We introduce set packing games as an abstraction of situations in which n selfish players select disjoint subsets of a finite set of indivisible items, and analyze the quality of several equilibria for this basic class of games. Special attention is given to a subclass of set packing games, namely throughput scheduling games, where the items represent jobs, and the subsets that a player can select are those jobs that this player can schedule feasibly. We show that the quality of three types of equilibrium solutions is only moderately suboptimal. Specifically, the paper gives tight bounds on the price of anarchy for Nash equilibria, subgame perfect equilibria of games with sequential play, and k -collusion Nash equilibria. Under the assumption that players are allowed to play suboptimally and achieve an $$\alpha $$ α -approximate equilibrium, our tight price of anarchy bounds are $$\alpha +1$$ α + 1 for Nash and subgame perfect equilibria, but less than $$\alpha +1/(e-1)$$ α + 1 / ( e - 1 ) for subgame perfect equilibria when games are symmetric. For k -collusion Nash equilibria, the price of anarchy equals $$\alpha +(n-k)/(n-1)$$ α + ( n - k ) / ( n - 1 ) , where $$1\le k\le n$$ 1 ≤ k ≤ n .

中文翻译：

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集合包装和吞吐量调度博弈的均衡质量

我们将集合打包博弈作为对 n 个自私玩家选择一组有限不可分项目的不相交子集的情况的抽象，并分析此类基本博弈的几个均衡的质量。特别注意集合包装游戏的一个子类，即吞吐量调度游戏，其中项目代表作业，而玩家可以选择的子集是该玩家可以可行调度的作业。我们表明三种类型的平衡解决方案的质量只是中等次优。具体而言，本文给出了纳什均衡、具有顺序博弈的子博弈完美均衡和 k-共谋纳什均衡的无政府状态价格的严格界限。在允许玩家进行次优游戏并达到 $$\alpha $$ α 近似均衡的假设下，对于纳什和子博弈完美均衡，我们对无政府状态边界的紧价格是 $$\alpha +1$$ α + 1，但小于 $$\alpha +1/(e-1)$$ α + 1 / ( e - 1 ) 用于当博弈对称时的子博弈完美均衡。对于 k -共谋纳什均衡，无政府状态的价格等于 $$\alpha +(nk)/(n-1)$$ α + ( n - k ) / ( n - 1 ) ，其中 $$1\le k\le n$$ 1 ≤ k ≤ n 。