We show that the problem of deciding whether in a multi-player perfect information recursive game (i.e. a stochastic game with terminal rewards) there exists a stationary Nash equilibrium ensuring each player a certain payoff is Existential Theory of the Reals complete. Our result holds for acyclic games, where a Nash equilibrium may be computed efficiently by backward induction, and even for deterministic acyclic games with non-negative terminal rewards. We further extend our results to the existence of Nash equilibria where a single player is surely winning. Combining our result with known gadget games without any stationary Nash equilibrium, we obtain that for cyclic games, just deciding existence of any stationary Nash equilibrium is Existential Theory of the Reals complete. This holds for reach-a-set games, stay-in-a-set games, and for deterministic recursive games.

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完全信息随机博弈中平稳纳什均衡实数完备性的存在论

我们表明，确定在多人完美信息递归博弈（即具有终端奖励的随机博弈）中是否存在确保每个玩家获得特定收益的稳定纳什均衡的问题是实数存在理论完备的。我们的结果适用于非循环博弈，其中可以通过反向归纳有效地计算纳什均衡，甚至适用于具有非负终端奖励的确定性非循环博弈。我们进一步将我们的结果扩展到纳什均衡的存在，其中单个玩家肯定会获胜。将我们的结果与没有任何平稳纳什均衡的已知小工具博弈相结合，我们得到对于循环博弈，只要确定任何平稳纳什均衡的存在就完成了实数存在理论。这适用于定场比赛、定场比赛、