Win and let win: the auction for super-star athletes

For many of us, football is no more than an entertainment. For others, it is almost a way of life. No matter where we find ourselves on this continuum, most of us have opinions as to whether our favorite club is well managed, or whether the league is exciting or just boring because the best players are all on one or two teams. Talk shows discuss infinitely whether Paul Pogba’s all-time highest football transfer fee, 105 million euros, is reasonable or not, and whether the revenue sharing between teams is insufficient or excessive.

Roberto Burguet and József Sákovics provide a tractable framework for analyzing these questions in their BSE Working Paper (No. 902), “To the Highest Bidder: The Market for Talent in Sports Leagues.”

A new function for club objectives

In order to perform this analysis, there are several difficulties that need to be overcome. First, it is unclear what football clubs’ or corporations’ objectives are: profit maximization or utility maximization of the club’s members.

The authors’ innovation on this front is to derive a new functional form for the clubs’ objective function that unifies both pecuniary and non-pecuniary benefits. In this framework, clubs trade off utility against money, and so their choices can be described as an interior solution to some optimization problem. This avoids the corner-solution nature of decisions that result from picturing clubs as simple spenders of a budget.

Comparing players based on units of talent

Another difficulty arises from the heterogeneity of players. Different players have different skill sets and styles. Burguet and Sákovics cut through this by assuming that what matters is the aggregate level of talent each club has, not how it is distributed across the players.

Based on these modelling choices, the authors arrive at their first result: all hired players are paid the same wage per unit of talent. This is consistent with what we observed in reality, namely, that a player with more talent usually gets paid more. So in this regard, Lionel Messi is paid higher than Bojan Krkic, because the former is endowed with more talent. Manchester United is willing to buy a single player for 105 million euros, simply because the price per unit of talent times the amount of talent in this single player equals 105 million euros.

Applying the model to the market for players

Next, Burguet and Sákovics posit the way the market for players operates. When competing for a player, to all effects that player auctions himself off to the highest bidder. So, competition for players results in a set of individual auctions for talent.

But clubs also compete on the pitch. Thus, a club’s willingness to pay for a player depends on whether other clubs are interested in him. Real Madrid may be willing to pay more for a player if Atlético is also bidding for him. Given the full description of the market, the authors can derive the equilibrium allocation and wages. They go on to show that the equilibrium allocation of talent across clubs is independent of the initial distribution, provided that the market-clearing wage exceeds the players’ reservation wage and there are no trade frictions.

The authors also prove that – when the talent supply is independent of the wage – the talent distribution is also independent of the presence of revenue sharing, which is the redistribution of the revenues. The only difference is that the prevailing wage in the market drops, which is not surprising since football clubs have less incentive to win when winning is rewarded less. In other words, revenue sharing – under some conditions – is just a way to transfer money from players to clubs.

However, when the supply of talent increases with the wage there is an additional effect of revenue sharing: it reduces the amount of talent hired as wages decrease. Since the market pressure for hiring talent can actually lead to exceed the efficient level of talent that is driven to the sport, the talent reducing effect of revenue sharing may in fact be beneficial.

An interesting conclusion that follows from the analysis is that, although the market select low reservation-wage units of talent, clubs do not tend to hire the players with low talent. The reason is that from the perspective of the model, high talent players are the ones with low per unit reservation wage, which is the reservation wage divided by the talent of the player. As the talent of excellent players is very large relative to the lowest of wages, their per unit reservation wage is very low.