In this paper, $2\times2$ zero-sum games are studied under the following assumptions: $(1)$ One of the players (the leader) commits to choose its actions by sampling a given probability measure (strategy); $(2)$ The leader announces its action, which is observed by its opponent (the follower) through a binary channel; and $(3)$ the follower chooses its strategy based on the knowledge of the leader's strategy and the noisy observation of the leader's action. Under these conditions, the equilibrium is shown to always exist. Interestingly, even subject to noise, observing the actions of the leader is shown to be either beneficial or immaterial for the follower. More specifically, the payoff at the equilibrium of this game is upper bounded by the payoff at the Stackelberg equilibrium (SE) in pure strategies; and lower bounded by the payoff at the Nash equilibrium, which is equivalent to the SE in mixed strategies.Finally, necessary and sufficient conditions for observing the payoff at equilibrium to be equal to its lower bound are presented. Sufficient conditions for the payoff at equilibrium to be equal to its upper bound are also presented.

翻译：本文在以下假设下研究 $2\times2$ 零和博弈：(1) 其中一方玩家（领导者）承诺通过采样给定的概率测度（策略）来选择其行动；(2) 领导者宣布其行动，而对手（跟随者）通过二进制信道观测该行动；(3) 跟随者基于对领导者策略的了解以及对领导者行动的噪声观测来选择自身策略。在这些条件下，证明均衡始终存在。有趣的是，即使存在噪声，观测领导者的行动对跟随者而言要么有利，要么无关紧要。具体而言，该博弈中均衡收益的上界为纯策略斯塔克尔伯格均衡（SE）收益，下界为等价于混合策略SE的纳什均衡收益。最后，给出了观测均衡收益等于其下界的充要条件，并提出了均衡收益等于其上界的充分条件。