The optimal size of a sports league.

Kesenne, Stefan

Introduction

The optimal size of a sports league is an issue that has been a constant concern of league authorities in European football. There are considerable differences in the number of teams in the top national divisions of European football, ranging from 10 to 20. Although the system of promotion and relegation allows top teams to move up to the first division, it is the monopoly league that decides how many teams compete in the top championship. Over the years, reductions and expansions of the league size have occurred in most European countries. Of the so-called Big Five: England, Spain, Italy, and France have 20 teams, and Germany, as the largest country of the Big Five, has only 18 teams. Among the other rich Western European countries, only the Netherlands and Belgium have 18 teams. In most European countries, teams are meeting each other twice in the league championship, in one home and one away game. In Switzerland or Austria, the top division consists of only 10 teams, but all teams meet the other teams four times. Scotland, among a few other countries, has only 12 teams meeting each other three times. A peculiarity is the fact that tiny Luxemburg has 14 teams and a population of only half a million and Russia, with a population of 144 million also has 14 teams. Over the years, England has come down from 22 to 20 teams, while France went up and down between 18 and 20 teams.

An interesting question is which league objective or considerations affect the leagues' decision on the numbers of teams. Although one can observe a small positive correlation between the league size and the country's population, it would be all too shortsighted to impose that the league size should be proportional to the size of the population. Applying this rule to Russia and Luxemburg might result in a spectacular league in Russia and a less than spectacular league in Luxemburg with only one or two teams.

In the recent sports literature, Vrooman (1997) and Szymanski (2003) have analyzed the decision of a monopoly league about the size of the league, assuming that total league revenue is shared among all participating teams, based on Buchanan's (1965) club theory. Noll (2003) has shown that revenue sharing among the teams creates a strong incentive for clubs to reduce the number of teams in a league. In contrast to the club theory, Kahn (2007) points out that, under specific conditions, the monopoly number of teams is the same as the welfare optimum number of teams, and furthermore, that, in a free-entry market, the number of teams entering the top division would be higher than the social optimum. The social optimum cannot be attained in a free-entry market because individual teams impose quality externalities on other teams and their fans by lowering the average team quality. Based on his approach, Kahn (2007) concludes that the welfare optimum will be closer to the free market equilibrium than to the monopoly league's decision if the supply of playing talent is more elastic. With an open player labor market in European football, this might well be the case in the national football championships, which we are concentrating on in this paper.

In this contribution, we want to compare the welfare optimal number of teams in a league with the number decided by the monopoly league. We start from a simple and aggregated demand model for spectator sport. The following section presents the model and the results, followed by the conclusion.

The Model

The derivation of the demand function for sport starts from the assumption that every citizen who is interested in the sport championship will watch the games if the average price of watching the championship's games in the stadium or on television is zero. If m is the size of the national market or a country's total population and a is the percentage of that population interested in the championship, which is an indicator of the popularity of the championship. If the price/quality ratio goes up, less people will be interested to watch the games, depending on the price elasticity, indicated by the parameter b as the slope of a linear demand curve. So, the demand function for the league championship can be written as:

d = am - b p / q 0 [less than or equal to] a [less than or equal to] 1 (1)

where p is the average price and q in an indicator of the championship attractiveness or quality. The elasticity parameter b is assumed to depend on the average income or the welfare level in the country; it is larger for a poor country and smaller for a rich country. Apart from the price, all variables, population, popularity, quality, and the welfare level have a positive effect on demand. If the price is zero, all interested fans (am) will watch the games. If the league fixes the average price in order to maximize total league revenue, which is: R = pd = p(am - b p / q), the first-order condition can be written as:

[partial derivative] R / [partial derivative] p = am 2b / q p = 0 (2)

So, the optimal price is [p.sup.*] = amq / 2b and corresponding demand is [d.sup.*] = am / 2.

Total revenue is then:

[R.sup.*] - [a.sup.2] [m.sup.2] q / 4b (3)

As can be seen, price and total league revenue are positively affected by the size of the population (m), the popularity of the championship (a), the quality of the league (q), and the welfare level of the country (b). However, the demand is only affected by population and popularity because quality and welfare differences are compensated by the level of the price. This demand curve and the optimal price are presented in Figure 1.

[FIGURE 1 OMITTED]

The crucial variable of this specification is the quality of the league q. What are the variables that affect league quality q or the attractiveness of the league championship? In the literature, the most important determinants seem to be: 1. the total number of teams in the league and the number of games in the championship; 2. the average talent level of the teams or the average quality of the games; 3. the winning percentages of the teams; and 4. the competitive balance.

1. The first variable is the number of games. The more games sport-loving spectators can watch, the better it is for the attractiveness of the championship. Also, supporters prefer to watch their favorite team playing in the top league. In the classical European setting, where each team plays one home and one away game against all other teams, without playoffs, that number of teams in a league determines the number of games. If the number of teams is n, the number of games is n(n-1).

2. The average talent level of the teams, and therefore the average quality of the games, will be higher with a lower number of teams in the league. Given that there is fixed talent supply (t) in a country during a championship, from both local and foreign origin, the playing talents will be allocated or spread out over a lower number of teams. So, the average quality of the game can be approached by the ratio of total available talent and the number of teams (t/n).

3. League quality is also positively affected by the winning percentages of the teams. Fans prefer to watch winning teams. With a reduction or an expansion of the number of teams in a league, or by promotion and relegation, it is always a weaker than average team that has to leave or join the closed major league or the top division. As a consequence, the winning percentages of the remaining teams are reduced by a contraction of the league size (unless the leaving team is a giant killer). It is a given fact that the sum of the winning percentages of all clubs always equals n/2.

4. How is the competitive balance affected by the number of teams? And does the competitive balance affects the quality of the league?

Regarding the first question, the elimination of the weakest teams does affect the competitive balance. In Table 1, a numerical example is presented to show this effect. Let's assume that the relative strength of the teams is given by: A>B>C>D>E>F and that there is a 50% chance that the stronger team beats the next one in the row and a 100% chance that it wins against the weaker teams.

Starting from a six-team league, the winning percentages can be calculated as presented in the first row. If the league is reduced to four teams, with the two weakest teams (E and F) leaving or being relegated to a lower division, the adjusted winning percentages are given in the second row. Both the range and the standard deviation, as simple indicators of the within-season competitive balance, are lower in the contracted league. So, the contracted league is more balanced. However, as observed by Noll (2003), the pennant race in the contracted league is not as close, which makes the competition less balanced. Given these observations, we can assume that the competitive balance is not strongly affected by the number of teams.

Moreover, the empirical research so far has not found very convincing evidence that spectators prefer a more balanced competition (e.g., Borland & Macdonald, 2003; Szymanski & Leach, 2005). Therefore, we leave out the competitive balance as an argument in the quality function.

From the analysis above, and dropping the competitive balance, we can conclude that the quality of the league is mainly affected by the number of games n(n-1), by the average talent level of the teams t/n, and by the sum of the winning percentages n/2. So, the quality q can be written as a function of n and t. We assume that the number of teams and talent have a positive effect on quality, with a decreasing marginal effect of the number of teams, so:

q [n, t] with [partial derivative] q / [partial derivative] n > 0 [[partial derivative].sup.2] q / [partial derivative] [n.sup.2] < 0 [partial derivative] q / [partial derivative] > 0 (4)

We can now derive what the number of teams is that will be fixed by the monopoly league in the interest of the participating or insider clubs. Assuming that the marginal cost of a team in the league is constant and equal to c and that the monopoly league tries to maximise the average net revenue of the teams in the league, or:

max (R - cn / n) - max [[a.sup.2] [m.sup.2] / 4bn q [n,t] - c) (5)

The first-order condition for a maximum of (5) is:

[partial derivative] q [n,t] / [partial derivative] n = q [n,t] / n (6)

As indicated by Szymanski (2003), this is equivalent to the solution of the optimal number of workers in a labor-managed firm. The optimal number of teams turns out to be independent of the size of the market, the popularity of the sport, and the income level.

In Figure 2, where marginal and average quality are drawn as a function of the number of teams, the league's optimal point is found in [n.sub.1], where the marginal quality and the average quality curves intersect.

We can also try to derive what the welfare optimal number of teams in a league will be in this model. Based on the demand equation in Figure 1, total welfare can be calculated as the sum of league revenue and consumer surplus. As can be seen, the consumer surplus (CS) is 50% of total league revenue and equal to: [a.sup.2] [m.sup.2] q [n,t] / 8b. So, the welfare level is:

W - R + CS - 3 [a.sup.2] [m.sup.2] q [n,t] / 8b (7)

Maximizing total welfare is then equivalent to maximizing quality (q), so the optimum condition is given by:

[partial derivative] q [n,t] / [partial derivative] n = 0 (8)

Again, the solution is independent of population, popularity, and income, but the available talents in the country do affect the league size. As can be seen in Figure 2, the number of teams will now be [n.sub.2], which is clearly well above the number of teams fixed by the monopoly league ([n.sub.1]).

Discussion and Conclusion

What this simple theoretical model shows is that the number of teams that monopoly league authorities allow to enter the top divisions in European football is well below the number that is optimal for the consumers and the industry from a welfare economic point of view. However, this model has not compared the quality of a championship with 20 teams and 380 games and a championship with only 16 teams and more than 480 games. In some leagues, there are fewer teams and more games. This is an important trade-off that should be considered by league authorities, as well, when deciding about the number of teams. One can opt for a lower number of teams and more games, as in Switzerland and Austria, or for more teams and fewer games as in two other small countries like Belgium and Holland. Fewer teams implies more high-quality games, which can increase demand, but too many games between the same quality teams in one championship might also decrease the marginal utility of the games and reduce demand.

[FIGURE 2 OMITTED]

Moreover, more teams in the top division will make more spectators happy because supporters like to see their home-town team play at the top level. Because it is unclear if this negative effect is stronger or weaker than the positive quality effect, we have assumed, for simplicity reasons, that the championship structure is a given fact, which implies that there is a fixed relationship between the number of teams and the number of games, as in the model above. So, even if a championship consists of a high number of games with a limited number of teams, in which case each team plays more matches, the basic conclusion of the model still holds that, with a given structure of the championship, in the presence or absence of playoffs, the number of teams, decided by the monopoly league, will be lower than the welfare maximizing number.

Kahn (2007) draws different conclusions; in his model, the expected decrease in average team quality has a strong negative impact in an expanded league, and his model neglects the fact that league expansion can also raise fan demand because it increases the clubs' winning percentages as in Noll (2003). In our model, the impact of expansion on the insider teams' winning percentages is taken into account, but the negative effect of expansion on average team quality might be underestimated because it is only captured by the decreasing marginal effect of the number of teams on league quality.

If so, the question can be asked if the decision about the number of teams can be left to a board of league authorities that only consists of representatives from the insider teams. Also, doesn't this monopoly justify some government interference in fixing the number of clubs in the top league? All too often, it is a deliberate policy of the largest and riches teams, which have an important say in the league's management and do not run a serious risk to be relegated, to keep as much as possible of the football money to themselves, neglecting the interests of supporters and consumers.

References

Buchanan, J. (1965). An economic theory of clubs. Economica, 32(125), 1-14.

Borland, J., & Macdonald, R. (2003). Demand for sport. Oxford Review of Economic Policy, 19(4), 478-503

Kahn, L. (2007). Sports league expansion and consumer welfare. Journal of Sports Economics, 8(2), 115-138.

Noll, R. (2003). The economics of baseball contraction. Journal of Sports Economics, 4(4), 367-388.

Szymanski, S. (2003). The economic design of sports contests. Journal of Economic Literature, 41(4), 1137-1187.

Szymanski, S., & Leach, S. (2005). Tilting the playing field: Why sports league planners would choose less, not more, competitive balance. Discussion paper, Tanaka Business School, Imperial College, January 2005.

Vrooman, J. (1997). Franchise free agency in professional sports leagues. Southern Economic Journal, 64(1), 191-219.

Stefan Kesenne (1)

(1) University of Antwerp, City Campus and Catholic University of Leuven Stefan Kesenne is a professor in the Economics Department at the University of Antwerp and in the Department of Human Kinesiology at Catholic University of Leuven. His research interests include sports economics and labor economics.

Table 1. League Contraction, Numerical Example

League              Winning Percentages             Sum   Range    SD
size
           A      B      C      D      E      F

6 teams   0.90   0.80   0.60   0.40   0.20   0.10    3    0.80    0.32
4 teams   0.83   0.67   0.33   0.17                  2    0.66    0.30