Existence of Nash equilibria in sporting contests with capacity constraints

Abstract

The main purpose of this paper is to demonstrate the existence of pure-strategy Nash equilibria in an “n” team sporting contest. Since the seminal papers of Szymanski (2003, 2004) and Szymanski and Késenne (2004), the Nash equilibrium model has been used in the analysis of professional team sports. However, many papers have been restricted to a two-team league model (Chang and Sanders, 2009; Cyrenne, 2009; Dietl et al., 2009). Dietl et al. (2008) that is considered a more general n-team league model; however, it isbased on the assumption that all teams have identical revenue generating potential and cost functions. Thus the sporting contest is symmetric. 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 of teams. In the present study, we adopt the share function approach introduced in Cornes and Hartley (2005) which extends the inclusive reaction function used by Szidarovszky and Yakowitz (1977) to study Cournot oligopoly games. The advantages of the approach are twofold: one is to avoid the proliferation of dimensions associated with the best response function approach and the other is to be able to analyze sporting contests involving many heterogeneous teams. Following the same steps as in Cornes and Hartley and Hirai and Szidarovszky (2013), we will prove that there exists a unique non-trivial Nash equilibrium in which at least two teams must be active in equilibrium. The uniqueness of Nash equilibrium is an important issue for the non-cooperative game. If the equilibrium is unique, then we have a self-constrained theory for predicting the game’s outcome. Moreover, uniqueness is crucial for comparative statics analysis which allows one to obtain qualitative results. 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 such as revenue sharing and salary caps. It has been recognized that unrestricted competition 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 no revenue 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 objectivesmaximize 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 the present study is in unifying and clarifying the results of these studies by putting them into a more general n-team model. The rest of the paper is organized as follows. Section 2 explains the basic model and the assumptions. In Section 3, we establish the existence of Nash equilibria in an n-team sporting contest. In this section, we also compare the winning percentage and playing talent of teams of different market sizes and win preferences. Concluding remarks are presented in Section 4.