In all pay auctions each player submits a bid (effort) for the object being sold, the player who submits the highest bid receives the object, but, independently of success, all players bear the cost of their bids. Common applications of all pay auctions include rent seeking, lobbying in organizations, R&D races, political contests, promotions in labor markets, and sport competitions. In the economic literature, all pay auctions are usually studied under complete information where the playersfvaluations for the object are common knowledge (see, for example, Hillman and Samet (1987), Hillman and Riley (1989), Baye et al. (1993) and Che and Gale (1998)), or under incomplete information where each playerfs valuation for the object is private information to that player and only the distribution of the players valuations is common knowledge (see, for example, Hillman and Riley (1989), Amman and Leininger (1996), Gavious et al. (2003) and Moldovanu and Sela (2006)).
Most of this literature has focused on all pay auctions with a unique prize that is awarded to the player with the highest effort. In the real world, however, we can find numerous contests with several prizes. For example, students compete for grades in exams (at least in U.S, the grades are Afs, Bfs, Cfs, Dfs and Ffs). Players in sport competitions may compete for a unique prize or they may compete for several prizes, i.e., gold, silver or bronze medals awarded in the Olympic games. In political races the winner may hold a position with several titles, or several winners may hold these titles separately. Large corporations (such as large banks) have, besides a single president, several executive vice presidents, tens of senior vice presidents , and several hundred "mere" vice presidents.
In the literature on all pay auctions only a few studies deal with the question of what is the optimal number of prizes in contests and particularly in all pay auctions. Moldovanu and Sela (2001) showed that in all pay auctions under incomplete information when cost functions are linear or concave in effort, it is optimal to allocate the entire prize sum to a single first prize, but when cost functions are convex, several positive prizes may be optimal. This explanation, however, cannot be generalized to the case of all pay auctions under complete information.
In symmetric all pay auctions under complete information, Barut and Kovenock (1998) showed that the revenue maximizing prize structure allows any combination of k-1 prizes, where k is the number of players. That is, the contest designer is indifferent to whether he should allocate one prize or several prizes. In this paper we show that in asymmetric all pay auctions under complete information allocation of several prizes might be profitable for the contest designer who maximizes the total effort. Consequently, this paper offers a rationale for multi prize contests in a single, integrated model.
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Allocation of Prizes in Asymmetric All Pay Auctions
