Optimal allocations of prizes and punishments in Tullock contests
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Optimal allocations of prizes and punishments in Tullock contests Aner Sela1 Accepted: 12 July 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract We study Tullock contests with n symmetric players. We show that in a contest without an exit option, if prizes and punishments (negative prizes) have the same cost, it is optimal for the designer who wants to maximize the players’ total effort to allocate the entire prize sum to a single punishment without any prize. On the other hand, in a contest with an exit option, it is optimal to allocate the entire prize sum to a single prize and a single punishment, where independent of the costs of the prize and the punishment, the optimal value of the prize is larger than the optimal value of the punishment. We also show that allocating a prize and a punishment in a twostage contest yields a higher expected total effort than in a one-stage contest. Keywords Tullock contests · Prizes · Punishments JEL Classification D44 · J31 · D72 · D82
1 Introduction During the last several decades, the contest literature has focused on the optimal prize structure. This issue has especially been studied for the Tullock contest (see Tullock 1980) which is probably the most commonly studied contest.1 For example, Berry (1993) assumed that in Tullock contests the players exert their efforts once and that the probability of player i to win one of k identical prizes is equal to the
1 Several studies have provided axiomatic justification for this contest form (see, for example, Skaperdas 1996). In addition, Baye and Hoppe (2003) have identified conditions under which a variety of rent-seeking contests, innovation tournaments, and patent-race games are strategically equivalent to this contest.
* Aner Sela [email protected] 1
Department of Economics, Ben Gurion University of the Negev, 84105 Beer Sheva, Israel
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total effort in all possible outcomes of k winners which include this player’s effort, divided by the total efforts in all possible outcomes of k winners. He showed that in such a symmetric Tullock contest, the contestants’ total effort is maximized only when one prize is awarded. For the Tullock contest with multiple prizes, Clark and Riis (1996, 1998) suggested another contest success function where the contestants exert their efforts once in the first stage and then the prizes are awarded sequentially. These authors found that in such a symmetric Tullock contest with multiple prizes and linear cost functions, the contestants’ total effort is maximized only when one prize is awarded. Schweinzer and Segev (2012) demonstrated that in the symmetric Tullock with non-linear cost functions, the contestants’ total effort is maximized also when only one prize is allocated.2 While most of the contest literature has concentrated on the incentive role of prizes, punishments (negative prizes), which are also part of many existing incentive contracts, have been ignored. This is because a contest that has some punishments can be replicat
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