The Instability of Win Maximizing Professional Sports Leagues 3rd IMA International Conference on Mathematics in Sport 2011 Alexander Konovalov University of Gothenburg Wolfgang Höchtl University of Innsbruck

Motivation Difficult financial situation in many european football leagues, debt, financial instability of clubs (Barros, 2006; Bosca et al, 2008; Dimitropoulos, 2010). Football clubs in Europe are win maximizers rather than profit maximizers (Garcia-del-Barrio and Szymanski, 2009). The paper seeks to explain the current crises by looking at the stability properties of win maximizing equilibria and consider the possible remedies to the problem.

Win maximization vs profit maximization Talent level of the team Revenue function Cost function

Win maximizing vs profit maximizing equilibria Talent level of the team i Talent level of the team j

The model

The assumptions Convex

Two cases

Stability issue

Stability problems

Profit maximization An equilibrium is always stable (A) by definition. An equilibrium, once unique, is also stable (B).

Salary cap Solves the problem of A-instability (may require restrictive limits). Solves the problem of B-instability (even mild restrictions will do).

Reaction function of i Reaction function of j Shock occurs Salary caps and instability of an equilibrium

Reaction function of i Reaction function of j A constrained equilibrium Shock occurs Salary caps and instability of an equilibrium (cont.)

Salary cap on foreign players

Further research: the impact of revenue sharing (?) Improves competitive balance, may help to get rid of “downslide” equilibria. (?) Decreases marginal revenues of the teams, may help to solve the problem of instability (A).

Conclusion The equilibria in win maximizing small scale professional sports leagues may violate stability properties. The problem of stability can be solved through the introduction of salary caps and (possibly) by other measures.