Everybody knows the famous all-pay auction, where the highest bidder wins the prize but everbody pays, or perhaps the highest M bidders all win M prizes. Political lobbying is the classic example. But many real-life situations do not involve bids, but rather involve cost functions for each player. Lobbying firm A may be able to generate some interest among politicians without exerting any effort at all, whereas lobbying firm B may have to expend significant resources to generate any interest.
Siegel models this as an “all-pay” contest. Bidders have complete information; everyone knows everyone else’s valuations and cost structures. There are N bidders and M identical prizes, and each bidder has a value from winning a prize of v(i). Each bidder’s strategy involves selecting a “score” s with a cost c(s) which may be different for different players, and which is not necessarily differentiable. There turns out to be a very nice Nash equilibrium in this game. Let the “reach” of a player be the s which represents the final time that v(i)-c(s) is positive; an assumption on the cost function ensures that bidding infinity is never optimal. Order the players by their reach. Let the reach of the M+1th player be the “threshold”. Let the “power” of a player be v(i)-c(s) when s is precisely the threshold; the M+1th player will have power of zero.
Solving for equilibrium strategies is very difficult, but they almost certainly will be mixed strategies. But we can solve for payoffs in equilibrium. The unique expected payoffs (not necessarily unique equilibrium, as different strategies may give the same expected payoffs) give every player the maximum of 0 and their power. With some (very minor) assumptions about the game, at least N-M players will win an object with zero probability; since they can guarantee no worse than zero payoff by simply not participating in the game (by bidding zero), this means that (at least) N-M players will [correction, 11/23/10: either not participate or participate and earn zero payoff in expectation.]
The proof is actually surprisingly straightforward mathematically: no crazy functional analysis is necessary. The only tricky bit is guaranteeing that an equilibrium exists at all, since as with most auction games, the payoff functions are discontinuous in the strategies. Siegel uses a nice application of a old result from Simon and Zane to guarantee existence. Essentially, Simon and Zane note that in many auctions, the only discontinuities we need to worry about occur when players bid the same amount (a “tie”). They then show that there exists some tie-breaking rule which, when this rule is added the game, equilibrium is guaranteed. Siegel uses this result, then shows ties occur with probability zero in the all-pay contest, meaning that for a measure-one set of games, the exact tie-breaking rule won’t matter, but it allows us to know an equilibrium exists.
The characterization of all-pay contest equilibrium is really useful in practice. For one, it shows that, no matter what the cost function of the bidders who lose, expected payoffs do not change. That is, a first-price auction, and a first-price auction where losers pay some fraction of their bids, will give precisely the same expected payoffs for all players; this is a weak form of revenue equivalence. Also, recruiting a “weak” additional bidder, meaning a bidder with reach lower than the threshold, will not increase seller revenue since that bidder will bid zero in equilibrium. This literature is still fairly wide-open: there is no Myerson-like result on “optimal contest design,” for instance. I was thinking a bit today about an extension where we let v(i) be a function of all player strategies; consider two fighter jet manufacturers who are lobbying a government to make a purchase. The advertising of the losing bidder may increase the value of winning for the winning bidder if it causes the government to make a larger purchase of jets. These types of “advertising contests” are pretty widespread, but as far as I know, there is no characterization of optimal design or of equilibrium in the literature.
One extension is finished, however: Ron mentioned in a seminar the other day that he has a similar result for games where costs and values are non-monotonic. An example is writing an economics paper: the initial effort is pleasurable, but the revising is costly. A minor modification of the 2009 paper is enough to guarantee existence of an equilibrium and to characterize payoffs.
http://faculty.wcas.northwestern.edu/~rsi665/all-pay-contests.pdf (Final Econometrica version)