On the Existence of Nash Equilibria in Asymmetric Sporting Contests with Managerial Efficiency

This paper considers a contest model of an n-team professional sports league. Teams can have different drawing potentials and different managerial skills to transform a given set of p laying talents into play ing performance. The analysis demonstrates that there exists a unique non-trivial Nash equilibrium under the general conditions (i.e., the revenue functions of the teams are concave, the production functions of the teams are strictly increasing and concave, etc). The proof uses the share function approach with the following two reasons: 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.


Introduction
This paper provides a general proof of the existence of pure-strategy Nash equilibria in an n-tea m sporting contest with heterogeneity of market size and of managerial efficiency among the teams. Since the seminal paper of [21], the Nash equilibriu m concept has been used in the analysis of professional team sports. However, there has been no attempt in the literature to provide a general proof of equilibriu m existence and uniqueness for economic modeling of team sports. Most papers have been restricted to a two-team league model. i Diet l et al. [3] that are considered a more general n-team league model; however, it is based on the assumption that all teams have identical revenue generating potential and cost functions. Thus the sporting contest is symmetric. Moreover, the existing theoretical studies implicit ly assume that all team managers/coaches have same managerial skills such as train and motivate individual player to achieve higher levels of playing performance. ii So me emp irical studies, however, have found evidence that managerial quality and experience is positively related to team and player performance ( [10], [18]); in addit ion, some managers are more efficient than others at transforming a given set of player inputs into team wins ( [14], [8]).
These restrict ions most probab ly app ly to the Nash equilibriu m model in sports because of the difficulty in managing non-identical teams with respect to their market size and/or managerial efficiency by conventional means, which treat the Nash equilibriu m as a fixed point o f the best response mapping. This entails working in a dimension space equal to the number of teams. In this paper, we adopt an alternative approach introduced in [2], which allows us to work co mpletely with functions of a single variable, considerably simplify ing the analysis. In a general asymmetric sporting contest, this paper will p rove that under general conditions, there exists a unique non-trivial Nash equilibriu m in which at least two teams must be active in equilibriu m.
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-tea m sporting contest. Concluding remarks are presented in Section 4.

The Model
We consider a professional sports league consisting of n(≥2) teams where each team i(=1,⋯,n) independently chooses a level of talent, t i (≥0), to maximize its profits. By assuming a co mpetitive labor market and fo llowing the sports economic literature, talent can be hired in the players' labor market at a constant marginal cost c>0; hence, the cost function can be written as C i (t i )=ct i .
(1) On the revenue side, the season revenue function of a team is defined as R i =R i (w i ).
(2) R i is total season revenue of team i, w i is the winning percentage of the team. It is co mmon in the sports economics literature to assume the follo wing.  [13]) that consumers in aggregate prefer a close match to one that is unbalanced in favor of one of the teams. Following [15, p. 272], we define the marginal revenue of a win for team as the market size or drawing potential fo r the team. iii A particularly well-studied form for R i is R i =m i w i -bw i 2 , where m i >0 represents the market size of team i and b≥0 characterizes the effect of co mpetit ive balance on team revenues.
The win percentage is characterized by the contest success function (CSF). The most widely used functional form in sporting contests is the logit that can be written as where T -i = ∑ t j n j≠i . 1 The factor n/2 results fro m the fact that winning percentages must average to 1/2 within a league during any one year; that is, that for the two-team models, the logit CSF (3) does not place a restraint on the teams' choices. Ho wever, for the n-team models this is not the case with the logit CSF (3). More precisely, the winning percentage can be larger than one if a team holds more than 2/n per cent of total league talent (with normalizat ion of ∑ t j n j=1 to one). 2 To avoid this, we can define the winning percentage as As already mentioned in the Introduction, we consider an asymmetric production technology describing the relationship player talent and player performance as follows: where y i is the level of p layer performance of team i. We call f i (⋅) the player-performance production function of team i. It represents the team i's production technology by which levels of talents are translated into a level o f the actual playing performance. We assume that Assumpti on 2. For all i the function f i satisfies the following conditions: The logit CSF is explicitly adopted in the seminal work of [4]. See also the excellent survey by [20]. 2  Notice that teams' production functions do not necessarily have to be identical. For examp le, a functional of , where a i >0 and γ i ∈(0,1]. This functional form was used by [3] and [6] but assuming identical parameters, i.e., a i =1 and γ i =γ for all i . Since f i is monotonic, it has a well-defined inverse function, g i (y i )=f i -1 (y i ). Then, A.2 imp lies that The function g i (y i ) times c describes the total cost to team i of generating the level y i of performance.
Fro m the p layer-performance production function (5), the logit CSF (3) and (4), we can define the win percentage of team i as follo ws: where Y -i = ∑ y j n j≠i . Then, the profit of team i is described by π i (y i ,Y -i )=R i (w i )-cg i (y i ). (8) Team i's original maximization problem is equivalent to the one of maximizing (8) with respect to y i . This defines a simu ltaneous-move game and the solution concept we use throughout the paper is that of a pure-strategy Nash equilibriu m of this game. 3

Existence Analysis
We can now calculate the best response of team i. Assume first that Y -i =0 in order that the other teams do not spend any resources on playing talent. Then, if y i >0 , the profit is negative in light of A.1, A.2, and (7). If team i sets y i =0, the profit becomes zero. Therefore, this game always has a trivial equilib riu m point ȳ1 =ȳ2 =⋯=ȳn=0. Ou r concern is with the non-trivial equilibriu m (i.e., ∑ ȳi n i=1 >0) and thus no further consideration is given to the trivial point.
If Y -i >0, it follows fro m (8) that we have ∂ ∂y i As the second-order condition we get ∂ 2 ∂y i ) Under A.1 and A.2, the second-order condition (10) is satisfied. Hence, it follows fro m (9) that given Y -i >0, team i's best response function y i =ϕ i (Y -i ) is given by 3 It is occasionally assumed that the total supply of talent is fixed in the analysis of sports leagues. Authors who have made this assumption have used a non-Nash conjecture to reflect this scarcity in each team's first-order condition ( [5], [22]). In this case and fo r a two-team we have dt 2 dt1 =-1. Indeed, although opinion is divided among sports economists on this subject, we use the Nash conjecture in this paper (see e.g., [11]).
where y i * is the unique solution of the strictly monotonic Observe that due to A.1 and A.2 the left-hand side of (12) strictly decreases and is continuous in y i and positive at y i =0; therefore there is a unique solution. It is well known that a strategy profile (ȳ1 ,⋯,ȳn ) is an equilibriu m if and only if for all i, ȳi is the best response with fixed values of Ȳ -i .
Further, we can rewrite the best responses of the teams in terms of aggregate player performance, which we will where y i ** solves equation Note that in the second case of (13), the left-hand side of (14) is positive at y i =0 and strictly decreasing, because it has a negative derivative given by ∂ where the sign comes fro m A.1 and A.2. Therefore there is a unique solution of equation (14), wh ich is a continuously differentiable function of Y>0 by the implicit function theorem. Following[23, p. 91], we call Φ i (⋅) the inclusive reaction function of team i, which is proposed by [19]. Rather than use the inclusive reaction function directly, we will examine properties of player 's share function s i (Y)=Φ i (Y)/Y, which is proposed by [2]. It can be readily checked that Nash equilibriu m values of occur where the aggregate share function equals unity. That is, ∑ s i n i=1 (Ȳ )=1. Given Ȳ , the corresponding equilib riu m (ȳ1,⋯,ȳn ) is found by mult iplying Ȳ by each team's share evaluated at Ȳ : ȳi =Ȳ s i (Ȳ ). This result enables us to prove the existence of a unique equilibriu m by demonstrating that the aggregate share is equal to one at a single value of . We can now define a share function for each team and denote team i's share value by σ i =y i /Y.
where σ i * is the unique solution of Proof. Using σ i =y i /Y, we can rewrite (14) as (16). Recall that a team's winning percentage in (7) is determined by the ratio of its performance to aggregate performance in the league. Therefore, team i 's revenue can be written as a function of σ i .
Let us denote the left-hand side of (16) by G i (σ i ) and the right-hand side by H i (σ i ). An intersection of these two functions, if any, which is a solution of (16) The proof is co mp leted by observing that The following lemma gives the crucial qualitative properties of the share function derived under A. 1 (16) demonstrates that the share must approach one as Y approaches zero, g iving Part 2. To justify Part 3, we investigate the slope of . The total differential of (16) has the following form: We can then express the slope of s i as follo ws: The inequality follows since the denominator is negative by A.1 and the numerator is positive by A.2. We may deduce that the positive shares are strictly decreasing in Y, establishing Part 3.
This completes the proof. It follows fro m Lemma 2 that the aggregate share function is continuous, exceeds 1 for s mall enough Y, is less than 1 for large enough Y, and is strictly decreasing when positive. Therefore, the equilibriu m value is unique. Finally, recall that a unique Ȳ implies a unique strategy profile (ȳ1,⋯,ȳn ), and we have the following result. Theorem 1. Under A.1 and A.2, the sporting contest has a unique non-trivial Nash equilibriu m in pure strategies.
Notice that for all team i and any fixed value of Y -i , the solution y i =0 always gives zero profit for this team. Therefore, at the best response, team i's profits must not be negative. Hence, under A.1 and A.2, each team enjoys nonnegative profits at the equilibriu m.

Conclusions
This study has proven that under general conditions, a unique non-trivial Nash equilibriu m exists in a contest model of an -team sports league with different drawing potentials and different managerial skills among the teams. Over the past few years, the Nash equilibriu m concept has been used in the analysis of professional team sports. A particularly great deal of attention has been focused on revenue sharing's effects on competitive balance. However, when the number of teams exceeds by two, revenue sharing's effects on the competitive balance are not clearly described. This study applies the share function approach to a general -team professional sports model, an approach that avoids the dimensionality problem associated with the best response function approach. We believe that the present paper may serve as a basis for further research on the effects of competitive-balance rules, such as revenue sharing and salary caps.