A new model of sports leagues

by

John G Treble[1]

(University of WalesSwansea)

February 2005

  1. Introduction

In this paper I construct a new model of sports leagues. The main difference between this model and the well-known work of El-Hodiri and Quirk(1971), Vrooman(1995), and Szymanski and Késenne(2004) lies in the specification of the revenue function. These authors agree that a defining characteristic of sports industries is that demand should be stronger for closer competitions. Yet none of these papers adopt revenue functions that are founded on such a demand structure. Revenue function aside, the model is very conventional, following the usual procedure of the industrial economics literature in specifying a demand structure for an industry’s output, a specification of industry structure, and a cost structure for each firm. These elements then are used to construct revenue and cost functions, the manipulation of which generates behavioural predictions, which may or may not accord with observations from the world around us.

The key feature of the model is that it takes the ‘peculiar economics’ (as Neale(1964) called them) of sports leagues seriously. Neale distinguishes clearly between sporting competition and economic competition, arguing that sports leagues are natural monopolies, and that it is a mistake to treat individual teams as if they are firms. In what he calls the ‘Louis-Schmelling paradox’(sic), he asks the reader to:

“consider the position of the heavyweight champion of the world. He wants to earn more money, to maximise his profits. What does he need in order to do so? Obviously, a contender, and the stronger the contender the larger the profits from fighting him. ... Pure monopoly is a disaster: Joe Louis would have had no one to fight and therefore no income....The first peculiarity of the economics of professional sports is that receipts depend upon competition among the ... teams, not upon business competition among the firms running the contenders.”

Neale goes on to argue that sports leagues are de facto monopolistic organisations, and this may well be the case. Nonetheless, even monopolies will sometimes find it useful to provide incentives for performance among their staff, and one of the best motivators known to man is a financial gain. Thus, even though a sports league may be a monopoly, it can still find it worthwhile to organise its business in such a way that its constituent teams engage in economic as well as sporting competition.

In the model developed here, there is a single organisation – the league – which consists of a given number, n, of teams. Each team has a fan base, and indeed the only ex ante distinction between teams is the size of their fan bases. A high quality product is generated by two factors that are valued by consumers: the closeness of competition and the absolute quality of the teams. If it is indeed true that ‘receipts depend upon competition among the teams’, then useful models of the industry will incorporate it. To our knowledge, no previous attempt has been made to model this distinctive characteristic of demand, although nearly all writers in the field mention it. In this paper I undertake this task, using the simplest formulation possible. The stripped-down model is nonetheless rich in insights, and generates results that are at odds with the existing economics literature, but not with the views of many in the industry.

If a close competition is valued by consumers then profit-maximising behaviour by teams in a league is unlikely to be individualistic. There is a real sense in which the provision of entertaining games, while it necessarily involves sporting competition, requires a generous measure of economic co-operation. Not surprisingly, then, the equilibria discussed in this paper turn out to be generalisations of Hirshleifer’s ‘weakest-link’ equilibrium, which is one of the key explanations of co-operative behaviour in the literature on private provision of public goods.

  1. The existing literature

The literature on sports leagues is dominated by the model first presented by el-Hodiri and Quirk(1971), and later amended by Vrooman(1995)[2]. In constructing their model, el-Hodiri and Quirk were mainly interested in evaluating the draft, which is a system used in North American leagues (but not elsewhere) designed to give teams with small home fan bases (small-drawing teams) an advantage in the market for players. Their model is a rather complicated attempt to model the dynamics of leagues, which, since the drafting rules are essentially dynamic in nature, makes sense for them. One aspect that is perhaps less complex than it ought to have been is their assumption that the supply of ‘talent’ is fixed. This assumption was, presumably a reasonable one in an analysis of the issues that they focus on in their paper, but is almost certainly not reasonable as a general proposition. Even baseball players have alternatives.

A second aspect of their model that is simpler than it should be is their specification of the revenue function. Here they assert that revenues rise with the winning percentage, but at some point greater than 50% fall, because having too many wins falls foul of the Louis-Schmeling paradox. It is worthwhile citing el-Hodiri and Quirk verbatim:

“The essential economic fact concerning professional team sports is that gate receipts depend crucially on the uncertainty of outcome of the games played within the league. As the probability of either team winning approaches 1, gate receipts fall substantially. Consequently, every team has an economic motive for not becoming ‘too’ superior in playing talent compared with other teams in the league. On the other hand, gate receipts of the home team are an increasing function of the probability of the home team winning for some range beyond a probability of .5, so that every team also has an economic incentive to be somewhat superior to the rest of the league.”

The last sentence of this paragraph is a non sequitur, and the main business of the present paper is to show why and to put it right. In particular, I show that if it is indeed the case that consumers value not only quality, but also closeness of competition[3], the maximum of a small team’s profit function may well be at less than 50% winning percentage. Indeed, in the equilibrium derived below, not only is this the case, but it is possible that the smallest team is the only team that would not seek to increase the quality of its players, even though its winning percentage can be well below 50%.

The literature based on El-Hodiri and Quirk’s model maintains the second of these two contentious assumptions with few exceptions. For instance, Szymanski and Késenne(2004) relax the assumption of fixed labour supply, and arrive at the counterintuitive conclusion that gate-sharing will make competitive balance worse. A notable exception is the paper by Késenne(1996), in which he shows generally that Quirk and El-Hodiri’s(1974) claim that competitive balance is invariant to gate-sharing is wrong, but fails pin down why. Késenne attempts to analyse a revenue function that is sensitive both the quality and quality difference, but his example[4] fails to meet the requirement that demand should increase both in quality and in competitive balance. The task undertaken in the present paper is to reinvent the model with a revenue function firmly based in the Louis-Schmeling paradox: “the stronger the contender the larger the profits from fighting him”, bearing in mind that to each competitor, the other is the contender.

The idea that competitive balance is an important component of demand has been a theme of the literature on professional sports ever since Rottenberg(1956). More recent literature attempting to subject the idea to empirical test has established competitive balance as a component of demand, but it appears dubious as to its importance relative to other factors, such as quality of play. Forrest et al.(2005) conclude: … “although outcome uncertainty is a significant determinant of audience size, the magnitude of its impact appears to be modest relative to the prominence of the issue in discussion of sports policy.” In what follows, the demand function used is perhaps more extreme that is justified by the empirical findings, but it is certainly in keeping with the persistent view, shared by the literature since its inception, that competitive balance is crucial in generating revenue. The results also turn out to be consistent with the view often expressed in the industry, but contradicted by much of the theoretical economic literature, that gate-sharing encourages competitive balance and increased quality.

3.The model

Consider a league that consists of n distinct teams, each of which is run by a profit-maximising entrepreneur. Let denote quality (or ‘managerial effort’) of team i. This variable is best thought of as the mean of a probability distribution of performance[5], and relative performance determines the outcome of each game. If two teams of equal quality meet, the probability of either of them winning is 0.5. Suppose that , then i has a lower probability of winning than j, and the larger the difference, the smaller that probability will become. This formulation has the agreeable property that no ‘adding-up constraint’ is necessary. Adding-up is a natural feature of the model.

Suppose that each team has a fixed fan base, , and that they are indexed in such a way that . The fan-base for any particular team may be simply thought of as the size of the city in which it is based, but may be more elaborate than that, taking into account such factors as local cultural differences, the presence or absence of competing events in a city, and so on. If , we refer to team j as the larger team and team i as the smaller team. At the start of a season, the manager of team i can choose at a cost. The quality of the team is then fixed for the duration of the season. Once a team of quality has been chosen, each member receives a fixed fee per game. We make the simplest possible assumption as to the structure of this cost, that is that the cost per game of a team with quality is . This implies marginal cost equal to .

Let the demand for a game between teams i and j by any individual consumer depend on the closeness of the competition and on the quality of the teams. A simple functional form that reflects this idea is . As indicated in the plot below, this ‘demand’[6] function increases in the quality of both teams, and is weakly maximal when team qualities are equal.[7] Viewed as a three-dimensional object the function looks like the corner of the roof of a house (See Figure I). Assume initially that there is no gate-sharing, so that home-teams take the entire revenue from each game.

4.Optimal team quality decisions without gate-sharing

Using these specifications, the profit functions for each team can be formulated. For clarity, I do this initially for a 2-team league that plays one game on each home ground. The revenue functions depend on the size of the fan base and the individual demand function that captures the idea that both closeness of competition and quality are valued.

(0.1)

We seek to characterise the Nash equilibria of this system. For the 2-team case, diagrams are useful.

In Figure II, the left hand panel shows the profit function for team 1, including the constraint on its optimisation implied by the minimum function. The red curve is the part of the profit function relevant when , the blue is relevant otherwise. The profit function is thus given by the red curve up to the intersection at , and by the blue curve to the right of the intersection. In the right-hand diagram, the profit function for team 2 is shown. The blue curve is the part of the profit function relevant when , the red is relevant otherwise. The profit function is thus given by the blue curve up to the intersection at , and by the red curve to the right of the intersection. The configuration shown is the Pareto-dominant Nash equilibrium[8], characterised by . The optimal quality of both teams is determined by the fan base of the team with the smaller fan base (which in this case is team 1), and the league is competitively balanced.

To see why this is an equilibrium, suppose that team 2 were to choose a lower quality of team, this would shift the blue curve on the left hand panel to the left and lower the profits of both teams. If team 2 were to choose a higher level of quality, the blue curve would shift to the right on the left hand panel, but team 1 would not change its chosen quality, since it is already at the global maximum of its unconstrained profit function. Therefore team 2’s profits would fall (along the red curve in the right hand panel). Now consider team 1. If it were to choose a lower level of quality, the red curve on the right hand panel would shift to the left and the profits of both teams would fall. By choosing a larger level, it can shift the red curve on the left hand panel to the right, and firm 2 will then choose a higher quality level, but this would not increase firm 1’s profits since the right shift of the blue line on the left diagram does not enable firm 1 to reach anything other than its global optimum.

The economics of the equilibrium work as follows: The smaller team chooses the quality level that equates marginal revenue and marginal cost. But there is nothing to be gained by the larger team from increasing its quality above that provided by its opponent, because the consumers would not be prepared to pay as much for the reduced competitive balance, and the extra quality adds to costs. Note that in this situation, it may be in team 2’s interests to give a side payment to team 1, enabling it to increase quality, in turn enabling team 2 to increase quality as well. Another way to think of the economics of the equilibrium is as an example of the ‘weakest-link’ equilibrium first described by Hirshleifer(1983). ‘Weakest-link’ public goods are ones where total provision is determined by the provision made by the smallest of a number of providers. Sea-defences are the most commonly cited example, where the effectiveness of the whole system is determined by the height of the lowest dyke. Keeping secrets is another. The same idea crops up in the current context, because we have assumed that the demand for each game, and therefore the revenue that can be derived from it, is determined by the quality of the lesser quality team. The 2-team model is exactly the same as the Hirshleifer model. Issues to do with the possibility of transfers from larger teams to smaller teams in order to raise the quality of both are analysed by Vicary(1990). The model with more than two teams is a generalisation in a direction that has no obvious interest for public economics, but which is crucial for the analysis of competitive sports leagues.

Now suppose that team 3 joins the league, so that the three teams are characterised by . The profit functions are:

Team 1 can still achieve its global optimum at . Team 2’s revenues from its games with team 1 remain the same, but team 2 now possibly has an incentive to raise its quality so that it can take the best advantage of its games with team 3. (Its revenue from games with team 1 are unaffected.) The global optimum for team 2 is . If this is less than team 1’s effort then team 2 will continue to match team 1’s effort, but if it is greater, it will be worthwhile for it to incur the additional costs. Team 3 will be constrained by the quality of team 2, for the same reasons as constrained team 2 in the 2-team league. The equilibrium in this 3-team model is:

(0.2)

More generally the following Proposition is true.

Proposition I: Let a league have n teams indexed by with fan bases such that . Let demand per fan be and costs per game given by . Then profit maximisation by each of the teams implies a unique Pareto-dominant Nash equilibrium such that:

(0.3)

Proof: See Appendix I.

Roughly speaking, the equilibrium of this model (which has no gate-sharing) in a general n-team structure is that each team chooses quality in accordance with the size of its fan base. “Big cities have winning teams and small cities have losing teams”[9]. The description is‘rough’, because even though the ranking of fan bases may be strict, the ranking of optimal effort may be weak. This arises because the higher the position of a team in the hierarchy of fan base size, the lower its marginal revenue. Investments in extra effort do not increase revenues from games played against teams with lower effort choices. It is possible, therefore, that marginal revenues are smaller than marginal costs at the effort choice of the next smallest team. In this case, effort will be restricted to that of the team with the next lowest effort choice. This effect is strongest for the team with the largest fan base, whichalways sets effort equal to that supplied by the next smallest team.

Table I illustrates some equilibrium configurations in a 5 team league:

Team Number / Fan base / Effort supply / Fan base / Effort supply / Fan base / Effort supply
1 / 20 / 20 / 20 / 20 / 20 / 20
2 / 30 / 22.5 / 25 / 20 / 25 / 20
3 / 50 / 25 / 40 / 20 / 60 / 30
4 / 110 / 27.5 / 75 / 20 / 110 / 30
5 / 120 / 27.5 / 120 / 20 / 120 / 30

Table I: Some equilibrium effort configurations

In each case, the range of fan base sizes is the same, but the resulting optimal effort decisions are quite different. This illustrates the importance of subtle distributional issues in the model. In the first example, the fan base sizes are spread more or less evenly through the range. The result is a wide spread of effort supplies. In the second example, bunching at the bottom of the distribution of fan base sizes generates perfect competitive balance, while in the third the league splits into two groups, one with high, but balanced, effort, the other with low, but balanced effort.