 729

not always be available or can be too costly. It may, however, be feasible to make mean-

preserving changes in abilities, for example, by transferring resources between players

or through ability-specific sorting of players into groups. The discussion of competitive

balance in the sports economics literature (see, e.g., Szymanski, 2001; Zimbalist, 2002;

Sanderson, 2002, Fort and Maxcy,2003; Szymanski and K ´

esenne, 2004; Grossmann et al.,

2010), is focused on the effect of parity or heterogeneity in players’ abilities on sports

leagues’ outcomes. The general consensus in that literature is that, given the average

ability level, more parity is better.1

In this paper, we address theoretically the following question: how does player

heterogeneity, or competitive balance, affect aggregate effort in contests? In the the-

oretical literature on tournaments and contests, the effect of heterogeneity of players

on aggregate contest outcomes has only been studied in a few specialized settings.

Stein (2002) considers the contest model of Tullock (1980) with linear costs of effort and

heterogeneous prize valuations and finds that “...introducing variation into the [prize

valuations] will decrease the rent dissipation assuming [the number of active players]

is not affected.” Cornes and Hartley (2005) extend this result to contests with contest

success functions (CSFs) with a power-law input production technology. It is still un-

clear to what extent this result is universal, however, as the analysis of contests with

heterogeneous players and more general CSFs is complicated.2

Additionally,most existing studies make the assumption that players’ abilities are

common knowledge. In applications, it is often the case that others’ abilities are not

observable to players.3The behavior of players in contests of incomplete information

has also received relatively little attention in the literature (for most recent results and

a literature review see, e.g., Fey, 2008; Ryvkin, 2010), and there is not a single study, to

the best of our knowledge, on the effect of player heterogeneity on aggregate contest

outcomes under incomplete information.

In this paper, we propose an approachthat allows us to analyze aggregate effort in

contests in a fairly general framework that provides a “natural” measure of heterogene-

ity. The approach relies on the assumption that players’ heterogeneity is not too strong

(the “weak heterogeneity” assumption) and uses quadratic approximation around the

symmetric equilibrium point to explore aggregate effort. Effectively, we find the expan-

sion of aggregate effort in the moments of the distribution of players’ abilities. These

are the true moments of the underlying stochastic abilities in the case of incomplete

information, or the sample moments of the observed abilities in the case of complete

information. Truncating the expansion after the second order, we obtain the impact of

heterogeneity in the form of the variance of abilities. We note that the previous studies

of contests, auctions, and other games under the weak heterogeneity assumption (see,

e.g., Fibich et al., 2002,2004,2006; Fibich and Gavious, 2003; Ryvkin, 2007,2009) were

restricted to the linear approximation where there is no impact of heterogeneity on ag-

gregate effort. Quadratic is the lowest order of approximation wheresuch an impact can

be identified.4

1. This statement should be understood as a ceteris paribus result. When parity is endogenous, for example,

when it is achieved by sharing revenue, the overall effect can be ambiguous, see, for example, Szymanski and

K´

esenne (2004).

2. See, for example, Rinott et al. (2012) for a discussion of a related problem in the operations research

literature.

3. Even in such cases, abilities are often observed by contest organizers. For example, managers typically

have access to employee background and performance records, whereas other employees do not.

4. Ryvkin (2011) uses the quadratic approximation to study the effect of sorting of players in contests

between groups.