We study contests where the designer's objective is an extension of the widely studied objective of maximizing the total output: The designer gets zero marginal utility from a player's output if the output of the player is very low or very high. We model this using two objective functions: binary threshold, where a player's contribution to the designer's utility is 1 if her output is above a certain threshold, and 0 otherwise; and linear threshold, where a player's contribution is linear if her output is between a lower and an upper threshold, and becomes constant below the lower and above the upper threshold. For both of these objectives, we study (1) rank-order allocation contests that use only the ranking of the players to assign prizes and (2) general contests that may use the numerical values of the players' outputs to assign prizes. We characterize the optimal contests that maximize the designer's objective and indicate techniques to efficiently compute them. We also prove that for the linear threshold objective, a contest that distributes the prize equally among a fixed number of top-ranked players offers a factor-2 approximation to the optimal rank-order allocation contest.

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具有阈值目标的竞赛设计

我们研究了竞赛，其中设计师的目标是广泛研究的最大化总输出目标的延伸：如果玩家的输出非常低或非常高，则设计师从玩家的输出中获得零边际效用。我们使用两个目标函数对此进行建模：二元阈值，如果玩家的输出高于某个阈值，则其对设计者效用的贡献为 1，否则为 0；和线性阈值，如果玩家的输出在下限和上限阈值之间，则她的贡献是线性的，并且在低于下限和高于上限阈值时保持不变。对于这两个目标，我们研究了 (1) 仅使用玩家排名来分配奖品的排名顺序分配竞赛和 (2) 可能使用玩家的数值的一般竞赛 输出分配奖品。我们描述了最大化设计师目标的最佳竞赛，并指出了有效计算它们的技术。我们还证明，对于线性阈值目标，在固定数量的排名靠前的玩家之间平均分配奖品的竞赛提供了最佳排名顺序分配竞赛的因子 2 近似值。