Profit-maximizing gate revenue sharing in sports leagues.

Peeters, Thomas

I. INTRODUCTION

The professional team sports industry is one of the scarce industries in which coordination between competing firms (clubs) is widely accepted. Clubs in almost all sports have organized themselves in legal cartels (leagues). It seems obvious that a certain degree of coordination is needed to produce a team sports competition, for example, for scheduling games. However, leagues have introduced regulations, which clearly go beyond the purely practical issues involved in producing games. The most relevant examples include gate revenue sharing (the home team gives a part of its match-day income to the visiting team), collective sales of media rights (the league monopolizes media rights and distributes the revenues), and salary caps (the league limits the amount teams may spend on player wages). These regulations would probably be classified as restrictive practices in any other industry. Yet, the fact that sports leagues openly communicate their application provides an opportunity to examine how these cartels have used their devices to coordinate behavior and increase joint profits.

Table 1 gives an overview of gate revenue sharing arrangements in sports leagues around the world. Almost all American major leagues engage in gate revenue sharing. The National Football League (NFL), Major League Baseball (MLB), and Major League Soccer (MLS) share revenues through a central pool. Each club contributes a fixed percentage of its gate revenues to this pool, which is distributed equally among all clubs. The National Hockey League (NHL) has a more complicated arrangement, where only teams that have revenues below the median and small media markets are eligible to receive support from revenue sharing. Introducing local revenue sharing was also reported to be one of the issues on the table during the recent lockout in the National Basketball Association (NBA). (1) To the best of my knowledge, however, the details of the new NBA revenue sharing rule have not been made public yet. In contrast, European soccer clubs, along with the Australian Football League (AFL), share (almost) no gate revenues. Both the Bundesliga and the English Premier League (EPL) have arrangements to share revenues from cup games, yet these constitute a minor portion of the teams' schedules. Interestingly, both the EPL and the AFL had sizeable sharing arrangements in the past but, contrary to the U.S. leagues, chose to abolish these.

These observations raise the question why some league cartels share gate revenues while others have chosen not to. I examine this question using a theoretical model of a sports league with profit-maximizing teams. A crucial innovation in my model is that teams serve two types of consumers. In each team's local market, fans are committed and prefer to see their team win. In the nationwide market, consumers are neutral TV viewers, who like to watch a tense and high-level competition. A team's local or stadium revenue increases in its on-field performance, whereas media revenue depends on the competitive balance and overall quality of play for the league as a whole.

My results first show that local (or gate) revenue sharing decreases talent investments. (2)

Initially lower talent investments boost club profits, because total costs go down. Too little investment, however, reduces revenues, which harms profits at high levels of sharing. To find a profit-maximizing sharing rule, the league has to balance these two effects. The dampening effect of sharing on investments is weaker in leagues whose teams have homogeneous local markets. Consequently, homogeneous leagues maximize profits by setting a more extensive gate revenue sharing rule than leagues with less equal teams. This may explain the observation from Table 1 that the more unequal European soccer leagues share less local revenues than the more homogeneous U.S. major leagues. (3) A somewhat striking implication of this result is also that gate revenue sharing is more interesting for clubs who have more equal revenue potentials. In other words, leagues who are less likely to suffer from a "competitive balance problem," (4) are more likely to engage in local revenue sharing.

To grasp the intuition for this result, it is helpful to develop the following reasoning. First, observe that gate revenue sharing has two distinct effects on the incentive of a team to invest in playing talent. Gate revenue sharing first acts as a taxation on talent-related revenues, because each club has to hand over part of its stadium revenues, which are increasing in on-field performance. On top of this, gate revenue sharing also creates an additional source of income, that is, receipts from the local markets of rival teams. Investing in talent negatively affects this revenue source, because it decreases the on-field success of competing teams and, therefore, the willingness-to-pay of committed fans in their local markets. Both effects, dubbed the taxation and free-riding effect in the literature, imply that gate revenue sharing leads teams to lower their talent investments. The "free-riding" effect is, however, stronger in heterogeneous leagues, because the additional revenues from large local markets in the league are more important for small market teams. They consequently shrink their investments more rapidly than teams in homogeneous leagues, which allows the large clubs to do the same. Now, consider the effect gate sharing has on both costs and revenues at the level of the league as a whole. As explained above, the league can use sharing to change club incentives in such a way that they lower their individual talent investments. For the league as a whole, sharing leads talent-related costs to go down and this has a positive impact on total costs. Local revenue sharing, therefore, functions as a cost-cutting tool. In terms of league-wide revenues, lower overall talent investments have two effects. They first reduce the interest of neutral fans, which directly harms media revenues. On top of this, a high level of sharing disrupts revenues from local fans, because it takes away the large teams' incentives to dominate the league, while this would maximize league-wide profits. The league has to balance this initial positive effect on costs (lower talent investments) with the potential negative effect on income (lower revenues) to find its profit-maximizing sharing rule. I explained above that gate revenue sharing has ceteris paribus larger effects on investments in heterogeneous leagues. These leagues, therefore, reach their profit-maximizing level of sharing at a lower level of sharing than homogeneous leagues, which explains the intuition for my results.

This article is further organized into five sections. In Section II, I relate this article to the existing literature. Then, Section III presents the model setup in full detail. Section IV uncovers the impact of revenue sharing on talent investments and competitive balance. In the subsequent Section V, I explore the characteristics of the profit-maximizing level of revenue sharing for the league cartel. In the final section, I provide concluding remarks, discuss the robustness of my results, and propose avenues for future research.

II. RELATED LITERATURE

A large part of the economic literature on professional sports has been devoted to the relationship between revenue sharing and competitive balance. In one of the seminal contributions, Fort and Quirk (1995) argue that revenue sharing has no impact on competitive balance, a result dubbed the invariance principle in the subsequent literature. A number of authors have assessed the robustness of this result to alternative modeling assumptions. (5) For example, Kesenne (2000) looks at the influence of club objectives by introducing win-maximizing instead of profit-maximizing clubs. Szymanski and Kesenne (2004) employ a model based on Nash-conjectures to replace the Walras model used in Fort and Quirk (1995). Fort and Win-free (2009) examine the impact of alternative functional forms for the contest success function. Feess and Stahler (2009) investigate the effects of sharing under different specifications for the club revenue functions. Finally, Dietl, Lang, and Rathke (2011) introduce the interaction between revenue sharing and salary caps. These contributions find that sharing may decrease competitive balance (e.g., Szymanski and Kesenne 2004), may increase the balance (e.g., the win-maximizing models in Kesenne 2000), or might do both (e.g., Feess and Stahler 2009). In line with Feess and Stahler (2009), the competitive balance effect in this article turns out to be ambiguous. This is a result of the complex nature of team revenues in my model, which depend on the relative and total talent stock in the league.

A crucial novelty with respect to the existing literature is that my model analyzes gate revenue sharing in a setting that explicitly distinguishes between multiple types of sports consumers. Szymanski (2001) first contrasted the utility of committed local fans and neutral TV viewers, while Forrest, Simmons, and Buraimo (2005) were the first to test this notion empirically. It is now well established in the empirical literature that the demand for stadium seating has different characteristics from that for televised games. Coates, Humphreys, and Zhou (2014), for example, argue that competitive balance has less impact on demand for stadium seating than previously expected. At the same time, recent studies by Tainsky and McEvoy (2012) and Feddersen and Rott (2011) show that uncertainty of outcome and team quality are important determinants of TV viewership. By introducing these insights into the analysis of local revenue sharing, I highlight the importance of interaction effects between different club revenue sources. In my model, gate revenue sharing ultimately leads to a decrease in media revenues, which in turn (partly) explains why leagues select different profit-maximizing levels of gate revenue sharing. This result cannot be shown if revenues from different sources are treated as homogeneous, as the majority of the previous literature has done.

This article also takes the competitive balance literature one step further in the question, which it addresses. Whereas the existing literature has successfully analyzed how sharing may affect competitive balance, this article aims to explain the level of sharing as a function of underlying league characteristics. In this sense, it also relates to a second strand of literature, which analyzes the restrictive practices of sports leagues to learn about the functioning of business cartels. Both Kahn (2007) and Farmer and Pecorino (2010) study the effects of labor market restrictions in college sports. Palomino and Sakovics (2004) and Peeters (2012) look at the design of sharing rules for media rights revenues. Ferguson, Jones, and Stewart (2000) focus on the implementation of salary restrictions in MFB. Within this strand of literature, the articles by Atkinson, Stanley, and Tschirhart (1988), Kesenne (2007), and Salaga, Ostfeld, and Winfree (2014) are most closely related to mine. Atkinson, Stanley and Tschirhart (1988) build a model to show how revenue sharing may function as a coordination device to steer talent investments and increase joint profits in the NFF. Kesenne (2007) argues that revenue sharing may have an adverse effect on the profits of large market teams. In line with a recent article by Salaga, Ostfeld, and Winfree (2014), my analysis extends these previous studies by building a richer theoretical model, which incorporates different revenue streams from media and stadiums. Salaga, Ostfeld, and Winfree (2014) focus primarily on the relative attractiveness of sharing media revenues versus stadium revenues. They further discuss ways to mitigate the negative effects of gate sharing on stadium investments. My focus on the other hand is on explaining the observed differences in local revenue sharing arrangements between leagues based on underlying market characteristics. (6) As explained above, I find that cartel behavior is mostly efficient, in that leagues select levels of gate sharing, which my model suggests are profit-maximizing given their distribution of local markets.

Only a handful of articles have analyzed the use of sharing rules in cartels outside of the sports industry. This is mainly explained by a lack of legal cartels (and therefore data) in other industries. Both Sorgard and Steen (1999) and Roller and Steen (2006) use a unique data set on Norwegian cement cartels to show how an inefficient sharing rule led to excess capacity investments. In a wider sense, this article ties into the literature on semi-collusion. Contrary to the models of Fershtman and Gandal (1994) and Brod and Shivakumar (1999), however, firms attempt to collude in quality provision, rather than pricing. To my knowledge, the use of revenue sharing to dampen excess investments in quality or capacity has not been looked at in this literature.

III. MODEL SETUP (7)

The model in this article consists of three stages. In the first stage, the league sets a sharing rule for local revenues. In the second stage, clubs simultaneously decide on their investments in talent. In the final stage, the clubs set ticket prices and collect media revenues.

In the model, two clubs (i and j) form a sports league. (8) Both clubs act to maximize their individual profits. (9) The league, acting as a cartel of clubs, aims to maximize the joint profits of its member teams. Clubs have the option to share the revenues from their local markets through a central pool revenue sharing arrangement. In the first stage of the model, the league decides on a sharing rule ([alpha]), where a reflects the part of local revenues a club is allowed to keep. Full sharing occurs when [alpha] = 0, while no sharing implies [alpha] = 1. Each team receives half of the proceeds collected in the central pool, such that full sharing ([alpha] = 0) implies that club i receives half of its own revenues and half of team j's revenues. (10) There is no player union in the model, such that the league does not have to negotiate on the level of sharing. I further assume that media revenues are shared equally among the clubs, as is done in the U.S. major leagues. (11)

In the second stage, clubs decide on talent investments. Talent is available at constant marginal cost, but as in Palomino and Sakovics (2004) and Peeters (2012) has a discrete nature. Clubs either make a high investment at cost h or a low investment at cost l. These talent investments translate into win probabilities given by a simple contest success function

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where [beta] > 1/2. (12) Observe that Equation (1) adheres to the adding-up constraint, because in each case [w.sub.i] + [w.sub.j] = 1. A second point to note is that the total playing talent in the league ([t.sub.i] + [t.sub.j]) is not fixed, because the league may employ a total amount of talent equal to either 2l, h + l, or 2h. In the literature, this assumption is often referred to as an open talent market to contrast it with the situation of a closed market in some of the U.S. major leagues. (13) The assumption that the market supply of talent is fixed implies that higher demand increases the price of talent. In such a setting, revenue sharing should again be even more advantageous for team profits, so the current assumption is the most conservative.

As first proposed by Szymanski (2001), consumers of sports contests are not a homogeneous group. In this model, I distinguish demand from hard-core club fans and neutral sports fans. Both groups have a different appreciation for the characteristics of sports contests. Neutral fans favor tension and a high level of play in a game. As a result, a higher level of play only increases their perceived quality (b([t.sub.i], [t.sub.j])), when it does not result in a deterioration of competitive balance, as such I assume

(2) b (h, h) > b(h, l) = b(l, h) = b (l, l).

Specification Equation (2) is consistent with the empirical literature on TV demand for sporting events. For example, Forrest, Simmons, and Buraimo (2005) find that both the combined wage bills and relative wages are significant determinants of TV audiences in the EPL. Tainsky and McEvoy (2012) show the importance of both quality of the teams and uncertainty of outcome to TV viewership for NFL games. Feddersen and Rott (2011) obtain similar results for German national football team fixtures. (14)

Committed club fans prefer to see their team win. Consequently, they rank possible outcomes according to their team's expected winning percentage. I assume that a decline in winning percentage leads to a similar decline in quality, whether it is on the "winning" or the "losing" segment. This means that (team-specific) quality ([f.sub.i]([t.sub.i], [t.sub.j])) may be expressed by (15)

(3) [f.sub.i](h, l) > [f.sub.i](h,h) = [f.sub.i](l, l) > [f.sub.i](l, h),

where,

[f.sub.i](h, l) - [f.sub.i](l, l) = [f.sub.i](l, l) - [f.sub.i](l, h).

Note that Equation (3) implies that stadium visitors do not care about the competitive balance in the league. This notion is (perhaps surprisingly) consistent with most recent contributions on the demand for stadium tickets (e.g., Coates, Humphreys, and Zhou 2014). (16)

In the final stage, clubs set ticket prices ([p.sup.f.sub.i) and sell media rights. Clubs are assumed to be monopolists in their local market of size [m.sup.i] where they face demand for stadium seating from committed fans. One club (Club 1) has a strictly larger market than the other (Club 2), so [m.sub.1] > [m.sub.2] > 0. Because talent investments are sunk in this stage and all other costs are neglected, the price-setting problem boils down to a revenue-maximization problem.

To meet demand for media appearances, the league pools media rights and sells them to the highest-bidding broadcaster. The broadcasting market is assumed to be competitive, so the league is able to extract all relevant profits. The league splits the proceeds equally among the clubs, as is common in the U.S. major leagues. (17) The broadcaster who obtains the rights faces demand from neutral fans. The size of the neutral consumer market is given by n, which may be interpreted as the number of TV households. The cost of broadcasting is neglected, so all relevant costs to the broadcaster are sunk. The broadcaster too should, therefore, set a pay-per-view price ([p.sup.b]) that maximizes total revenues.

Demand of both groups follows from Equations (3) and (2) by assuming that consumers have individual-specific preferences [x.sup.b.sub.v] and [x.sup.f.sub.v], which are uniformly distributed along the interval [0,1]. The consumers' utility-maximization problem is

max {0, [x.sup.f.sub.v] [f.sub.i] (t.sub.i], [t.sub.j]) - [p.sup.f.sub.i]}

max {0, [x.sup.b.sub.v]b ([t.sub.i], [t.sub.j]) - [p.sup.b]},

for committed and neutral fans, respectively. It follows that market demand is linearly decreasing in prices and increasing in quality at a decreasing rate,

(4) [D.sup.f.sub.i] = [m.sub.i] [f.sub.i]([t.sub.i], [t.sub.j]) - [p.sup.f.sub.i]/[f.sub.i]([t.sub.i], [t.sub.j])

(5) [D.sup.b] = n b([t.sub.i], [t.sub.j]) - [p.sup.b]/b([t.sub.i], [t.sub.j]).

Given the demands Equations (4) and (5), revenues from both fan groups are maximized by setting

[p.sup.f.sub.i] = f([t.sub.i], [t.sub.j])/2

[p.sup.b] = b([t.sub.i], [t.sub.j])/2.

Substituting this leads to revenue functions of the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where [R.sup.f.sub.1] and [R.sup.f.sub.2] represent the revenues of the game at the stadium of Team 1 and Team 2, respectively, and [R.sup.b] is the media revenue for both games combined. Given that talent investments are the only relevant costs to clubs, it is now straightforward to express each club's profits as a function of these separate revenue sources

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

This can be rewritten as

(6) [[pi].sub.1] = n/4 b([t.sub.1], [t.sub.2]) + 1/8 ((1 + [alpha]) [m.sub.1]f([t.sub.1], [t.sub.2]) + (1 - [alpha])[m.sub.2]f([t.sub.1], [t.sub.2])) - [t.sub.1]

(7) [[pi].sub.2] = n/b([t.sub.1], [t.sub.2]) + 1/8 ((1 + [alpha])[m.sub.2]f([t.sub.1], [t.sub.2]) + (1 - [alpha])[m.sub.1]f ([t.sub.1], [t.sub.2])) - [t.sub.2].

The league's objective is to maximize joint profits of both clubs. This means its objective function boils down to

(8) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

IV. TALENT INVESTMENTS

A. Nash-Equilibrium

In the second stage, clubs decide on their talent investments. Because each club has two options to choose from, four different outcomes may arise, mutually high investments ([t.sub.1] = h, [t.sub.2] = h), mutually low investments ([t.sub.1] = l, [t.sub.2] = l), and domination by the large ([t.sub.1] = h, [t.sub.2] = l) or the small team ([t.sub.1] = l, [t.sub.2] = h).

PROPOSITION 1. There is no sharing rule a on the interval [0,1] for which [t.sub.1] = l, [t.sub.2] = h is a Nash-equilibrium of the talent investment stage.

Proof See Appendix.

To provide some intuition for the result in Proposition 1, it is helpful to first analyze the situation in which both clubs share no local revenues (a = 1). Observe that when [t.sub.1] = l, [t.sub.2] = h, the small team prefers to incur high investment costs instead of responding to the low investment of the large team by an equally low investment. This means that it is able to offset the cost (h - l) with the increased revenues from its (small) market of committed fans. At the same time, the large market team has the option to match high investments, but chooses not to. Therefore, it somehow judges the same cost difference (h - l) too high to be recouped by higher earnings from its (large group of) committed fans and half of the increased revenues in the neutral market. Obviously, this is impossible, so it pays for the large market team to match high investments, if it pays for the small market team to make them.

When local revenues are shared, the importance of the team's own local market diminishes. At full sharing, both teams receive half of their revenues from each local market. However, the same reasoning applies, because even with full sharing, the large team still has the additional income from the neutral market to offset its high talent investment, which is lacking for the small team. Therefore, the result of Proposition 1 holds for every sharing rule on the relevant interval [0,1],

To characterize the Nash-equilibrium in the talent investment stage, I define two thresholds ([[theta].sub.h]; and [[theta].sub.l]) on the values of the model parameters. These thresholds describe regions in which each of the three remaining outcomes is the unique Nash-equilibrium. In order to ensure that a unique Nash-equilibrium exists at all parameter values (except for the thresholds), it is necessary to impose that both clubs have sufficiently different local markets, that is,

(9) [m.sub.1] - [m.sub.2] [greater than or equal to] n b(h,h) - b (h,l)/f(h,l) - f (h,h).

PROPOSITION 2. The Nash-equilibrium of the talent investment stage is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Proof See Appendix.

As a direct consequence of Proposition 1, it is always the large team that deviates from a Nash-equilibrium in mutually low investments. Mutually low investments are, therefore, a Nash-equilibrium if the large team's local market is small enough to discourage a deviation toward unbalanced investments. Consequently, the threshold 0/ puts an upper limit on the size of the large local market. The reverse is true for mutually high investments. Here, the small team's local market should be large enough to allow high investments to be profitable. Therefore, 0;i defines a lower limit for the size of the small local market. Unbalanced investments are the Nash-equilibrium at all intermediary values. Intuitively, a larger media market (n) and a larger step in the utility of the neutral fans increase the probability of an equilibrium in mutually high investments. Further, Proposition 2 shows that higher investment costs (h - l) result in lower incentives to invest, that is, an increase in (h -1) increases [[theta].sub.l], as well as [[theta].sub.h].

[FIGURE 1 OMITTED]

To illustrate the implications of Proposition 2 for revenue sharing, Figure I depicts [[theta].sub.h] and [[theta].sub.l] in an ([m.sub.1], [m.sub.2])-diagram for [alpha] = 1 (clubs share no gate revenues) and [alpha] = 0.5 (clubs share 50% of gate revenues through the central pool). (18) Because under condition of Equation (9) [m.sub.1] is strictly larger than [m.sub.2], the white region to the lower right is not feasible in this model. Revenue sharing has two distinct effects on talent investments. First, it effectively functions as a taxation on talent investments. Instead of being able to retain the revenues from winning, clubs see part of the gain flowing to their competitor. In response, teams decline their talent investments. Feess and Stahler (2009) show the robustness of this "dulling effect" in various models such as Szymanski and Kesenne (2004) and Fort and Quirk (1995). In Figure 1, this effect is responsible for the outward shift of both threshold lines. Second, with revenue sharing, clubs obtain part of their revenue from their competitor's local market. As such, they have an incentive to cut back their talent investments to protect their competitor's local revenue. Szymanski and Kesenne (2004) label this the "free-riding" effect. This effect is always stronger for the small team, because the revenues it obtains from sharing are more important. In Figure 1, this shifts the slope of both threshold lines up, because clubs internalize the size of each other's local market. The overall effect of sharing in Figure 1 is clearly a shift from region h, h to region h, l and from h, l to l,l, a decrease in talent investments.

[FIGURE 2 OMITTED]

B. Competitive Balance

In the framework of this model, a Nash-equilibrium in unbalanced investments can be interpreted as a measure of competitive (im)balance in the league. As is clear from Figure 1, revenue sharing influences this outcome in two ways. On one hand, sharing promotes unbalanced investments instead of mutually high investments. On the other hand, it mitigates unbalanced investments in favor of mutually low investments.

Figure 2 explicitly depicts the effect of moving from [alpha] = 1 to [alpha] = 0.5 on the balance in the league using the same parameter values as in Figure 1. As Figure 2 shows, sharing reduces the balance in the league for intermediate values of [m.sub.2] and sufficiently large values of ml. The small market team cuts back its investment more sharply than the large market team in order to free ride on the proceeds of the large local market and the balance in the league drops. This effect coincides with the result of Szymanski and Kesenne (2004).

In a smaller region where [m.sub.2] and [m.sub.1] are both lower, sharing improves the balance. Here, only the large team lowers its investment, because the small team cannot drop its investments further. Consequently, sharing improves the balance.

An even smaller region with intermediate values of [m.sub.1] and [m.sub.2] sees the league moving from [t.sub.1] = h, [t.sub.2] = h directly to [t.sub.1] = l, [t.sub.2] = l. In this case, the taxation and free-riding effect exactly offset each other and we end up with equal balance, but lower talent investments. This is the situation described by Fort and Quirk (1995) as the invariance principle. Finally, sharing appears to have no effect on the Nash-equilibrium in leagues with either (a) a comparatively large [m.sub.2] or (b) a comparatively large [m.sub.1] and small [m.sub.2] or (c) small [m.sub.1] and [m.sub.2], because it has too little influence on [[theta].sub.l] and [[theta].sub.h]. Note, however, that a sufficiently small choice for the talent investment difference (h - l) ensures that sharing always has an effect on the thresholds, so this result should be interpreted with caution.

In conclusion, it is difficult to make general statements on the competitive balance effect in this model. In a sense, this confirms the findings of Feess and Stahler (2009), who show that the balance effect of local revenue sharing is indeterminate, when competitive balance, total playing quality, and relative quality influence club revenues (as is the case in this model).

V. SETTING A PROFITABLE SHARING RULE

A. League Optimum

To characterize the talent investment outcomes, which maximize total profits in the league, I again define thresholds ([[lambda].sub.u,h] and [[lambda].sub.u,l]) on the model parameters. The subscripts indicate which situations are being compared, where h stands for [t.sub.1] = h, [t.sub.2] = h, l for [t.sub.1] = l, [t.sub.2] = l, and u for [t.sub.1] = h, [t.sub.2] = l.

PROPOSITION 3.

[t.sub.1] = l, [t.sub.2] = h never maximizes league profits. League profits are maximized with:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Proof. See Appendix.

Proposition 3 defines both thresholds in terms of the cost differential in the talent market, h--l, because the league weighs the cost of talent against the benefits in terms of increased willingness-to-pay. Mutually high investments are preferred when h--l is comparatively low, so [[lambda].sub.u,h] defines an upper limit to the talent cost. Further, [[lambda].sub.u,h] increases in the size (n) and the willingness-to-pay (b(h. h) - b(h, l)) in the neutral market. As moving from unbalanced to mutually high investments benefits the small team's fans, [m.sub.2] also has a positive impact, while [m.sub.1] lowers [[lambda].sub.u,h]. Mutually low investments maximize profits if talent costs are comparatively high, so [[lambda].sub.u,l] sets a lower bound on h -1. A larger local market for the large team increases [[lambda].sub.u,l], because this renders unbalanced investments more attractive. The opposite is true for the small local market [m.sub.2]. Note also that each talent investment outcome, except small market domination, may maximize league-wide profits. This implies that balanced talent investment is not necessarily the desirable outcome for the league. If talent costs are at an intermediate level, total profits are maximized if the large market team dominates the competition, especially when both local markets are sufficiently different.

B. Is Sharing Profitable?

In the framework of this model, the league maximizes the joint profits of its teams. This implies that it should set a sharing rule to match the league thresholds defined in Proposition 3 with the Nash-equilibrium thresholds in Proposition 2. To illustrate this problem graphically, Figure 3 depicts the Nash-equilibrium regions in an (n, h - l)-diagram for different values of the sharing rule [alpha]. (19) Figure 3 also shows the league thresholds with [[lambda].sub.u,h] being the upward sloping line and [[lambda].sub.u,l] the horizontal line. As observed before, sharing decreases talent investments. It promotes a Nash-equilibrium in [t.sub.1] =l, [t.sub.2] = l in favor of [t.sub.1] = h, [t.sub.2] = l and an equilibrium in [t.sub.1] = h, [t.sub.2] = l in favor of [t.sub.1] = h, [t.sub.2] = h. As such, a rationale for positive gate revenue sharing exists, if talent investments are too high without sharing. The upper left panel of Figure 3 suggests that this is indeed the case, because the league thresholds clearly lie below the Nash-equilibrium thresholds for all values of n.

[FIGURE 3 OMITTED]

PROPOSITION 4. When no gate revenue sharing takes place ([alpha] = 1):

* h - l [less than or equal to] [[lambda].sub.u,h] [??] [t.sub.1] = h, [t.sub.2] = h

* [t.sub.1] = l, [t.sub.2] = l [??][[lambda].sub.u,l] [less than or equal to] h - 1.

Proof. See Appendix

Proposition 4 formally establishes that without local sharing, talent investments always exceed the league's profit-maximizing level, an observation I already made above in relation to Figure 3. Specifically, the first line of Proposition 4 guarantees that without sharing, mutually high investments always arise in equilibrium if this maximizes league profits, whereas the second line states that a Nash-equilibrium in mutually low investments only arises if this corresponds to the league optimum. In other words, private incentives are strong enough to push the individual teams' talent investments to a level that more than suffices from the league's perspective of joint profit-maximization. In the absence of sharing, the league, therefore, wants to push teams toward lower talent investments. As local revenue sharing decreases investment incentives, it could be a tool to achieve a Nash-equilibrium in talent investments, which is closer to the league's profit-maximizing solution. Following this line of reasoning, it is clear that introducing "some" local revenue sharing is a profitable strategy for the league.

The question remains what level of gate revenue sharing the league should choose, because selecting an overly high percentage of sharing leads clubs to underinvest in talent. The bottom right panel of Figure 3 depicts a situation where this is the case. In this graph, clubs engage in full local revenue sharing. Clearly this drives their investments down too far, because [t.sub.1] = l, [t.sub.2] = l is the Nash-equilibrium outcome, while profits are maximized under [t.sub.1] = h, [t.sub.2] = l or even [t.sub.1] = h, [t.sub.2] = h. However, as long as the result of Proposition 4 holds for a given level of revenue sharing a, the league should not be worried about this type of underinvestment.

COROLLARY 1. The results of Proposition 4 are guaranteed to hold for all values of [alpha] > [m.sub.1] - [m.sub.2]/[m.sub.1] + [m.sub.2] = [[alpha].sup.*].

Proof. See Appendix.

Corollary 1 implies that on the interval [[m.sub.1] - [m.sub.2]/[m.sub.1] + [m.sub.2] = 1], the league is sure that increasing snaring cannot lead to underinvestment in Nash-equilibrium. The league has a clear rationale to introduce a local sharing rule of at least [[alpha].sup.*]. This policy guarantees to induce lower investments, which is profitable for the league as a whole, without the risk of underinvestment. The bottom left panel of Figure 3 illustrates this property. At [[alpha].sup.*], the league and Nash-equilibrium thresholds on [t.sub.1] = l, [t.sub.2] = l exactly coincide, a result that holds for all relevant choices of the parameter values. Pushing local revenue sharing further inevitably brings about partial underinvestment. Observe, however, that there is still some overinvestment at [[alpha].sup.*] with respect to [t.sub.1] = h. [t.sub.2] = h versus [t.sub.1] = h, [t.sub.2] = l. This implies that local revenue sharing is not a fully adequate tool to reach the optimal outcome for the league.

In order to further explore the effects of revenue sharing on the profitability of the clubs, it is instructive to compare profits under [alpha] = 1 and [alpha] = [[alpha].sup.*]. I denote the profits of team i by [[pi].sup.1.sub.i] and [[pi].sup.*.sub.i] for [alpha] = 1 and [alpha] = [[alpha].sup.*], respectively, and similarly league profits by [[pi].sup.1.sub.L] and [[pi].sup.*.sub.L].

PROPOSITION 5. Under a sharing rule [[alpha].sup.*] = [m.sub.1] - [m.sub.2]/[m.sub.1] + [m.sub.2].

* total league profits are at least as high as without gate revenue sharing, [[pi].sup.1.sub.L] and [[pi].sup.*.sub.L]

* the small market team profits are at least as high as without gate revenue sharing, [[pi].sup.l.sub.2] and [[pi].sup.*.sub.2].

This result does not hold for the profits of the large market team ([[pi].sup.1.sub.1] [??] [[pi].sup.*.sub.l]).

Proof. See Appendix.

Proposition 5 explicitly establishes that the total profits in the league are higher under [[alpha].sup.*] than without local revenue sharing, a result implied by Proposition 4 and Corollary 1. Proposition 5 further shows that the small team always benefits when local revenues are shared at the level [[alpha].sup.*]. This is intuitive because its smaller local market and the result of Proposition 1 together imply that the small team contributes less to the central pool than it receives. Obviously this cannot simultaneously be true for the large club. For the large club, any benefit from the sharing arrangement has to come in the form of a reduction in talent costs. In other words, if sharing has no effect on the Nash-equilibrium in the talent investment stage, there is no gain for the large team. This observation touches on the issue of the incentive compatibility of local sharing for the large club, that is, why would the club enter an agreement that may not be beneficial to its private objectives? Peeters (2012) explores this issue in more detail for media revenue sharing. In the current context, it is noteworthy that the incentive compatibility problem may put additional restrictions on the league's choice of sharing rule. Finally, note that the results of Proposition 5 are in line with the findings of Kesenne (2007).

C. Characteristics of the Profitable Sharing Rule

Corollary 1 leads to three interesting observations, which merit further discussion. First, [[alpha].sup.*] = [m.sub.1] - [m.sub.2]/[m.sub.1] + [m.sub.2] relates the level of local revenue sharing to the relative size of the large local market. Corollary 1 suggests, therefore, that leagues with more heterogeneous local markets need to share less revenues to be profitable. To grasp the intuition for this result, think about the free-riding effect of revenue sharing. In a league with heterogeneous local markets, the revenue the small club obtains from local revenue sharing is more important for its overall profits. As such, the small club quickly internalizes the damage, which its talent investments inflict on the large team's local revenues. This leads it to shrink its investments faster than would be the case in a homogeneous league and allows the large team to do the same. A somewhat striking implication of Corollary 1 is that leagues in which teams differ strongly in their revenue-generating potential should share less gate revenues. From Section 4, we know that leagues with large differences in local market sizes also have a higher chance of a Nash-equilibrium in unbalanced investments. Consequently, sharing is less likely to be employed by leagues with lower competitive balance. Clearly, this finding is hard to reconcile with the defense of revenue sharing based on competitive balance arguments.

Second, it is instructive to discuss the two types of underinvestment that arise if the league pushes for a sharing rule above [[alpha].sup.*]. First, too much sharing undermines local revenues, because it incentivizes the large market team to revert to low investments, while the league would prefer it to dominate the competition. Second, lower talent investments also diminish the overall quality of play. Remember, however, that I have assumed that this is of no concern to local fans and, therefore, cannot harm gate revenues themselves. Only neutral fans care about overall quality of play, so only media revenues go down in response to an overall decline of playing quality. In other words, overly generous gate sharing hits the league partly through a decline in media revenues. This shows that sharing one revenue source may have important ramifications on revenues from other sources and underlines the importance of including multiple revenue streams in the analysis.

Finally, the results of Corollary 1 provide a further explanation for the observations of Table 1. The dp measure indicates that European soccer teams differ substantially in local market size. Consequently, gate revenue sharing is a rare phenomenon in these leagues. (20) Rather than being a shortcoming of league authorities or stemming from a desire to comply with competition policy, this may simply constitute an example of profit-maximization. On the other side of the Atlantic, the NFL, MLB, NHL, and MLS force their teams to share local and/or gate revenues. Again, this may constitute a profit-maximizing strategy, especially because the most homogeneous league (NFL) shares more than the other U.S. major leagues. The NBA has had no gate sharing in the recent past. This is somewhat at odds with the results of the model, as it appears to have quite homogeneous markets. Yet, the new collective bargaining agreement may be expected to include an increase in some form of local revenue sharing. (21) Also, the NBA directly taxes high payrolls through its luxury tax, the proceeds of which are redistributed. Another puzzle is the AFL, which appears to have relatively homogeneous teams, but abolished gate sharing in 1999.

VI. CONCLUDING REMARKS

This article has analyzed how a sports league may use gate revenue sharing to coordinate talent investments. Revenue sharing depresses talent investments by all teams, which has an initial profit-enhancing effect. Leagues with more unequal local markets have less to gain from high amounts of revenue sharing than more homogeneous leagues. This implies that revenue sharing persists more, when there is less chance of a low competitive balance.

An important question with regard to the robustness of these results is how far these effects would also prevail in a model of an 77-team league, as opposed to a two-team league. Intuitively, the basic strategic effects, which drive the main results in this article, would generalize to such a context. It seems straightforward that clubs competing in an 77-team league would reduce investments in talent when local revenue sharing is introduced. Likewise, sharing would result in both a taxation and free-riding effect for all 77 teams in the league. For small market teams in an 77-team league, the revenue received from local sharing also constitutes a more important part of their overall revenue potential if the larger teams in the league serve larger local markets. As in a two-team league, this will lead small teams in heterogeneous leagues to free ride more than small teams in homogeneous leagues. The talent-reducing effects of sharing would, therefore, be more pronounced in 77-team leagues where the gap between large and small teams is larger. As in the model of this article, this would lead to the result that heterogeneous leagues tend to maximize profits by sharing a lower percentage of local revenues. Clearly, the optimal sharing rule I derive for the case of a two-team league is more readily interpretable than its potential 77-team equivalent, simply because it is harder to capture the distribution of 77 local markets in a single comprehensive metric. Furthermore, providing formal proof for all propositions in an 77-team context would probably require moving away from the assumption of discrete talent investments, and the tractable assumptions in terms of consumer utility applied here. I will, therefore, restrict my modeling to a two-team league, leaving these extensions to be picked up in future research.

Several other issues put forward in my analysis also provide opportunities for future research. First, the results obtained in this article could be tested empirically. A potentially fruitful way to make progress in this issue would be to construct a panel dataset of sharing arrangements in the American major leagues. The American leagues have a long history of local revenue sharing, which provides variation in sharing arrangements within one league over time, as well as between leagues in the same time period. According to the predictions of my theoretical model, local sharing should ceteris paribus be higher if teams in the league serve more homogeneous local markets. The distribution of local markets can be assessed empirically by calculating distributional measures (e.g., the Gini coefficient) for the population of the metropolitan statistical areas in which franchises are located with appropriate corrections for teams sharing markets. Again, there would be variation over time and between leagues, as both franchise movements and demographic factors lead local markets to shift. Alternatively, one could envision a setup as in Peeters (2011), which compares sharing mechanisms across European soccer leagues. Here, the dp measure presented in Table 1 is a potential measure for the distribution of local markets. Note that most other variables in my model also have a natural empirical interpretation. For example, the size of the neutral TV market n corresponds to the number of TV households in the league's home country.

Second, the model could be enriched by the interaction between local revenue sharing and other coordination devices. This topic is pioneered by Dietl, Lang, and Rathke (2011), who look at the competitive balance effect of sharing when leagues combine local revenue sharing with salary caps.

Finally, the analysis in this article raises questions on the applicability of revenue sharing as a coordination device in other industries. A similar setup may be applied to the entertainment industry, where the quality of production also depends on a small group of highly talented and highly rewarded employees. Further, advertising and R&D expenditures often function in a similar way to talent investments in the present model. In a competitive equilibrium, firms invest more in advertising and R&D than the collusive optimum. Yet, these investments also raise demand for the industry as a whole, as talent investments do for sports leagues. These issues seem worthwhile to be explored, especially in the context of the semi-collusion literature.

APPENDIX

In this Appendix, I provide proof of all propositions.

Proposition 1

Proof.

[t.sub.1] = l, [t.sub.2] = h is a Nash-equilibrium if both

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

by using Equations (3) and (2).

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], by using Equation (3).

Equating both expressions leads to:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], this is a contradiction.

Proposition 2

Proof.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], using Equation (3).

At both thresholds, multiple equilibria are possible. In order to have unique Nash-equilibria at all other values for the model parameters, it is necessary that both thresholds are never met simultaneously. Observe this is the case when [m.sub.1] < [[theta].sub.l] and [m.sub.2] > [[theta].sub.r], are not simultaneously fulfilled. This is guaranteed by [m.sub.1]- [m.sub.2] [greater than or equal to] n(b)(h,h)-b(l,l)/f(h,l)-f(l,l). To see this, plug this into the second condition to obtain 1/1 + [alpha] (8 h-l/f(h,l)-f(l,l) - 2 ([m.sub.1] - [m.sub.2]) + (1 - [alpha][m.sub.1]) < [m.sub.2].

From which straightforward derivation shows that 1/1+[alpha] (8 [h-l/f(h,l)-f(l,l)] + (1 - [alpha])[m.sub.2]) < [m.sub.1], which violates the first condition.

Proposition 3

Proof.

I derive thresholds on the level of talent costs for each talent investment outcome to maximize league profits. League profits in each situation are given by:

(A1) [[pi].sup.h,h.sub.L] = 1/4 (([m.sub.1] + [m.sub.2]) + m2) f (h,h) + 2nb(h,hj) -2h

(A2) [[pi].sup.h,l.sub.L] = 1/4 (([m.sub.1]f (h,l) + [m.sub.2]f (l,h) + 2nb(h,l) - (h + l)

(A3) [[pi].sup.l,h.sub.L] = 1/4 (([m.sub.1]f (l,h) + [m.sub.2]f (h,l) + 2nb(l,h) - (h + l)

(A4) [[pi].sup.l,l.sub.L] = 1/4 (([m.sub.1] + [m.sub.2])f(l,l) + 2nb (l,l) -2l

* [t.sub.1] = l, [t.sub.2] = h maximizes league profits only if [[pi].sup.l,h.sub.L] [greater than or equal to] [[pi].sup.h,l.sub.L]. From Equations (A2) and (A3), this may be filled out to obtain 1/4 (([m.sub.1]f (l,h) + [m.sub.2]f (h,l) + 2nb(l,h) - (h + l) [greater than or equal to] 1/4 (([m.sub.1]f (h,l) + [m.sub.2]f (l,h) + 2nb(h,l) - (h + l). Simplifying leads to [m.sub.2] [greater than or equal to] [m.sub.1]. a contradiction. Consequently, [t.sub.1] = l, [t.sub.2] = h; never maximizes league profits and it is not necessary to compare it to the other outcomes.

* [t.sub.1] = [t.sub.2] = h maximizes league profits if both:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

again because of Equation (3).

Because Equation (9) implies that 1/4 (([m.sub.1] - [m.sub.2]) (f(h,l) - f(h,h) > 1/4 n (b(h,h) - b(l,l), [[lambda].sup.u,h] [??][[lambda].sup.u,h] is the only relevant threshold.

* [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

since Equations (2) and (3) may be applied.

Observe that Equation (9) implies that [[lambda].sup.h,l] < [[lambda].sup.u,l] [??][[lambda].sup.u,l] is the binding threshold in this case.

* [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] as shown above.

Joining both conditions gives [[lambda].sup.u,h] [less than or equal to] h-l [less than or equal to] [[lambda].sup.u,l] which shows that [t.sub.1] = h, [t.sub.2] = l maximizes league profits for intermediate levels of the talent cost.

Proposition 4 Proof.

The first statement of Proposition 4 implies that for [alpha] = 1:

(A5) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

First, observe that if [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

So Equation (A5) holds if:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

This holds because of Equation (9).

The second statement of Proposition 4 implies that for [alpha] = 1:

(A6) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], which proves Equation (A6).

Corollary 1

Proof.

To allow for any a in the proof of Equation (A5), [[theta].sub.h] < [m.sub.2], may be rewritten as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Therefore, Equation (A5) holds as long as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

This holds because of Equation (9) if:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

To show the same for Equation (A6), rewrite 9 (> nq for any [alpha]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Therefore, Equation (A6) holds as long as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Proposition 5

Proof.

I first prove that [[pi].sup.*.sub.L] [greater than or equal to] [[pi].sup.l.sub.l] by showing that the reverse leads to a contradiction. Observe that the difference in joint profits may be written as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(A7)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Given that [t.sup.*.sub.1] [less than or equal to] [t.sup.l.sub.i], four talent investment outcomes are feasible:

1 [t.sup.1.sub.l] = [t.sup.*.sub.1] [t.sup.l.sub.2] = [t.sup.*.sub.2]

Filling out Equation (A7) leads to 0 < 0, which is a contradiction.

2. ([t.sup.l.sub.1] = h, [t.sup.l.sub.2] = h) [right arrow] ([t.sup.*.sub.1] = h, [t.sup.*.sub.2] = l)

Filling out Equation (A7):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Plug this into the expression to obtain:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], which violates Equation (9) and is, therefore, a contradiction.

3. ([t.sup.l.sub.1] = h, [t.sup.l.sub.2] = h) [right arrow] ([t.sup.*.sub.1] = l, [t.sup.*.sub.2] = l)

Filling out Equation (A7):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], which implies a contradiction.

4. ([t.sup.l.sub.1] = h, [t.sup.l.sub.2] = l) [right arrow] ([t.sup.*.sub.1] = l, [t.sup.*.sub.2] = l)

Filling out Equation (A7):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], which implies a contradiction.

A similar logic can be applied to show that [[pi].sup.*.sub.2] [greater than or equal to] [[pi].sup.1.sub.2]. Observe that:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(A8)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

There are again four cases:

1 [t.sup.1.sub.l] = [t.sup.*.sub.1], [t.sup.l.sub.2] = [t.sup.*.sub.2]

Filling out (A) leads to 0 < [m.sub.2]/4([m.sub.1] + [m.sub.2]) ([m.sub.2]f ([t.sup.*.sub.2], [t.sup.*.sub.1]) - [m.sub.1]f ([t.sup.*.sub.1], [t.sup.*.sub.2])), which is a contradiction.

2. ([t.sup.l.sub.1] = h, [t.sup.l.sub.2] = h) [right arrow] ([t.sup.*.sub.1] = h, [t.sup.*.sub.2] = l)

Filling out Equation (A7):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Plug this in to obtain:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], which is a contradiction.

3. ([t.sup.l.sub.1] = h, [t.sup.l.sub.2] = l) [right arrow] ([t.sup.*.sub.1] = l, [t.sup.*.sub.2] = l) Filling out

Equation (A7):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

I can again proceed as above, and further apply Equation (3) to obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], which is a contradiction.

4. ([t.sup.l.sub.1] = h, [t.sup.l.sub.2] = l) [right arrow] ([t.sup.*.sub.1] = l, [t.sup.*.sub.2] = l) Filling out

Equation (A7):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] because of

Equation (2). This is a contradiction, because f(l, l) > f(l,h) and [m.sub.1] > [m.sub.2].

To see that [[pi].sup.*.sub.l] [greater than or equal to] [[pi].sup.l.sub.1] does not always hold, simply observe that in case ([t.sup.l.sub.1] = [t.sup.*.sub.1], [t.sup.1.sub.2] = [t.sup.*.sub.2]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

ABBREVIATIONS

AFL: Australian Football League

EPL: English Premier League

MLB: Major League Baseball

MLS: Major League Soccer

NBA: National Basketball Association

NFL: National Football League

NHL: National Hockey League

doi: 10.1111/ecin.12184

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(1.) See Washington Post, June 29, 2011, "Lengthy NBA lockout looms, with owners and players deeply divided."

(2.) In the sports economics literature, talent investments are usually taken to mean wage spending, training costs, and/or transfer fees. The underlying assumption is that clubs will choose an optimal mix of spending on wages, training, and transfers, given the total level of talent investment they aim for.

(3.) The last column of Table 1 presents a measure for the distribution of a league's local markets. A more unequal distribution corresponds to a higher value of this "dp" measure.

(4.) The main justification for introducing gate revenue sharing has traditionally been the perceived need to protect the uncertainty of outcome in sporting competitions. This line of reasoning is usually referred to as the competitive balance argument in the sports economics literature.

(5.) Naturally, this enumeration is far from exhaustive. It simply serves as an illustration of the variety of papers found in this literature.

(6.) I do not model the internal decision-making in the cartel, as is done in Easton and Rockerbie (2005) or Peeters (2012).

(7.) When I encounter several alternative modeling options, I choose the assumption implying the lowest incentives to invest in talent. In doing so, I avoid that the result of proposition 4 (overinvestment in talent without sharing) could come about as an artifact of overly generous modeling assumptions.

(8.) I provide a discussion on the transferability of the results to an n-team model in the final section. Still, the assumption of a two-team league is fairly standard in the theoretical literature on sports leagues (see, e.g., Szymanski and Kesenne 2004). Using a two-team model allows to clarify the strategic effects at play in a straightforward way, because it provides closed-form solutions, which are more easily interpretable. Furthermore, it greatly reduces the mathematical burden to obtain meaningful results in any type of contest model and thus improves the readability of the paper. In the specific setup used in this paper, it also avoids the need to make several ad hoc assumptions on consumer utility and so forth.

(9.) Alternatively, authors (e.g., Kesenne 2000) have assumed that teams maximize their winning percentage or a combination of winning and profits. Doing so in this model would strengthen the incentive to invest in talent, which I have opted to avoid.

(10.) Note that this is not necessarily the case, as the league could decide to implement another way of sharing the pool, for example, performance-based sharing. In that case, the results obtained here need not hold.

(11.) As is shown in Peeters (2012), this leads to lower talent investments than performance-based sharing, which is often applied in European football leagues.

(12.) Obviously, this constitutes a very basic representation of the relationship between talent investments and wins. For a study that examines the influence of the choice of contest success function, see Fort and Winfree (2009).

(13.) See, for example. Fort and Quirk (1995) for a model with a closed talent market.

(14.) Alternatively, I could assume that b(h, h) > b(h, l) = b(l, h) > b(l, l) or b(h, h) > b(l, l) > b(h, l) = b(l, h), such that neutral fans have a larger preference for either absolute quality or tension. However, to my knowledge, there is no empirical evidence to guide the choice between these alternatives. Furthermore, assuming b(h, h) > b(h, l) = b(l, h) > b(l, l) again implies that talent investment incentives would rise, which I aim to avoid. Alternatively, b(h, h) > b(l, l) > b(h, l) = b(l, h) would shrink the occurrence of unbalanced investments. Since all results in the paper are shown for any value of h-l, one could simply assume a lower value of the talent cost to again arrive at unbalanced investments.

(15.) Again, one may alternatively assume that [f.sub.i](h, l) > [f.sub.i](h, h) > [f.sub.i](l, l) > [f.sub.i](l, h) such that local fans care for the absolute quality in the league as well as the relative quality of the home team. This would imply that teams have an additional incentive to invest in talent.

(16.) One might argue that other forms of local revenues, such as the value of local sponsorship deals, are also responsive to the utility of committed fans and, therefore, may be analyzed in a similar fashion.

(17.) In the United States, the league usually sells national media rights, while local rights are sold by individual clubs (the exception being the NFL, which has no local TV rights deals). Under an alternative interpretation of this model, local media revenues (stemming from local, committed fans) are shared under the gate revenue sharing arrangement.

(18.) To draw this picture, the other parameter values in the model were fixed to h - l = 2,000,000, n = 1,000,000, f(h, l) -f (l, l) = 20, and b(h, h) - b(h, l) = 0.2.

(19.) The graph is drawn for [m.sub.1] = 1,000,000, [m.sub.2] = 500,000, f(h, l) -f (l, I) = 20, and b(h, h) - b(h, l) = 0.2. The right-hand border of the graphs is given by condition (9).

(20.) As noted before, first division teams in the EPL and Bundesliga usually play a low amount of cup games each year, meaning that sharing only applies to a small fraction of games.

(21.) See Washington Post, June 29, 2011, "Lengthy NBA lockout looms, with owners and players deeply divided."

THOMAS PEETERS, I would like to thank the editor, Jeff Borland, one anonymous referee, Stefan Kesenne, Jan Bouckaert, Mathias Reynaert, Stefan Szymanski, Jason Winfree, Steven Salaga, Brian Mills, Iwan Bos, Martin Grossman, Stephen Layson, Ignacio Palacios-Huerta, Wilfried Pauwels, Bruno De Borger, and Eric van Damme for comments and suggestions. This article also benefited from discussions with seminar and conference participants at the University of Antwerp, the University of Michigan, the EARIE conference in Rome, the NAASE sessions in San Francisco, the HOC, and the Southern Economic Association Conference in Washington, DC. I gratefully acknowledge the financial support from the Flanders Research Foundation (FWO).

Peeters: Assistant Professor, Erasmus School of Economics, Applied Economics, Erasmus University Rotterdam, Rotterdam 3000DR, Netherlands. Phone +32 494124936, Fax +31 104089141, E-mail peeters@ese.eur.nl

TABLE 1
Gate Revenue Sharing in Sports Leagues

Country          Sport                 League

United States    American football     NFL
United Staes     Baseball              MLB
United States    Basketball            NBA
United States    Hockey                NHL
United States    Soccer                MLS
Germany          Soccer                Bundesliga
United Kingdom   Soccer                EPL
Spain            Soccer                La Liga
Australia        Australian football   AFL

Country          Local Revenue         Market Size
                 Sharing               Distribution
                                       (dp) (a)

United States    60-40 (b)             0.103
United Staes     69-3 l (c)            0.248
United States    Unclear1 (d)          0.101
United States    yes                   0.106
United States    70-30                 0.238
Germany          Cup games             0.403
United Kingdom   Cup games             0.367
Spain            No                    0.615
Australia        No (e)                0.208

(a) As a proxy for the market size distribution, I calculate
the "dp" measure on the 2001-2010 seasons. This measure
gives the average of the standard deviation of 10-year
average attendances divided by the league average attendance
on a season-by-season basis. See Peeters (2011) for more on
this.

(b) Since 2001, sharing is organized through a central pool.

(c) Since 1996, gate revenue sharing proceeds through a
central pool. Percentages have varied over time; since 2007,
31% of revenues are to be shared.

(d) At the time of writing, the details of the new revenue
sharing rule were not publicly available.

(e) Source: Booth (2006).