sports economics

oxfordhb-Competitive20Balance.pdf

Home >Business & Finance homework help >Economics homework help >sports economics

Club Objectives, Competitive Balance, and the Invariance Proposition

Page 1 of 15

PRINTED FROM OXFORD HANDBOOKS ONLINE (www.oxfordhandbooks.com). © Oxford University Press, 2018. All Rights Reserved. Under the terms of the licence agreement, an individual user may print out a PDF of a single chapter of a title in Oxford Handbooks Online for personal use (for details see Privacy Policy and Legal Notice).

Subscriber: Ohio State University; date: 17 August 2018

Abstract and Keywords

This chapter considers the analysis of the competitive balance in a formal model that focuses attention on the nature of the market for talent and the nature of the club owner's objective function, and derives the conditions which have led to the dissenting results about the competitive balance. It then addresses the optimal competitive balance in a league theoretically, and whether a win- or profit-maximizing club comes closest to the social optimum. Next, the chapter deals with the invariance proposition, analyzing the effects of restrictions on player mobility and revenue-sharing arrangements. The most unequal competitive balance can be expected in a league in which the large-market clubs are win maximizers and the small-market clubs are profit maximizers. It is noted that the invariance proposition no longer holds if one of the teams in a league is a win maximizer.

Keywords: competitive balance, invariance proposition, club owner, win maximizers, profit maximizers, player mobility, revenue sharing

Club Objectives, Competitive Balance, and the Invariance Proposition Stefan Kesenne The Oxford Handbook of Sports Economics: The Economics of Sports Volume 1 Edited by Leo H. Kahane and Stephen Shmanske

Print Publication Date: Apr 2012 Subject: Economics and Finance, Industrial Organization, Business Economics Online Publication Date: Sep 2012 DOI: 10.1093/oxfordhb/9780195387773.013.0003

Oxford Handbooks Online

Club Objectives, Competitive Balance, and the Invariance Proposition

Page 2 of 15

Subscriber: Ohio State University; date: 17 August 2018

1. Introduction There has been some controversy in the sports-economics literature regarding the differences in competitive balance between a league in which all teams are maximizing profits and a league in which the objective of the teams is to maximize their season winning percentage. Kesenne (1996, 2000), using a simple fixed-supply Walrasian model, has shown that the distribution of playing talents in a win-maximization league will be more unequal than in a profit-maximization league. However, Fort and Quirk (2004) have questioned the validity of this result, showing that nothing can be derived about the difference in competitive balance between a win- and a profit-maximization league, if no other assumptions are made on the club-revenue functions beyond concavity. In section 2 of this chapter, we will further investigate this issue by deriving the conditions that have led to these dissenting results about the competitive balance. In section 3, we try to derive the optimal competitive balance in a league theoretically, and whether a win- or

profit-maximizing club, comes closest to the social optimum. Section 4 deals with the invariance proposition, analyzing the effects of restrictions on player mobility and revenue-sharing arrangements. Section 5 concludes.

2. Competitive Balance The conclusion of Fort and Quirk (2004) that nothing can be derived regarding the competitive balance between a win- and a profit-maximization league if no further assumptions are made on the club-revenue function beyond concavity is, no doubt, correct. This can be illustrated graphically in Figure 3.1 for a simple two-team league with linear marginal- and average-revenue curves. The large-market team x is portrayed from left to right, and the small-market team y from right to left. The distance between the two origins is determined by the (constant) supply of playing talent. Under the profit- maximization assumption, both clubs' demand curves for talent are given by their marginal revenue curves with the competitive market equilibrium in point (T ). Under the win maximization assumption, both clubs' demand curves for talent are given by the average-revenue curves, and the market equilibrium is found in point (T ) (see Kesenne, 1996, 2000).

(p. 39)

Club Objectives, Competitive Balance, and the Invariance Proposition

Page 3 of 15

Subscriber: Ohio State University; date: 17 August 2018

In this particular representation, which shows a very steep slope of the MR-curve of large- market team x and a very flat slope of the small- market team y, it can be seen that that the competitive balance under profit maximization (T ) is more unequal than under win maximization (T ).

However, there might be good reasons to make further assumptions about a club-revenue function in team sports that will allow drawing some valuable conclusions on the competitive-balance issue. The slope of the MR-curve is affected by the preferences of the spectators regarding the win percentage of the team and the competitive balance in the league. We try to show here that there should be a relationship between the slope and the intercept of the MR-curves in order to become well-behaved revenue functions.

Most sports economists agree that five major variables should enter a club's revenue function:

1. The market size or drawing potential of the team 2. The winning percentage 3. The competitive balance in the league 4. The ticket price 5. The absolute quality of the league (see Scully, 1989; Noll, 1974; Zimbalist, 1992; Vrooman J., 1995; Downward and Dawson, 2000; Dobson and Goddard, 2001; Szymanski, 2003; Sandi, Sloane, and Rosentraub, 2004; Gerrard, 2006; Fort, 2006; Kesenne, 2007).

The absolute quality, however, is constant in a fixed-talent supply model. Because we want to concentrate on the player labor market, we also start from an exogenously given and constant ticket price. Based on the extensive empirical evidence (see Simmons, 1996; Forrest and Simmons, 2001; Garcia and Rodriguez, 2002), the positive impact of the market size is unquestioned, but there is some reasonable doubt about the relative importance of the winning percentage and the competitive balance. In particular, the importance of the uncertainty of outcome on attendances and public interest is doubtful (see Borland and Macdonald, 2003; Szymanski and Leach, 2006). This is a crucial issue regarding the competitive balance between win- and profit-maximization leagues.

In the following two-club model, we start from a specification of a club-revenue function that leaves open the relative importance of winning and competitive balance. As in

Click to view larger

Figure 3.1 Profit and Win Maximization Equilibrium with Ill-Behaved MR-Curves.

(p. 40)

Club Objectives, Competitive Balance, and the Invariance Proposition

Page 4 of 15

Subscriber: Ohio State University; date: 17 August 2018

Vrooman (2008), we assume that a linear combination of winning w and uncertainty of outcome (uo) positively affects club revenue, with unspecified weights β .

(1)

In this specification, the uncertainty of outcome is approached by the product of the winning percentages of the two teams: u0 = w w = w (1−w ). This variable reaches its maximum value for w = 0.5. If x is the large-market team and y the small-market team; m 〉m . We also allow the weights β to be different for the large- and the small-market teams, because it is possible that the supporters of small-market teams attach a different value to winning percentage and competitive balance than the supporters of large-market teams.

Rearranging the revenue functions from Eq. (1), they can be written as:

(2)

What appears are the simple quadratic club-revenue functions and linear marginal- revenue functions that are often used in the sports economics literature (Fort and Quirk, 1995; Vrooman, 1995; Szymanski, 2004; Dietl, Lang, and Rathke, 2009). However, it is important the notice here that, based on specification (1), a positive relationship emerges between the absolute value of the slope and the intercept of the marginal revenue functions. This condition is necessary to get well-behaved quadratic club-revenue functions, meaning that, ceteris paribus (all else equal), the large-market team is at least as talented as the small-market team. Without this condition, the large-market team can turn out to be less talented than the small-market team in the market equilibrium. Without this condition, there is also no guarantee that a market equilibrium will be found with positive talent demands and a positive unit cost of talent. By only assuming concavity, club-revenue functions can be ill-suited, so that nothing can be derived regarding the competitive balance between a win- versus a profit-maximization league, as asserted by Fort and Quirk (2004). With well-suited revenue functions, one will always find more talents in the large-market club, all else equal, for whatever values of the market sizes, as can be shown later.

On the cost side, the total season cost of a team consists of the player labor cost (ct ) and the capital cost (c ). In the long term, one can assume that the capital cost is proportional to the number of talents in a team (with the same proportionality factor

x y x x

x x

y i

(p. 41)

i 0

Club Objectives, Competitive Balance, and the Invariance Proposition

Page 5 of 15

Subscriber: Ohio State University; date: 17 August 2018

v≥0), because successful teams with top talents need a larger stadium, and well-paid top players are not content with poor stadium facilities. So, the cost function can be written as: .

The market equilibrium can be found by equalizing the marginal revenue function in Eq. (2):

(3)

All else equal (β = β = β ), the winning percentages are:

(4)

With 0〈β≤1, it is clear from Eq. (4) that w ≥w . In the extreme case with β = 1, meaning that supporters do not care at all about the winning percentage of their team but only value a balanced competition, we find that w = w = 0.5.

If the winning percentages are given by the ratio of the playing talents of a team and the total supply of talents in the league:

(5)

The MP of talent can then be derived as:

(6)

1 2

(p. 42) x y

x y

Club Objectives, Competitive Balance, and the Invariance Proposition

Page 6 of 15

Subscriber: Ohio State University; date: 17 August 2018

Because the controversy in the literature about the competitive balance between a profit- and a win-maximization league refers to the fixed-supply Quirk/Fort model (1992) of the player labor market, we concentrate on that model in this paper. If the supply of talent is constant and normalized to equal unity, and the externalities of talent hiring are internalized—that is, club managers take into account, in calculating their marginal revenue, that if they hire an extra talent, the other team is losing a talent—it follows that ∂t /∂t = −1, which is the so-called Walras fixed-supply conjecture. So, the MP of talent simplifies to:

(7)

Under these hypotheses, the winning percentage in the revenue functions of Eq. (2) can simply be replaced by the number of talents. A somewhat odd feature of this Walrasian fixed-supply conjecture model is that team managers have full control of their winning percentage, and also that one team has no choice of talents because t = 1−t . See Szymanski and Kesenne (2004) and Szymanski (2004) for a critical discussion of this model. With the Nash conjecture ∂t /∂t = 0, the MP of talent becomes:

(8)

If teams are profit maximizers, the non-cooperative Nash-Cournot equilibrium can be found by solving the two teams' reaction functions:

(9)

However, if the number of teams in a league is large enough, the non-cooperative Nash equilibrium approaches the competitive equilibrium. It follows that with a large number of teams, the Walrasian competitive equilibrium model can be used to derive the competitive balance, also under the Nash conjecture. Given these hypotheses, the demand curves for talent of two profit maximizing teams can be written as:

y x

(p. 43)

Club Objectives, Competitive Balance, and the Invariance Proposition

Page 7 of 15

Subscriber: Ohio State University; date: 17 August 2018

(10)

In order to derive the number of talents of the large-market team in the market equilibrium (talent demand equals talent supply) under profit maximization, we have to solve the following equation:

(11) So that:

(12)

The equilibrium number of talents of the large-market team in a win-maximization league, taking into account that the demand curves are now determined by the average revenue curves, can be found by solving the equation:

(13)

This results in the following number of talents in the large market team:

(14)

Comparing Eq. (14) to Eq. (12), one can see that , whatever the size of the

weights β in the large and the small market club. It follows that, under the specified conditions above, the distribution of talent will be more unequal under win maximization, whatever the differences in the spectators' preferences for winning and competitive balance.

(p. 44)

Club Objectives, Competitive Balance, and the Invariance Proposition

Page 8 of 15

Subscriber: Ohio State University; date: 17 August 2018

It is also possible that the teams in one league have different objectives. If we consider these mixed leagues and start from the scenario with a profit-maximizing large-market club and a win-maximizing small-market club, the solution is found by solving:

(15) So that:

(16)

If the large-market club is a win maximizer and the small-market team is a profit maximizer, we have to solve the following equation:

(17) so that the number of talents in the large-market team is:

(18)

Comparing Eq. (18) to Eq. (16), we can see that , meaning that a more

unbalanced competition can be expected if the large-market club is a win maximizer and the small-market club is a profit maximizer.

On can also derive, by comparing Eq. (16) to Eq. (12) and Eq. (18) to Eq. (14) that

and that , as long as . In case and , which is

impossible in reality, both are truncated to equal unity. Summarizing, the following proposition holds, whatever the spectators' preferences for winning and uncertainty of outcome:

(19)

In other words, the highest uncertainty of outcome will be reached in a league in which the large-market clubs are profit maximizers and the small-market clubs are win maximizers. The most unequal competitive balance can be expected in a league in which the large-market clubs are win maximizers and the small-market clubs are profit

Club Objectives, Competitive Balance, and the Invariance Proposition

Page 9 of 15

Subscriber: Ohio State University; date: 17 August 2018

maximizers. The competitive balance will be more unequal in a pure-win-maximization league than in a pure-profit-maximization league.

These results are presented graphically in Figure 3.2.

3. Optimal Competitive Balance Another interesting question is which one of these four competitive balances from section 2 is optimal? One possible optimality criterion is the maximization of total league revenue. It is clear that, in a league in which all clubs are profit maximizers, total league revenue will reach

its highest possible level, because only in this market equilibrium is MR , = MR , so that the allocation of talents over the teams is optimal. That is, every talent is playing on the team in which its marginal productivity is highest. In the three other leagues, it is clear from Figure 3.2, that there is some misallocation of resources, causing a welfare loss in terms of total league revenue. This welfare loss can be measured by the size of the triangular area between the two MR-curves and between the profit-maximizing equilibrium and the actual market equilibrium. This loss is largest in a league in which the large-market club is a win maximizer and the small-market club is a profit maximizer (T ), as indicated by the shaded area in Figure 3.2.

However, this welfare criterion does not take into account the preferences of the public regarding the competitive balance. If the public prefers a perfectly balanced competition, which means that the win percent of every team is 0.50, it is clear from the foregoing analysis that the league in which the large-market clubs are profit maximizers and the small-market clubs are win maximizers comes closest. It is doubtful, however, from both theoretical and empirical research, that this is what spectators want (see Rasher and Solmes, 2007). Theoretically, starting from the uncertainty of outcome specification, u0 = w (1−w ), maximizing the spectators' utility function over winning percentage and competitive balance, U = w·u0 = w (1−w), it is clear that the optimal winning percentage can be found from ∂U/∂W = 2w−3w = 0 so that w = 2/3 = 0.67 Also, graphically, it can be seen in Figure 3.3 that the point of tangency between the constraint given by the uo specification and the highest possible indifference curve will result in a winning percentage of their favorite team well above 0.5.

Click to view larger

Figure 3.2 Comparing Competitive Balances with Well-Behaved MR-Curves.

(p. 45)

x y

i i 2

2 *

Club Objectives, Competitive Balance, and the Invariance Proposition

Page 10 of 15

Subscriber: Ohio State University; date: 17 August 2018

Obviously, given this specification of the spectators' utility function, the supporters of both teams will prefer a winning percentage of 0.67 for their favorite team, which is impossible. What we need is a social-welfare function that takes into account the preferences of the supporters of both teams. If

the large-market team has more supporters, the supporters of the large-market team should get a larger weight in the welfare function. The weight α could be set equal to the ratio of the market sizes: α = m /m 〉1 If the social-welfare function is specified as:

(23)

The competitive balance, which maximizes welfare, can be found by solving the first- order condition:

(24)

If the two markets are of the same size (α = 1), it does not come as a surprise that both teams should have the same winning percentage, equal to 0.50. However, if the large market is 50 percent larger than the small market (α = m /m = 1.5), the ratio of the

winning percentages should be w /w = 4/3.5 = 1.14, with and .

The larger the difference in market size, the larger should be the welfare optimal winning percentage of the large-market team. For whatever difference in market size, the optimal winning percentage of the large team stays between 0.5 and 0.67.

Figure 3.3 Optimal Winning Percentage.

(p. 46)

x y

Club Objectives, Competitive Balance, and the Invariance Proposition

Page 11 of 15

Subscriber: Ohio State University; date: 17 August 2018

Which league comes closest to this welfare optimum? We try to find out using a simple numerical example.

Assume that the market size of club x is 15,000 and the market size of club y is 10,000, or α = m /m = 1.5. It is reasonable to assume that a club's marginal revenue approaches zero if its winning percentage approaches 100 percent. So MR = m − 2β m w = 0 if w = 1; that is, if m 2β m = 0 or β = m /2m = 0.5. So, a reasonable β -value for both clubs in this numerical example is β = β = 0.5.

With these values in mind, we can now calculate the competitive balance for the four leagues discussed in section 2, that is, Eqs. (12), (14), (16), and (18), respectively:

(25)

Given the optimum under these conditions, , the mixed league with a profit-

maximizing large-market club and a win-maximizing small-market club comes closest, followed by the league with all profit-maximizing clubs. The two other leagues are too unbalanced to reach the optimum. Based on this competitive equilibrium model, the competitive balance in a win-maximization league is too unbalanced to reach the optimum, so some revenue sharing among clubs would be preferable.

4. The Invariance Proposition The well-known invariance proposition, which goes back to Rottenberg's (1956) seminal article, and which was formally proven by Quirk and El-Hodiri (1974), states that the restrictions on player mobility, imposed by a reservation or a retain-and-transfer system, do not affect the competitive balance if clubs are profit maximizers. The distribution of talent remains unchanged because the marginal revenue of talent is not affected by paid or received transfer fees. With or without a transfer system, a player will end up on the team where his marginal revenue is highest. However, profit- maximizing team owners in a monopsonistic player market that is created by a transfer system will exploit players. Scully (1974, 1986) has observed a high rate of monopsonistic exploitation of baseball players in de United States before the abolition of the reserve clause in the mid-1970s. After the abolition, baseball players were paid according to their marginal revenue (see Scully, 1999). However, in a win-maximization league, things are different. Under a retain-and-transfer system, the small-market club, as a net-seller of talent in the transfer market, will use the received transfer fees to increase its demand for talent, which is now

determined by the net average revenue of talent ( ).

The large-market club will reduce its demand for talent so that a more equal distribution of talent will emerge. This positive effect on the competitive balance will be very limited,

x y (p. 47) i i i i i

i i i i i i i i

x y

Club Objectives, Competitive Balance, and the Invariance Proposition

Page 12 of 15

Subscriber: Ohio State University; date: 17 August 2018

though, because a team in a small market will not be able to attract many talents that can be sold to the large-market team. Also, a win-maximizing club will not exploit its players, who would be rather overpaid, that is, above their marginal revenue, even in a monopsonistic player market (see Kesenne, 2002, 2007).

Later on, the invariance proposition has been extended to gate sharing. Also, gate- revenue sharing among clubs does not affect the competitive balance in a profit- maximization league, because both clubs, large and small, will reduce their demand for talent if the home club has to share its marginal revenue of talent with the visiting team. The only effects of revenue sharing will be lower salary levels (see Quirk and Fort, 1992), and higher profits in the small-budget clubs. The profits of the large-budget clubs can be reduced by sharing if the presharing profits of a club are larger than the average budget in the league (see Kesenne, 2007).

As distinct from a profit maximization league, in a win-maximization league, gate-revenue sharing among teams does improve the competitive balance because the small-market team will increase and the large-market club will reduce its demand for talent. Because the upward shift of the small-market club's demand for talent will also be stronger than the downward shift of large-market club's demand for talent, revenue sharing can have an increasing effect on player salaries (see Kesenne, 2007). Therefore, the invariance proposition does not hold in a league where teams are win maximizers.

In a mixed league where the large-market club is profit maximizer and the small-market club is a win maximizer, gate revenue sharing will slightly improve the distribution of talent, because the large team will reduce its demand for talent (MR) and the small team will increase its demand (AR). In the mixed league where the small-market club is a profit maximizer and the large-market club is a win maximizer, the impact of revenue sharing on the competitive balance is theoretically indeterminate, because both teams will reduce their demand for talent. Consequently, revenue sharing will now reduce the player salary level. The invariance proposition no longer holds if one of the teams in a league is a win maximizer.

Moreover, Szymanski and Kesenne (2004), criticizing the constant-supply Walrasian model, have shown that, in both the flexible supply model and the fixed-supply model under the Nash conjecture, revenue sharing worsens the competitive balance under profit maximization, because the negative external effects from hiring talents are larger for the high-revenue teams, so that they are better off if revenues are shared.

5. Conclusion In this contribution, we have argued that, under some reasonable assumptions, such as well-behaved club-revenue functions, it can shown that the competitive balance in a win maximization league will be more unbalanced than in a profit maximization league. We

(p. 48)

Club Objectives, Competitive Balance, and the Invariance Proposition

Page 13 of 15

Subscriber: Ohio State University; date: 17 August 2018

have also compared the competitive balances in the so-called mixed leagues, in which some clubs are profit maximizers and others are win maximizers. In a first and simple welfare-economic exercise, we have shown that, in a league with profit-maximizing large-market clubs and win-maximizing small-market clubs, the expected competitive balance will be closest to the welfare optimum. Whereas little can be expected from a player reservation system to improve the competitive balance in both a profit- and a win- maximization league, revenue sharing will only affect the distribution of talent in a win maximization league.

References

Andreff, W., and S. Szymanski. 2006. Handbook of sports economics. Cheltenham, UK, and Northampton, MA: Edward Elgar.

Borland, J., and R. MacDonald. 2003. Demand for sport. Oxford Review of Economic Policy, 19(4): 478–503.

Dobson, S., and J. Goddard. 2001. The economics of football. Cambridge: Cambridge University Press.

Downward, P., and A. Dawson. 2000. The economics of professional team sports, London and New York: Routledge.

Fort, R., and J. Quirk. 1995. Cross-subsidization, incentives and outcomes in professional team sports leagues. Journal of Economic Literature XXXIII: 1265–1299.

Fort, R., and J. Quirk. 2004. Owner objectives and competitive balance. Journal of Sports Economics 3(1).

Forrest, D., and R. Simmons. 2002. Outcome uncertainty and attendance demand in sport: The case of English soccer. The Statistician, 51(2): 229–241.

Garcia, J., and P. Rodriguez. 2002. The determinants of football match attendance revisited: Empirical evidence from the Spanish Football League. Journal of Sports Economics, 3(1): 18–38.

Gerrard, B., ed. 2006. The economics of association football. Cheltenham, UK, and Northampton, MA: Edward Elgar.

Jones, J. 1969. The economics of the national hockey league. Canadian Journal of Economics, 2(1): 1–20.

Kesenne, S. 1996. League management in professional team sports with win maximizing clubs. European Journal for Sports Management, 2(2).

Kesenne, S. 2000. Revenue sharing and competitive balance in professional team sports. Journal of Sports Economics, 1(1).

(p. 49)

Club Objectives, Competitive Balance, and the Invariance Proposition

Page 14 of 15

Subscriber: Ohio State University; date: 17 August 2018

Kesenne, S. 2007. Revenue sharing and owner profits. Journal of Sports Economics, 8(5).

Kesenne, S. 2007. The economic theory of professional team sports: An analytical treatment. Northampton, MA: E. Elgar.

Noll, R., ed. 1974. Government and the sport business. Washington, DC: The Brookings Institution.

Quirk, J., and R. Fort. 1992. Pay dirt, the business of professional team sports. Princeton: Princeton University Press.

Rascher, D., and Solmes, John Paul G. 2007. Do fans want close contests? A test of the uncertainty of outcome hypothesis in the National Basketball Association. International Journal of Sport Finance, 1(4).

Sandy, R., P. Sloane, and M. Rosentraub. 2004. The economics of sport: An international perspective. New York: Palgrave Macmillan.

Scully, G. 1974. Pay and performance in Major League Baseball. American Economic Review 64(6).

Scully, G. 1989. The business of Major League Baseball. Chicago: University of Chicago Press.

Scully, G. 1999. “Free agency and the rate of monopsonistic exploitation in baseball.” In Competition policy in professional sports, edited by C. Jeanrenaud and S. Kesenne. Antwerp: Standard Editions Ltd.

Simmons, R. 1996. The demand for English League Football: A club-level analysis. Applied Economics, 28: 139–155.

Szymanski, S. 2003. The economic design of sporting contests. Journal of Economic Literature, 41(4): 1137–1187.

Szymanski, S. 2004. Professional team sports are only a game: The Walrasian fixed-supply conjecture model, contest—Nash equilibrium and the invariance principle. Journal of Sports Economics, 5(2): 111–126.

Szymanski, S., and S. Kesenne. 2004. Competitive balance and gate revenue sharing in team sports. Journal of Industrial Economics, 51(4): 513–525.

Szymanski, S., and S. Leach. 2005. Tilting the playing field: Why a sports league planner choose less, not more, competitive balance? Working paper, Tanaka Business School, Imperial College, London.

Vrooman, J. 1995. A general theory of professional sports leagues. Southern Economic Journal, 61(4): 971–990.

(p. 50)

Club Objectives, Competitive Balance, and the Invariance Proposition

Page 15 of 15

Subscriber: Ohio State University; date: 17 August 2018

Vrooman, J. 2008. Theory of the perfect game: Competitive balance in monopoly sports leagues. Review of Industrial Organization 31: 1–30.

Zimbalist, A. 1992. Baseball and billions. New York: Basic Books.

Stefan Kesenne

Stefan Kesenne, Department of Economics, University of Antwerp.