We consider two-player games with imperfect information and the synthesis of a randomized strategy for one player that ensures the objective is satisfied almost-surely (i.e., with probability 1), regardless of the strategy of the other player. Imperfect information is modeled by an indistinguishability relation %that describing the pairs of histories that the first player cannot distinguish, a generalization of the traditional model with partial observations. The game is regular if it admits a regular function whose kernel commutes with the indistinguishability relation. The synthesis of pure strategies that ensure all possible outcomes satisfy the objective is possible in regular games, by a generic reduction that holds for all objectives. While the solution for pure strategies extends to randomized strategies in the traditional model with partial observations (which is always regular), we show that a similar reduction does not exist in the more general model. Despite that, we show that in regular games with Buechi objectives the synthesis problem is decidable for randomized strategies that ensure the outcome satisfies the objective almost-surely.

翻译：我们考虑不完全信息下的双人博弈，以及为一方参与者综合一种随机策略，确保无论另一方参与者采取何种策略，目标都能几乎必然（即概率为1）得到满足。不完全信息通过一种不可区分关系建模，该关系描述了第一方参与者无法区分的历史对，这是对传统部分观测模型的推广。若博弈承认一个正则函数，且该函数的核与不可区分关系交换，则此博弈是正则的。在正则博弈中，通过适用于所有目标的通用归约，可以综合出确保所有可能结果满足目标的纯策略。虽然纯策略的解法可扩展到传统部分观测模型（该模型总是正则的）中的随机策略，但我们证明，在更一般的模型中不存在类似的归约。尽管如此，我们表明，在具有Büchi目标的正则博弈中，对于确保结果几乎必然满足目标的随机策略，其综合问题是可判定的。