Jersey-number assignment equilibrium: When one buyer pays another, and the seller gets nothing

Background

In the National Football 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. So long as everyone holds up his end of the bargain, this usually goes off without a hitch.


Problem

But what to do if a team signs two 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 the number is awarded to the player with the higher bid, who then pays the other player 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.


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 theoretically the value each places 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 come out ahead.

Let's relax the assumption that each player bids the value he places on wearing 19. If 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.


Conclusion

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 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.



UPDATE: Professor Walt Wessels suggests that the best solution uses the same bidding technique, but has the higher bidder get the number and pay the lower bidder the amount of the lower bid.