We consider normal-form games with $n$ players and two strategies for each player, where the payoffs are i.i.d. random variables with some distribution $F$ and we consider issues related to the pure equilibria in the game as the number of players diverges. It is well-known that, if the distribution $F$ has no atoms, the random number of pure equilibria is asymptotically Poisson$(1)$. In the presence of atoms, it diverges. For each strategy profile, we consider the (random) average payoff of the players, called Average Social Utility (ASU). In particular, we examine the asymptotic behavior of the optimum ASU and the one associated to the best and worst pure Nash equilibria and we show that, although these quantities are random, they converge, as $n\to\infty$ to some deterministic quantities.
翻译:我们用美元玩家和每个玩家的两种策略来考虑正常的游戏形式,这种游戏的回报是:d.随机变量,有些分布是F$,我们把与游戏中纯平衡有关的问题视为玩家数目的不同。众所周知,如果分配的F$没有原子,那么纯平衡的随机数量是暂时的Poisson(1)美元。在原子存在的情况下,它存在差异。对于每个战略配置,我们考虑玩家的平均回报(随机),称为“平均社会效用 ” ( ASU ) 。特别是,我们审视了最佳ASU的无约束行为,以及与最佳和最差的纯纳什平衡相关的行为,我们表明,尽管数量是随机的,但它们与某些确定性数量相趋近,即为$n\to\infty$。
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