Existence of the uniform value in zero-sum repeated games with a more informed controller
Existence of the uniform value in zero-sum repeated games with a more informed controller
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具有更明智的控制器的零和重复博弈中存在统一值
DOI：
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发表时间：
2014
期刊：
影响因子：
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通讯作者：
Xavier Venel
中科院分区：
文献类型：
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作者：
Fabien Gensbittel;Miquel Oliu;Xavier Venel
我们证明了在两个玩家零和重复游戏中，其中一个玩家，比如玩家$1 $，比他的对手更知情，并控制着状态信息的演化，存在统一值。
这一结果推广了部分观测马尔可夫决策过程的已有结果（Rosenberg，Solan，Vieille
[15])，并与知情的控制器重复游戏（雷诺[14]）。我们对更知情的参与者的正式定义比包含信号更一般，因此允许对行动的不完善监控。我们构造了一个辅助随机对策，其状态空间是局中人$2 $的二阶信念（局中人$1 $对原对策状态变量的信念）的集合，并利用Renault [14]的一个结果证明了它有一个值.在这项工作中的一个关键因素是证明参与人$1 $可以在我们的一般框架中使用原始游戏中的辅助游戏的策略，
我们利用经典的论证推导出辅助博弈的值也是原博弈的值。
We prove that in a two-player zero-sum repeated game where one of the players, say player $1$, is more informed than his opponent and controls the evolution of information on the state, the uniform value exists.
This result extends previous results on Markov decision processes with partial observation (Rosenberg, Solan, Vieille
[15]), and repeated games with an informed controller (Renault [14]). Our formal definition of a more informed player is more general than the inclusion of signals, allowing therefore for imperfect monitoring of actions. We construct an auxiliary stochastic game whose state space is the set of second order beliefs of player $2$ (beliefs about beliefs of player $1$ on the state variable of the original game) with perfect monitoring and we prove it has a value by using a result of Renault [14]. A key element in this work is to prove that player $1$ can use strategies of the auxiliary game in the original game in our general framework, from which
we deduce that the value of the auxiliary game is also the value of our original game by using classical arguments.