This paper describes a market model in which property rights for the desired good are not assignable ex ante, using the example of professional athletes and the assignment of jersey numbers when two players have an equal inherent right to the number.
Background
In the major sports leagues (the National Football League, National Basketball Association, Major League Baseball, and National Hockey League), when a high-profile player is signed to a contract by a new team, he oftentimes wants to keep using the jersey number he wore with his previous team. If the player wearing that number on the new team is suitably low-profile, a deal is sometimes struck, in which the new player buys the right to that number from the existing player for an amount that is relatively insignificant to the high-profile player, but may represent a nontrivial 'bonus' to the marginal player.(1) So long as both players honor the agreement, this process is relatively routine and straightforward. (2)
Problem
But what to do if a team signs two high-profile players who each desire the same number? Since the team is prohibited from selling the number to the highest bidder, how can we resolve this dilemma?
Solution
The utility-maximizing approach would be to have each player submit a blind bid for the number, and then award the number to the player with the higher bid, who then pays the player with the lower bid the mean of the two bids.
The player who gets the number gets it for less than he was willing to pay, and the player who doesn't get the number gets more than he was willing to pay. Also, the players are disincentivized to either over- or under-bid, as over-bidders are wasting money and under-bidders are leaving money on the table. This solution ensures that the surpluses of each player are equal, assuming that both players bid their true value.
A potential alternative solution would be for the player with the higher bid to pay either the amount of his bid to the player with the lower bid, or to pay the amount of the lower bid. Both of these approaches are suboptimal, however, as one player is left with no surplus in each scenario.
Examples
Smith and Jones are both signed by the same team and both very much want to wear number 19, which is currently unassigned. Smith submits a blind bid for $10,000 and Jones submits a blind bid for $12,000. These amounts are taken to be the value placed by each on wearing number 19. In this example, Jones gets number 19 and pays Smith $11,000. Smith, who was willing to pay $10,000 for the right to wear 19 gets $11,000 instead. Jones, who was willing to pay $12,000, gets the number for $11,000. Both players have an equal, positive surplus of $1,000.
If we relax the assumption that each player bids the value he places on wearing 19, then we can explore rational and irrational over- and under-bidding. If, for example, Smith underbids by offering only $7,000, then he gets only $9,500, which is less than the true value he placed on number 19. If, on the other hand, Jones overbids by offering $15,000, then he has to pay $12,500, which is greater than the true value he placed on number 19.
If, on the other hand, Smith bids his true value ($10,000), but Jones bids an amount less than, but near to, his true value, say $11,000, then Jones will get the number for $10,500. This will give Jones a surplus of $1,500, but Smith only $500. A similar result can be obtained if the low bidder over-bids. The potential for such rational over- or under-bidding increases in proportion to the difference between the true value placed on the number by each player.
Conclusions
The example shows that, while it is possible that players could strategically over- or under-bid, doing so would run them the risk of dissatisfaction with the outcome. The key insight is that the player with the higher true value can only gain by underbidding, and the player with the lower true value can only gain by overbidding. A player has no way of knowing, however, whether his true value is higher or lower than that of the other player, so his expected value of over-bidding cannot be higher or lower than his expected value of under-bidding.
The only way that both players fail to come out ahead is if one or both fail to bid their true value, unless, of course, they both bid the same amount, a situation for which I don't yet have a satisfactory equilibrium, and is an avenue for further research.
This model is applicable to any similar scenario with exactly two buyers in which property rights to the desired good are not assignable ex ante.
1: It is customary, but not mandatory, for the recipient to donate the money received to his preferred charity. This paper will overlook this fact, on the assumption that utility is increasing in both the rights to the desired number and the money received, regardless of the intended use of the funds.
2: A summary of examples of this practice, including the notable brouhaha involving then-Washington Redskins running back Clinton Portis, can be found here: http://www.thestar.com/sports/football/2011/05/10/jersey_numbers_never_as_easy_as_123.html
See specifically Rickey Henderson (MLB), Jeff Feagles (NFL), and Portis.
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