Social welfare in sports leagues with profit-maximizing and/or win-maximizing clubs.

• Author: Dietl, Helmut M.

1. Introduction

   Welfare analysis is the heart of economics. There is a huge body of literature devoted to the welfare effects of regulations, institutions, policies, and the like. Surprisingly, there are hardly any welfare analyses in the professional team sports industry. We believe that the lack of welfare analysis in professional team sports is caused by the confusion created by the so-called uncertainty of outcome hypothesis (Rottenberg 1956; Neale 1964). According to this hypothesis, fans prefer to attend games with an uncertain outcome and enjoy close championship races. The uncertainty of outcome hypothesis describes one of the unique economic characteristics of the team sports industry. Unlike Toyota, Microsoft, and Wal-Mart, which benefit from weak competitors in their respective industries, Real Madrid and the New York Yankees need strong competitors to maximize their revenues. In sports, a weak team produces a negative externality on its stronger competitors.

   Based on the uncertainty of outcome hypothesis, professional team sports leagues have introduced a variety of measures to increase competitive balance. Two of the most prominent measures are reserve clauses (1) and revenue-sharing arrangements. Whether these measures actually increase competitive balance is the most disputed question in the sports economics literature. According to Rottenberg's invariance proposition, (2) the distribution of playing talent between clubs in professional sports leagues does not depend on the allocation of property rights to players' services. In particular, changes in property rights, such as the introduction of a reserve clause, will not alter the allocation of players and therefore have no impact on competitive balance. Quirk and El-Hodiri (1974), Fort and Quirk (1995), and Vrooman (1995) extended this invariance proposition to gate revenue sharing. Invariance propositions provide economists with tough challenges, both theoretically and empirically. Theoretically, it is important to identify the exact assumptions under which such propositions hold. The empirical challenge is to show whether these assumptions actually hold and lead to the predicted results. So far, the empirical challenge has proved to be too tough because apart from the problems of measuring competitive balance, it has been impossible to isolate the effect of single measures such as revenue sharing or free agency on competitive balance.

   A number of authors have taken on the theoretical challenge. Their analysis can be grouped along two dimensions of assumptions: profit maximization versus win maximization and fixed versus flexible supply of talent. Along the first dimension, club owners may be modeled either as profit maximizers or win maximizers. Profit maximizers do not care about winning percentages unless they affect profits. Win maximizers invest as much as they can into playing talent and are only constrained by zero profit. The second dimension concerns the elasticity of talent supply. Under the assumption of fixed supply, aggregate talent within the league is constant, and the race for talent is a zero-sum game between owners. Under flexible supply, owners can hire as much talent as they want at a constant (exogenous) wage rate. According to this categorization, the invariance proposition with regard to revenue sharing is derived under the assumptions of profit maximization and fixed supply. There is wide agreement that the invariance proposition does not hold in leagues with either win-maximizing owners or flexible talent supply (see, e.g., Atkinson, Stanley, and Tschirhart 1988; Kesenne 2000, 2005; Vrooman 2008). There is disagreement, however, on whether the invariance proposition holds in a league with profit-maximizing owners and fixed talent supply. Szymanski and Kesenne (2004) argue that increased gate revenue sharing results in a more uneven distribution of talent between large- and small-market clubs. This result contradicts the invariance proposition with respect to gate revenue sharing.

   Where does this disagreement come from? Obviously, Szymanski and Kesenne work from the same assumptions as do Quirk and El-Hodiri and others. The root of the disagreement is in the underlying model conjectures. As Szymanski (2004) has shown, the assumption of fixed talent supply is often used to justify Walrasian fixed-supply conjectures instead of contest-Nash conjectures. Under Walrasian fixed-supply conjectures, the quantity of talent hired by at least one club owner is determined by the choices of all the other club owners. In a two-club league, the Walrasian fixed-supply conjecture collapses the non-cooperative choice of talents into a choice of winning percentages by only one club owner. Under the Walrasian fixed-supply conjectures, the "game" between profit-maximizing owners loses its non-cooperative character and leads to results that are more in line with joint profit maximization.

   We believe that the invariance proposition and the related literature on competitive balance miss the point by raising the wrong question. In our view, it is much more important to analyze the welfare effects of different assumptions and issues of league design, such as club owner objectives and revenue sharing, than their effect on competitive balance. An exclusive focus on competitive balance would only be justified if the uncertainty of outcome hypothesis completely holds. If, on the other hand, social welfare does not monotonically increase as competitive balance increases, an exclusive focus on the effects of different assumptions and measures on competitive balance will result in inefficient policy conclusions.

   There is strong evidence that competitive balance is not a good proxy for social welfare. Theoretically, a fully competitive league does not maximize social welfare if clubs differ with respect to market size. Large-market clubs have, on average, higher marginal revenues of wins than do small-market clubs. As a result, league revenues (and profits) are maximized when the large-market clubs have higher winning percentages than do their small-market rivals. Empirical evidence supports the assumption that match attendance is maximized when the home team's winning probability is about twice as large as that of the visiting team (e.g., Forrest and Simmons 2002; for an overview, see Borland and Macdonald 2003).

   Given this evidence, we present a model that analyzes the welfare effects of heterogeneous club objectives. So far, most models have assumed that leagues were homogeneous in the sense that all clubs maximize identical objective functions.

   Traditionally, these objectives were either profit maximization or win maximization. Exceptions are Rascher (1997) and Vrooman (1997, 2000), who introduced a league in which owners maximize a combination of profits and wins. This objective function is more general than are the traditional assumptions because it allows club owners to trade off profits for wins. Even with this more general objective function, however, the league is still modeled as homogeneous because all the club owners maximize identical objective functions.

   Our major contribution in this respect is the introduction of heterogeneous objective functions. This extension allows us to compare mixed leagues in which club owners maximize different objective functions with homogeneous leagues in which all the club owners maximize identical objective functions. Mixed leagues have not yet been modeled in sports economics despite the fact that most major leagues are mixed leagues. For example, the most valuable team in 2008, according to Forbes, Manchester United, is fully owned by the Glazer family and may be regarded as a profit-maximizing club. In the prestigious Union of European Football Associations (UEFA) Champions League, Manchester United competes against clubs such as Real Madrid, F.C. Internazionale Milano, and F.C. Barcelona. Since these clubs are organized as (not-for-profit) members associations, they should be characterized as win-maximizing clubs.

   Second, and most importantly, we explicitly integrate the consumer (fan) into our analysis in order to compare social welfare in homogeneous and mixed leagues. We derive club-specific demand and revenue from a general fan utility function by assuming that a fan's willingness to pay depends on fan type, on the preferred team's winning percentage, and on competitive balance.

   Using this approach, we are able to extend the literature by providing an integrated framework to analyze welfare effects. In particular, we show that homogeneous leagues in which all clubs are profit maximizers dominate all other leagues; whereas, mixed leagues in which small-market clubs are profit maximizers and large-market clubs are win maximizers (type-I mixed leagues) are dominated by all other leagues. In addition, we show that, from a welfare perspective, large-market clubs win too often in (purely) win-maximizing and type-I mixed leagues; whereas, small-market clubs win too many games in (purely) profit-maximizing leagues and in mixed leagues in which the large-market clubs are profit maximizers and the small-market clubs are win maximizers (type-II mixed leagues).

   These results have important policy implications. For example, they show that---contrary to prevailing claims--social welfare would increase if clubs were reorganized from win-maximizing, non-profit members associations to profit-maximizing (public or private) corporations. Moreover, it is socially desirable to reorganize large-market clubs first because in mixed leagues it is better if the large-market clubs maximize profits instead of the small-market clubs. Furthermore, the efficiency of measures that increase the competitiveness of small-market clubs depends on the league type. If the large-market clubs are profit maximizers, for example, small-market clubs should win fewer rather than more games.

   Finally, we derive new insights...