Existence of Nash equilibria in sporting contests
Chuo UniversityShumei Hirai

The main purpose of this paper is to demonstrate the existence ofpure-strategy Nash equilibria in an “n” team sporting contest. Since theseminal papers of Szymanski (2003, 2004) and Szymanski and Késenne(2004), the Nash equilibrium model has been used in the analysis ofprofessional team sports. However, many papers have been restricted to a
two-team league model (Chang and Sanders, 2009; Cyrenne, 2009; Dietl etal., 2009). Dietl et al. (2008) that is considered a more general n-team league
model; however, it is based on the assumption that all teams have identicalrevenue generating potential and cost functions. Thus the sporting contest issymmetric. These restrictions most probably apply to the Nash equilibrium
model in sports because of the difficulty in managing non-identical teams
with respect to their market size or drawing potential by conventional means,
which treat the Nash equilibrium as a fixed point of the best response
mapping. This entails working in a dimension space equal to the number ofteams. In the present study, we adopt an alternative approach introduced inCornes and Hartley (2003, 2005), which allows us to work completely withfunctions of a single variable, considerably simplifying the analysis. We will
prove that there exists a unique non-trivial Nash equilibrium in which atleast two teams must be active in equilibrium.
In addition, this study demonstrates that at the non-trivial equilibrium,
each team's winning percentage and playing talent are determined by its
composite strength, its market size and win preference. The findings'
implications are significant for the premise of competitive-balance rules suchas revenue sharing and salary caps. It has been recognized that unrestrictedcompetition between teams will lead to a league dominated by a few
large-market teams with strong-drawing potential. In the theoretical
literature on sports contests, however, this situation is not self-evident.
Szymanski and Késenne (2004, p. 169) demonstrated that if there is norevenue sharing in equilibrium, a large-market team will dominate a small
one in a two-team league. However, Késenne (2005, p. 103) observed that
this result does not necessarily hold in an n-team model. Moreover, Késenne(2007, pp. 54-55) and Dietl et al. (2011) demonstrated that if team objectives

maximize a combination of profits and wins, as introduced by Rascher (1997),
a large-market team will not always dominate a small one in equilibrium,
but these studies are restricted to two-team models. The contribution of thepresent study is in unifying and clarifying the results of these studies byputting them into a more general n-team model.
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Topics in Microeconomics: IndustrialOrganization, Auctions, and Incentives，Cambridge: Cambridge University[28]Press.

Wolfstetter,E.

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