Title: Long Range Games Show Chinese title

Title: 远距离博弈

Abstract: We consider $N$-player games, in continuous time, finite state space and finite time horizon, on a geometrical structure possessing a macroscopic limit in a suitable sense. This geometrical structure breaks the permutation invariance property that gives rise to mean field games. The corresponding limit game is a variant of mean field games that we call {\em long range game}. We prove that this asymptotic scheme satisfies the following key properties: a) the long range game admits al least one equilibrium; b) this equilibrium is unique under a suitable monotonicity condition; c) the feedback corresponding to any equilibrium of the long range game is a quasi-Nash equilibrium for the $N$-player games. We finally show that this scheme includes several examples of interaction mechanisms, in particular Kac-type interactions and interactions on generalized Erd\"{o}s-Renyi graphs. Show Chinese abstract

Abstract: 我们考虑连续时间、有限状态空间和有限时间范围内的$N$人博弈，在一个在适当意义上具有宏观极限的几何结构上。 这个几何结构破坏了产生平均场博弈的置换不变性性质。 相应的极限博弈是一个我们称为{\em 长距离游戏}的平均场博弈的变体。 我们证明这个渐近方案满足以下关键性质：a) 长程博弈至少存在一个均衡；b) 在适当的单调性条件下，这个均衡是唯一的；c) 长程博弈任何均衡对应的反馈对于$N$人博弈来说都是准纳什均衡。 最后我们表明，该方案包括了多种相互作用机制的例子，特别是 Kac 类型的相互作用和在广义 Erdös-Renyi 图上的相互作用。