_{1}

Szymanski [1] and Szymanski and Késenne [2] showed that, in the standard model of a sports league, gate revenue sharing will tend to increase competitive imbalance between weak and strong teams, a seemingly perverse result. Dobson and Goddard [3] claim that “this analysis is flawed. If the revenue function is specified appropriately, gate revenue sharing always reduces competitive inequality.” This comment points out the analytical error in their paper which leads to their erroneous conclusion. Once their error is corrected, it is shown that the earlier results stand.

Competitive balance is considered an important issue in sports league. Many people believe that in the absence of a sufficient degree of competitive balance among teams the outcome of league competition will become too predictable; fans will lose interest; and the league will collapse. Redistributing revenues from strong teams to weak teams is widely advocated for overcoming this problem. However, the effectiveness of these schemes depends crucially on the revenue sharing mechanism. In a series of papers Szymanski [

Dobson and Goddard [

The standard model in the sports literature assumes that there are two profit maximizing teams that each chooses a quantity of talent which can be hired at a constant marginal cost per unit of talent. One team, the large market team, is assumed to have a larger revenue generating capacity for any given win percentage. DG (p412) define the revenue functions for teams 1 and 2 as _{ }is the total amount of playing talent in the league, w is win percentage,

A condition for equilibrium is that the marginal revenue of talent of team 1 and team 2 are equalized. This condition can be defined as

α is the degree of revenue sharing;

In DG the derivatives are stated as follows:

where

The second term in each of these equations is incorrect. The derivative we are looking for in the second term of

Likewise, the derivative we are looking for in the second term of

Correcting these errors, Equations (2) and (3) are restated as follows

These errors have important implication for the derivation of the equilibrium condition. After some manipulation it can be shown that DG Equation (3) should read

By inspection it should be clear that this formulation (3’) encompasses DG Equation (1), which emerges as a special case where absolute talent has no impact.

Once the correct specification (3’) is applied, the claims of the authors are no longer tenable. First, it is impossible to derive any general conclusions about the equilibrium values of

I have, however, attempted to find simulation solutions choosing particular values of

Thus the claim of the authors to have shown that “Competitive inequality is lower with equal revenue sharing

Intuitively the problem with the specification advanced by the authors is that both teams stand to gain by increasing total quality, but since the revenue of team 1 is larger than the revenue team 2 for any given value of own win percentage, team 1 typically has the greater incentive to invest in talent, and this turns out to be true even under revenue sharing.

Finally, it is worth commenting briefly on the authors assertion that the “If revenues depend upon relative team quality only, however, the equal revenue sharing solution

This comment addresses an error in the mathematical derivation of Dobson and Goddard [

Szymanski, S. (2016) Revenue Sharing in a Sports League with an Open Market in Playing Talent: A Comment. Theoretical Economics Letters, 6, 1337-1340. http://dx.doi.org/10.4236/tel.2016.66123