具有消失阶段时长和公共信号的零和随机博弈中的极限值
Limit Value in Zero-Sum Stochastic Games with Vanishing Stage Duration and Public Signals
AI总结：
研究具有公共信号的不完全观测零和随机博弈，发现在离散时间下无极限值，但在连续时间下存在极限值。
摘要：
我们考虑在具有阶段时长的博弈中，当折扣因子 $λ$ 趋近于 0 时（即参与者越来越有耐心），$λ$-折扣零和博弈的行为。在阶段时长为 h 的随机博弈中，参与者在时间 0、h、2h 等时刻行动。收益和离开概率与 h 成正比。当 h 趋近于 0 时，此类离散时间博弈逼近于连续时间博弈。在状态完全观测的随机博弈和状态盲视的情况下，值的渐近行为（当 $λ$ 和 h 同时趋近于 0 时）已被研究过。我们针对状态不完全观测的随机博弈考虑相同的问题。更准确地说，我们考虑此类博弈的一个特例，即具有公共信号的随机博弈，其中参与者在每个阶段都会获得一个仅取决于当前状态的公共信号。我们的主要结论表明，存在一个具有公共信号的随机博弈，当阶段时长为 1 时，不存在极限值（当折扣因子 $λ$ 趋近于 0 时），但当阶段时长 h 和折扣因子 $λ$ 同时趋近于 0 时，存在极限值。通俗来讲，这意味着离散时间下的极限值不存在，但连续时间下的极限值（即当 h 趋近于 0 时）存在。这种情况在状态完全观测的随机博弈中是不可能发生的。
英文摘要：
We consider the behaviour of $λ$-discounted zero-sum games as the discount factor $λ$ approaches 0 (that is, the players are more and more patient), in the context of games with stage duration. In stochastic games with stage duration h, players act at times 0, h, 2h, and so on. The payoff and leaving probabilities are proportional to h. When h tends to 0, such discrete-time games approximate games played in continuous time. The asymptotic behavior of the values (when both $λ$ and h tend to 0) was already studied in the case of stochastic games with perfect observation of the state and in the state-blind case.We consider the same question for the case of stochastic games with imperfect observation of the state. More precisely, we consider a particular case of such games, stochastic games with public signals, in which players are given at each stage a public signal that depends only on the current state. Our main result states that there exists a stochastic game with public signals, with no limit value (as the discount factor $λ$ goes to 0) if stage duration is 1, but with a limit value when stage duration h and discount factor $λ$ both tend to 0. Informally speaking, it means that the limit value in discrete time does not exist, but the limit value in continuous time (i.e. when h approaches 0) exists. Such a situation is impossible in the case of stochastic games with perfect observation of the state.
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