#### Explainer: The Tripartite Auction Theorem

by David K. Levine, Andrea Mattozzi and Salvatore Modica on April 23, 2017

In a political contest such as voting or lobbying groups offer costly effort - votes, money - and the side that offers the most wins. In voting effort is provided regardless of the outcome. With bribes only the winner pays. The tripartite auction theorem says that the groups do not care whether all pay or only the winner.

We analyze a political contest over a prize between two groups with costly effort provision. The group offering the greatest effort wins a prize. The tie breaking rule is endogenous: this means that a group that is willing to bid a little bit more may be assumed to win in case of a tie.

Such a contest is called an auction. We examine three different types of auctions in which the high bidder pays their own bid - these are called first price auctions. In the first two auctions only the winner pays: in the ascending bid auction each group sets a reserve price above which it drops out of the bidding; in the sealed bid auction both groups simultaneously submit bids. The third auction is the all-pay auction in which both groups simultaneously submit bids and each pays their own bid regardless of whether they win or lose.

Our model of behavior is that of Nash equilibrium. Nash equilibrium requires that each group makes the best decision for itself given its beliefs and that its beliefs about the other group's strategy is correct. We also require that neither group use a strategy that is weakly dominated. A strategy is weakly dominated if there is alternative strategy which never does worse and sometimes does better. Roughly speaking: only a fool would use a weakly dominated strategy.

A key concept in auction theory is the willingness to bid. This is the most that a group will bid for a certainty of getting the prize instead of a certainty of losing. We analyze only the case in which the willingness to bid of the two groups is not identical. We refer to the group with the higher willingness to bid as advantaged, to the one with the lower willingness to bid as disadvantaged. The surplus is the difference between the value of the prize to the advantaged group and the cost to the advantaged group of matching the willingness to bid of the disadvantaged group.

Tripartite auction theorem: For all three auctions the disadvantaged group gets zero and the advantaged group gets the surplus.

It is not difficult to explain why this is the case. Start with the fairly obvious fact that no group bids more than its willingness to bid. No matter what the other group does bidding your willingness to bid is at least as good as bidding more - and if the other group bids zero you would be better off bidding your willingness to bid rather than higher.  In technical terms bidding your willingness to bid weakly dominates any strategy of bidding higher.

Since the disadvantaged group will not bid more than its willingness to bid, the advantaged group does not get less than the surplus. It would be foolish to do so since it can do better by bidding just a bit more than the willingness to bid of the disadvantaged group.

By contrast the disadvantaged group gets zero. To understand why this is, think about the lowest bid by either group. Obviously such a bid is not terribly likely to win. In fact: one of the groups must lose for sure when it makes this bid - if  both groups had a chance of winning at this lowest bid the advantaged group should raise its bid a tiny bid to raise its chance of winning a lot. A group that loses for sure cannot earn more than zero. We already know the advantaged group earns at least the surplus so it must be the disadvantaged group that loses for sure and earns no more than zero. Since either group can guarantee zero by bidding zero the disadvantaged group must get exactly zero.

Because the disadvantaged group gets zero the highest bid by the advantaged group must be equal to the willingness to bid of the disadvantaged group. Otherwise the disadvantaged group could earn more than zero by slightly beating that highest bid.

When the advantaged group bids the willingness to bid of the disadvantaged group it cannot get more than the surplus. As we already know it cannot get less than the surplus we conclude that it gets at exactly this much.