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
Revenue function
Cost function
Talent level of the team
Win maximizing vs profit maximizing equilibria
Talent level of the team j
Talent level of the team i
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).
Salary caps and instability of an equilibrium
Reaction function of j
Reaction function of i
Shock occurs
Salary caps and instability of an equilibrium (cont.)
Reaction function of j
Reaction function
of i
A constrained equilibrium
Shock occurs
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.