Abstract
In this paper, we consider the sports industry and in particular we focus our study on the European Football. In this market, the more the league is balanced with good teams and the more it becomes profitable. However, empirical data suggest that the top teams involved in more balanced national championships have less probability to win the international competitions. Starting from these considerations, we propose a model which examines, from the point of view of a profit-maximizer sports league, the opportunity to introduce measures in order to foster a balanced internal championship, taking into due account the competitiveness of the internal teams in the international competitions. Then we extend the model in order to include strategic interactions between more leagues in a non-cooperative game setting, and compare the Nash Equilibrium of the game with the cooperative outcome, in which the leagues maximize their joint expected profits.
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Notes
For a survey of previous contributions, see Szymanski (2003).
For a discussion on the different measures of competitive balance, see Fort and Maxcy (2003).
The same result holds if we also consider the Europa League (UEFA Cup, until 2008–2009) winners, since in the considered time period the Spanish teams won it eight times (five times Sevilla, three times Atletico de Madrid), much more than every other league’s teams.
To formalize, denote with X the random variable which takes value b if the internal championship is balanced and \({{\bar{b}}}\) otherwise. Similarly, denote with Y the random variable which takes value c if an internal team wins the international cup and \({\bar{c}}\) otherwise. We are thus assuming that: \({\mathbb {P}}[X=b|Y=c]={\mathbb {P}}[X=b]=p(a),{\mathbb {P}}[Y=c|X=b]={\mathbb {P}}[Y=c]=q(a)\).
In particular, we have:
$$\begin{aligned}&\frac{\partial (p(a^*)-q(a^*))}{\partial {\overline{\delta }}}=-\frac{C\varDelta _\delta [4B^2\varDelta _\epsilon ^2 + BC\varDelta _\epsilon (3\varDelta _\delta + 2\varDelta _\epsilon ) + C^2\varDelta _\delta ^2]}{(B\varDelta _\epsilon + C\varDelta _\delta )^3}<0,\\&\frac{\partial (p(a^*)-q(a^*))}{\partial {\overline{\epsilon }}}=\frac{B^3\varDelta _\epsilon ^3 + B^2C\varDelta _\delta \varDelta _\epsilon (2\varDelta _\delta + 3\varDelta _\epsilon ) + 4BC^2\varDelta _\delta ^2\varDelta _\epsilon }{(B\varDelta _\epsilon + C\varDelta _\delta )^3}>0,\\&\frac{\partial (p(a^*)-q(a^*))}{\partial {\underline{\delta }}}= -\frac{B\varDelta _\epsilon ^2[b^2\varDelta _\epsilon - BC\varDelta _\delta - 2C^2\varDelta _\delta ]}{(B\varDelta _\epsilon + C\varDelta _\delta )^3},\\&\frac{\partial (p(a^*)-q(a^*))}{\partial {\underline{\epsilon }}}=-\frac{C\varDelta _\delta ^2[2B^2\varDelta _\epsilon + BC\varDelta _\epsilon - C^2\varDelta _\delta ]}{(B\varDelta _\epsilon + C\varDelta _\delta )^3}. \end{aligned}$$According to compactness and closeness hypothesis on strategy sets and by using the continuity and concavity of objective functions, the existence of a Nash Equilibrium is guaranteed (see, e.g., Gibbons 1992).
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The authors are grateful to two anonymous referees for their very helpful comments and suggestions.
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Bisceglia, M., Caputi, A.G., Grilli, L. et al. Internal Balance and International Competitiveness: Sports Leagues Decision Models. Ital Econ J 4, 567–578 (2018). https://doi.org/10.1007/s40797-018-0079-1
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DOI: https://doi.org/10.1007/s40797-018-0079-1