The All Pay. Common Value Auction as a Model of Contests

An all pay, common value auction is proposed as a model of contests. The common value of the prize is ex-ante unknown, but each contestant has private information about its true value. These private information are affiliated. Unlike in
the symmetric winner pay auction model of Milgrom and Weber (1982). The affiliation assumption is not sufficient to ensure the existence of an increasing equilibrium
bid. The all pay auction is shown to require a stronger condition; a joint restriction on the expected valuation of the prize and the beliefs bidders have of each other’s
estimates of the prize. Equilibrium bids are increasing if there is not too strong a degree of affiliation and the expected value of the prize is increasing sufficiently
rapidly in each bidder’s type. When there exists an increasing equilibrium, the equilibrium is unique even in the asymmetric bidder case. The results extend to the all pay private values auction which is obtained as a special case. The all pay and winner pay auctions are then compared at the symmetric equilibrium. With increasing
equilibrium bids, it is possible to order the expected revenues from the two auctions. This was first proposed by Amann and Leininger (1994). It is shown by way of an example that although sufficient, monotonicity of equilibrium bids is not a necessary condition for the ordering. Without monotonicity, however comparing the two auctions can lead to conclusions which are not robust. Two examples are provided to compare the role of affiliation and the effects of nonmonotonicity in the
all pay and winner pay auctions.