ORMAT
Funny Peculiar:
Modelling Sports Leagues
by
John Treble[1]
(University of Wales Swansea)
March 2005
ABSTRACT
The economics of sports leagues is distinctive, because demand for a contest is believed to increase in the closeness of the contenders involved. This belief, which has some empirical support, has rarely been reflected in the formal models of sports leagues found in the literature. Even those models that do include it weight closeness in such a way as to make its contribution to the outcome relatively unimportant.
This paper proposes a family of models that include closeness as an element in demand, and analyses the issue of gate-sharing using the two tractable members of the family. Some analysis of a league prize and the salary cap is also offered. The models’ predictions are more in keeping with the apparent beliefs of lawyers and professional sports managers, than those of existing models.
Funny Peculiar:
Modelling Sports Leagues
by
John Treble
1. Introduction
When Neale (1964) described the economics of sports leagues as peculiar, he meant that sports leagues have distinctive features setting them apart from other industries. Chief among these is the ‘Louis-Schmelling paradox’ (sic). Neale 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.”
In other words, the demand for the output of sporting competitions depends not simply on the quality of the contenders, but also on how closely matched they are.
The literature that followed Neale’s paper continued to stress this idea, but can only be described as odd, since none of the papers in it, with the exception of a number of recent contributions by Frederic Palomino and his co-authors[2], use a demand structure that actually reflects the Louis-Schmeling paradox. Furthermore, as I shall show, even Palomino’s work gives such small weight to the closeness of the match as to emasculate Louis-Schmeling, and deprive the resulting model of its economic distinctiveness.
The goal of this paper is to document this curious situation, and to consider how models incorporating the Louis-Schmeling idea may be designed. Two classes of models are considered: those that have revenue functions that are differentiable everywhere, and those that do not. We analyse two models (one belonging to each class) in some detail. The analysis concentrates particularly on the impact of gate-sharing on competitive balance, but we also briefly consider a league prize and the payroll cap.
The differentiable models turn out to be less tractable than their non-differentiable counterparts. This observation adds to the oddness of the existing sports literature, which contains no examples of this model, but teems with papers that introduce ‘general’ revenue functions that are assumed to be twice differentiable. The non-differentiable models lead to more interesting economics than the differentiable ones because the Louis-Schmeling paradox is reflected clearly in the predicted behaviour of the contending agents. There is also a clear link between these models and the literature on the private provision of public goods.
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). Thus El-Hodiri and Quirk (1971) claim that:
“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.”
and more recently, Szymanski(2004) refers to:
“the uncertainty of outcome hypothesis that consumers in aggregate prefer a close match to one that is unbalanced in favour of one of the teams.”
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, outcome uncertainty can be given various weights in the demand function, so that the models considered are in keeping both with the empirical evidence and 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.
So closeness is important. But how should it be measured and represented in a formal model? A key axiom of any measure of ‘closeness’ is that the measure should decrease when the characteristic(s) being measured get farther apart. This is certainly not true of many claimed measures. For example, Fort and Quirk(1995): “closeness between two teams i and j can be described by ”. In this expression, if , is increasing in . A second axiom of ‘closeness’, which one might call symmetry, is that if a changes in such a way as to bring it closer to b, then the same change in b, but in the opposite direction, should have the same effect on the closeness measure. Késenne (2000) constructs a “revenue function…(that)…is quadratic in the winning percentage of the home team because of the decreasing marginal impact of winning on revenue due to the impact of the uncertainty of outcome.” But it isn’t quadratic in the (negative) of the winning percentage of the away team, so it violates the second axiom.
The El-Hodiri and Quirk paper is important since they were the first to propose a formal model, and most subsequent authors take the model they constructed as a basis for elaboration. It is appropriate therefore that the adoption of unsuitable functional forms can be traced back to a passage in this paper. Immediately after the passage cited above, they write:
“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.”
This last sentence is a non sequitur, as I shall show, but it is a non sequitur that has been picked up and thrown about in subsequent writings with all the élan of the Welsh rugby team in the Stade de France. Szymanski (2004) again:
“a team’s revenue initially increases with winning, but then peaks and starts to decrease as the team achieves a high level of dominance in the league.”
Not surprisingly, the predictions that arise from this theoretical literature are curious, too. In a world in which many US[3] sports leagues have been able to convince judges that restrictive labour practices should be permitted because otherwise the loss of competitive balance would cause a damaging loss of demand, academic economists are debating whether the impacts are zero or negative.
Finally, I should point out that these resonances are not wholly lost on the protagonists in this debate. In a recent contribution, Szymanski (2004) having once again demonstrated with the use of a nice smooth demand function that sharing can make things less equal, finishes rather ruefully with the words:
“At the very least, it must be the case that the underlying model on which the previous literature has relied requires a more formal justification. It may be, however, that the model just makes no sense.”
The present paper aims to provide a formal justification for a model that is different from El-Hodiri and Quirk(1971), in the hope that it, unlike its predecessor, will make sense.
2. Closeness
My aim is to construct demand functions representing consumers who value not just the quality of the contenders in a sporting contest, but also the closeness of the contest. The first of these requirements is easy enough: Let and represent the quality of the contenders, then any function that is increasing in both arguments will suffice.
Closeness, though, is rather more tricky. Consider two quantities, x and y such that . As x increases, they initially become closer, so that ‘closeness’ increases with x. This is true as long as remains true, but there will come a point when , beyond which any further increases in x will cause ‘closeness’ to fall. Furthermore, there is no reason to suppose that the transition between increasing ‘closeness’ and decreasing ‘closeness’ is smooth. Suppose now that revenue is an increasing function of closeness. Such a revenue function will in general have a kink at points where , and the marginal revenue function will in general have a discontinuity. Interior solutions to firms’ optimizing problems rely on an equality between marginal revenue and marginal cost, so that the non-existence of marginal revenue can lead to significant analytical issues.
‘Closeness’ of contestants can be measured by the absolute difference between their respective qualities, so that any function will do, with increasing. To keep things simple, we will limit the present discussion to demand functions that are additive.
Demand will thus be represented by structures of the following sort:
1)
where is to be interpreted as the reservation price of a consumer for a contest between i and j. Because the main focus of this paper is on the function , I use only the simplest functional forms for , assuming for the most part that it takes the simple additive form .
For , I consider the following family of measures, indexed by :
2)
This family of functions is illustrated in Figure I, where is fixed at 20, a at unity, and the functions are plotted for . Clearly, not all members of this family of functions are differentiable everywhere. In fact, it is easy to show[4] that if , is non-differentiable where . This shows in the Figure as a crease along the line . Note that for , the function is simply an absolute value function, increasing with a slope of 1 as increases, where , and decreasing with a slope of -1 where . For , the function is the squared difference, and is differentiable everywhere.
In principle, one could base a demand function on any of the measures in , but only two of them, and , are tractable. In the next two sections, I look at models based on these two functions. Except for the specification of the demand, the two models are identical, and follow the rest of the literature in supposing that the i’th team has monopoly rights over a fixed fan base, . 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. Teams are indexed in such a way that . If , we refer to team j as the larger team and team i as the smaller team. A league 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.
The models dealt with in this paper are static for simplicity. (Nonetheless, the main message of the paper applies equally to dynamic formulations.) 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 .
3. Model I
In the first model is used in the demand function, giving: