We present a general existence result for a type of equilibrium in normal-form games, which extends the concept of Nash equilibrium. We consider nonzero-sum normal-form games with an arbitrary number of players and arbitrary action spaces. We impose merely one condition: the payoff function of each player is bounded. We allow players to use finitely additive probability measures as mixed strategies. Since we do not assume any measurability conditions, for a given strategy profile the expected payoff is generally not uniquely defined, and integration theory only provides an upper bound, the upper integral, and a lower bound, the lower integral. A strategy profile is called a legitimate equilibrium if each player evaluates this profile by the upper integral, and each player evaluates all his possible deviations by the lower integral. We show that a legitimate equilibrium always exists. Our equilibrium concept and existence result are motivated by Vasquez (2017), who defines a conceptually related equilibrium notion, and shows its existence under the conditions of finitely many players, separable metric action spaces and bounded Borel measurable payoff functions. Our proof borrows several ideas from (Vasquez (2017)), but is more direct as it does not make use of countably additive representations of finitely additive measures by (Yosida and Hewitt (1952)).

我们提出了正规形式博弈中一种均衡类型的一般存在结果，它扩展了纳什均衡的概念。我们考虑具有任意数量的玩家和任意动作空间的非零和正规态游戏。我们仅施加一个条件：每个玩家的收益函数是有界的。我们允许玩家使用有限加性概率测度作为混合策略。由于我们不假设任何可测量性条件，因此对于给定的策略配置文件，通常不会唯一定义预期收益，并且积分理论仅提供上限（上限）和下限（下限）。如果每个参与者都通过较高的积分来评估该轮廓，并且每个参与者都通过较低的积分来评估其所有可能的偏差，则该策略轮廓被称为合法均衡。我们证明合法均衡总是存在的。我们的均衡概念和存在结果受到Vasquez（2017）的启发，他定义了一个概念上相关的均衡概念，并在有限多个参与者，可分离的度量作用空间和有界Borel可衡量的收益函数的条件下显示了其存在。我们的证明借鉴了（Vasquez（2017））的一些观点，但更为直接，因为它没有利用（Yosida and Hewitt（1952））的有限累加测度的可数累加表示。