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Inter-league Competition and the Optimal Broadcasting Revenue-Sharing Rule

  • Yvon Rocaboy EMAIL logo
Published/Copyright: February 2, 2023
The B.E. Journal of Theoretical Economics
From the journal The B.E. Journal of Theoretical Economics Volume 23 Issue 2

Abstract

We propose a model where two sports leagues compete for sporting talent, and at the same time consider the competitive balance in their domestic championships. The allocation of broadcasting revenues by the league-governing body acts as an incentive for teams to invest in talent. We derive a strategic league authority’s optimal sharing rule of broadcasting revenues across teams in the league. While a weighted form of performance-based sharing is the best way of attracting talent, cross-subsidization from high- to low-payroll teams is required to improve competitive balance. The optimal sharing rule is then a combination of these two “sub-rules”. We show that the distribution of broadcasting revenues in two first divisions in European men’s football, the English Premier League (EPL) and the French Ligue 1 (L1), corresponds to the optimal sharing rule we discuss. We propose a new method to assess empirically the cross-subsidization impact of the sharing formula. As the impact of cross-subsidization is greater in the EPL than L1, we conclude that ensuring domestic competitive balance seems to be a more important target for the EPL than for L1.

Keywords: revenue sharing; sports economics; distribution of broadcasting revenues; national football leagues; inter-league competition; competitive balance; sport regulation
Corresponding author: Yvon Rocaboy, Univ Rennes, CNRS, CREM-UMR 6211, Condorcet Center for Political Economy, F-35000 Rennes, France, E-mail:

Acknowledgment

I am grateful to Jean-Pascal Gayant, Guillaume L’Oeillet and Fabien Moizeau for helpful comments on an earlier version of this paper. I also greatly benefitted from useful comments and suggestions from two anonymous referees and from the editors of this journal. Of course all remaining errors are mine.

Appendix A: The Impact of a Change in α on the Allocation of Talent Across and within Leagues and on w

From Eq. (6) of the within-league equilibrium allocation of talent in A, we have:

(33a) 1 + M R i A ′ ∂ t i A ∂ α i A = M R j A ′ ∂ t A ∂ α i A − ∂ t i A ∂ α i A
(33b) M R i A ′ ∂ t i A ∂ α j A = 1 + M R j A ′ ∂ t A ∂ α j A − ∂ t i A ∂ α j A

From the inverse demand function for talent of team i from league A (Eq. (14) for K = A and l = i), we obtain:

(34a) ∂ w ∂ α i A = 1 + M R i A ′ ∂ t i A ∂ α i A
(34b) ∂ w ∂ α j A = M R i A ′ ∂ t i A ∂ α j A

From the equation of the within-league equilibrium allocation of talent in B, we have:

(35) M R i B ′ ∂ t i B ∂ α l A = M R j B ′ ∂ t B ∂ α l A − ∂ t i B ∂ α l A for l = i , j

From the inverse demand function for talent of team i from league B, we obtain:

(36) ∂ w ∂ α l A = M R i B ′ ∂ t i B ∂ α l A for l = i , j

From Eq. (15) we obtain:

(37) ∂ t K ∂ α l K = ∂ t i K ∂ α l K + ∂ t j K ∂ α l K for K = A , B and l = i , j

Last, from Eq. (16):

(38) ∂ t A ∂ α l A + ∂ t B ∂ α l A = 0 for l = i , j

We have a system of 14 equations with 14 unknown variables. The solution to this system is as follows:[14]

(39) ∂ t i A ∂ α i A = − M R j A ′ M R i B ′ + M R j B ′ + M R i B ′ M R j B ′ M R i A ′ M R j A ′ M R i B ′ + M R j B ′ + M R i B ′ M R j B ′ M R i A ′ + M R j A ′ > 0
(40) ∂ t i A ∂ α j A = M R i B ′ M R j B ′ M R i A ′ M R j A ′ M R i B ′ + M R j B ′ + M R i B ′ M R j B ′ M R i A ′ + M R j A ′ < 0
(41) ∂ t j A ∂ α i A = M R i B ′ M R j B ′ M R i A ′ M R j A ′ M R i B ′ + M R j B ′ + M R i B ′ M R j B ′ M R i A ′ + M R j A ′ < 0
(42) ∂ t j A ∂ α j A = − M R i A ′ M R i B ′ + M R j B ′ + M R i B ′ M R j B ′ M R i A ′ M R j A ′ M R i B ′ + M R j B ′ + M R i B ′ M R j B ′ M R i A ′ + M R j A ′ > 0
(43) ∂ t A ∂ α i A = − M R j A ′ M R i B ′ + M R j B ′ M R i A ′ M R j A ′ M R i B ′ + M R j B ′ + M R i B ′ M R j B ′ M R i A ′ + M R j A ′ > 0
(44) ∂ t A ∂ α j A = − M R i A ′ M R i B ′ + M R j B ′ M R i A ′ M R j A ′ M R i B ′ + M R j B ′ + M R i B ′ M R j B ′ M R i A ′ + M R j A ′ > 0
(45) ∂ t i B ∂ α i A = M R j A ′ M R j B ′ M R i A ′ M R j A ′ M R i B ′ + M R j B ′ + M R i B ′ M R j B ′ M R i A ′ + M R j A ′ < 0
(46) ∂ t i B ∂ α j A = M R i A ′ M R j B ′ M R i A ′ M R j A ′ M R i B ′ + M R j B ′ + M R i B ′ M R j B ′ M R i A ′ + M R j A ′ < 0
(47) ∂ t j B ∂ α i A = M R j A ′ M R i B ′ M R i A ′ M R j A ′ M R i B ′ + M R j B ′ + M R i B ′ M R j B ′ M R i A ′ + M R j A ′ < 0
(48) ∂ t j B ∂ α j A = M R i A ′ M R i B ′ M R i A ′ M R j A ′ M R i B ′ + M R j B ′ + M R i B ′ M R j B ′ M R i A ′ + M R j A ′ < 0
(49) ∂ t B ∂ α i A = M R j A ′ M R i B ′ + M R j B ′ M R i A ′ M R j A ′ M R i B ′ + M R j B ′ + M R i B ′ M R j B ′ M R i A ′ + M R j A ′ < 0
(50) ∂ t B ∂ α j A = M R i A ′ M R i B ′ + M R j B ′ M R i A ′ M R j A ′ M R i B ′ + M R j B ′ + M R i B ′ M R j B ′ M R i A ′ + M R j A ′ < 0
(51) ∂ w ∂ α i A = M R j A ′ M R i B ′ M R j B ′ M R i A ′ M R j A ′ M R i B ′ + M R j B ′ + M R i B ′ M R j B ′ M R i A ′ + M R j A ′ > 0
(52) ∂ w ∂ α j A = M R i A ′ M R i B ′ M R j B ′ M R i A ′ M R j A ′ M R i B ′ + M R j B ′ + M R i B ′ M R j B ′ M R i A ′ + M R j A ′ > 0

As the marginal willingness to pay for talent is decreasing M R l K ′ < 0 , the sign of Eqs. (39)–(52) is unambiguous.

Appendix B: Maximization of Talent Supply and Maximization of the Wage Rate

Formally, league K’s talent-supply maximization problem may be written as:

Maximize α i K , α j K t K ( α ) subject to s K = α i K × t i K ( α ) + α j K × t j K ( α ) .

At the optimum we have:

(53) ∂ t K / ∂ α i K ∂ t K / ∂ α j K = t i K + α i K ∂ t i K ∂ α i K + α j K ∂ t j K ∂ α i K t j K + α i K ∂ t i K ∂ α j K + α j K ∂ t j K ∂ α j K

In the same vein, the first-order condition for the wage-rate maximization problem is as follows:

(54) ∂ w / ∂ α i K ∂ w / ∂ α j K = t i K + α i K ∂ t i K ∂ α i K + α j K ∂ t j K ∂ α i K t j K + α i K ∂ t i K ∂ α j K + α j K ∂ t j K ∂ α j K

It can easily be checked from Eqs. (43), (44), (51) and (52) in the Appendix that ∂ t K / ∂ α i K ∂ t K / ∂ α j K = ∂ w / ∂ α i K ∂ w / ∂ α j K = M R j K ′ M R i K ′ . The optimal conditions in both maximization problems are identical, showing that maximizing t K or w leads to the same distribution of broadcasting revenue: in order to attract as much talent as possible, a league governing body has to allocate the broadcasting revenue to teams such that the wage rate for talent is maximized.

Lemma 1

Maximizing the league’s supply of talent or maximizing the wage rate of sporting talent leads to the same distribution of broadcasting revenue across teams.

Appendix C: Cross-Subsidization and Gini Coefficients

When the n values of a variable x are placed in ascending order such that each x i has rank i, the Gini coefficient can be calculated as:

(55) G ( x ) = 2 n 2 x ̄ ∑ i = 1 n i × ( x i − x ̄ )

The Gini coefficient of the estimated distribution of broadcasting revenues can be written as:

(56) G ( s ̂ K ) = 2 c K 2 s ̄ K ∑ i = 1 c K i × s ̂ i K − s ̄ K = 2 c K 2 s ̄ K ∑ i = 1 c K i × m i K m K × s K + μ ̂ K m ̄ K − m i K − s ̄ K ,

which gives:

(57) G ( s ̂ K ) = 2 c K 2 s ̄ K μ ̂ K − s ̄ K m ̄ K ∑ i = 1 c K i × m ̄ K − m i K .

And the Gini coefficient of the payroll distribution is:

(58) G ( m K ) = 2 c K 2 m ̄ K ∑ i = 1 c K i × m i K − m ̄ K .

We then have:

(59) G ( s ̂ K ) G ( m K ) = − m ̄ K s ̄ K μ ̂ K − s ̄ K m ̄ K = 1 − μ ̂ K × m ̄ K s ̄ K

Given that R K = μ ̂ K × m ̄ K s ̄ K , we therefore have R K = 1 − G ( s ̂ K ) G ( m K ) , QED.

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Received: 2022-04-02
Accepted: 2023-01-13
Published Online: 2023-02-02

© 2023 Walter de Gruyter GmbH, Berlin/Boston

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  2. Research Articles
  3. Collusion, Shading, and Optimal Organization Design
  4. Software Cracking and Degrees of Software Protection
  5. The R&D Investment Decision Game with Product Differentiation
  6. An Urban Configuration with Online Competition
  7. Double Implementation in Dominant Strategy Equilibria and Ex-Post Equilibria with Private Values
  8. Risk Aversion and Uniqueness of Equilibrium in Economies with Two Goods and Arbitrary Endowments
  9. On Iterated Nash Bargaining Solutions
  10. Inter-league Competition and the Optimal Broadcasting Revenue-Sharing Rule
  11. The Weak Hybrid Equilibria of an Exchange Economy with a Continuum of Agents and Externalities
  12. Choosing Sides in a Two-Sided Matching Market
  13. Notes
  14. On the Relation between Private Information and Non-Fundamental Volatility
  15. Cost-Reducing Technologies and Labor Supply in a Krugman-type Model where Consumption is Time-Constrained: Some New Results
https://doi.org/10.1515/bejte-2022-0042
Keywords for this article
revenue sharing; sports economics; distribution of broadcasting revenues; national football leagues; inter-league competition; competitive balance; sport regulation

Articles in the same Issue

  1. Frontmatter
  2. Research Articles
  3. Collusion, Shading, and Optimal Organization Design
  4. Software Cracking and Degrees of Software Protection
  5. The R&D Investment Decision Game with Product Differentiation
  6. An Urban Configuration with Online Competition
  7. Double Implementation in Dominant Strategy Equilibria and Ex-Post Equilibria with Private Values
  8. Risk Aversion and Uniqueness of Equilibrium in Economies with Two Goods and Arbitrary Endowments
  9. On Iterated Nash Bargaining Solutions
  10. Inter-league Competition and the Optimal Broadcasting Revenue-Sharing Rule
  11. The Weak Hybrid Equilibria of an Exchange Economy with a Continuum of Agents and Externalities
  12. Choosing Sides in a Two-Sided Matching Market
  13. Notes
  14. On the Relation between Private Information and Non-Fundamental Volatility
  15. Cost-Reducing Technologies and Labor Supply in a Krugman-type Model where Consumption is Time-Constrained: Some New Results
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