奖金均分的同时竞赛：计算复杂性与无政府状态的无政府代价
Simultaneous Contests with Equal Sharing Allocation of Prizes: Computational Complexity and Price of Anarchy
摘要：
我们研究了一个一般情况下，同时比赛，分配奖品的基础上平等分享：每个比赛奖项的奖金，以满足一些最高的特定标准的所有玩家，这个奖项的价值，以赢家的数量增加而减少。参与者为一组活动产生输出，比赛的获胜标准基于这些输出。我们考虑模型的两种变化：（i）参与者有生产产出的成本;（ii）参与者没有成本，但有广义的预算约束。我们观察到，这些游戏是确切的潜在的游戏，因此总是有一个纯策略的纳什均衡。对于预算模型，无政府状态的无政府代价是$2$，但对于成本模型，无政府状态的代价可以是无限的。我们的主要结果是这些游戏的计算复杂性。我们证明，对于模型的一般版本，精确或近似计算最佳响应是NP难的。对于自然的限制版本，最佳对策是很容易计算的，我们发现，找到一个纯策略纳什均衡是PLS完全的，找到一个混合策略纳什均衡是（PPAD $\cap$ PLS）完全的。另一方面，可以在伪多项式时间内找到近似的纯策略纳什均衡。这些博弈是显式拥塞博弈的严格但自然的子类，但它们仍然具有相同的均衡硬度结果。
英文摘要：
We study a general scenario of simultaneous contests that allocate prizes based on equal sharing: each contest awards its prize to all players who satisfy some contest-specific criterion, and the value of this prize to a winner decreases as the number of winners increases. The players produce outputs for a set of activities, and the winning criteria of the contests are based on these outputs. We consider two variations of the model: (i) players have costs for producing outputs; (ii) players do not have costs but have generalized budget constraints. We observe that these games are exact potential games and hence always have a pure-strategy Nash equilibrium. The price of anarchy is $2$ for the budget model, but can be unbounded for the cost model. Our main results are for the computational complexity of these games. We prove that for general versions of the model exactly or approximately computing a best response is NP-hard. For natural restricted versions where best response is easy to compute, we show that finding a pure-strategy Nash equilibrium is PLS-complete, and finding a mixed-strategy Nash equilibrium is (PPAD$\cap$PLS)-complete. On the other hand, an approximate pure-strategy Nash equilibrium can be found in pseudo-polynomial time. These games are a strict but natural subclass of explicit congestion games, but they still have the same equilibrium hardness results.
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