Zero-Sum Games with Noisy Observations

In this paper, $2 \times 2$ zero-sum games (ZSGs) are studied under the following assumptions: (1) One of the players (the leader) publicly and irrevocably 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 and be often different from the Nash and Stackelberg equilibria. Even subject to noise, observing the actions of the leader is either beneficial or immaterial to the follower for all possible commitments. When the commitment is observed subject to a distortion, the equilibrium does not necessarily exist. Nonetheless, the leader might still obtain some benefit in some specific cases subject to equilibrium refinements. For instance, $\epsilon$-equilibria might exist in which the leader commits to suboptimal strategies that allow unequivocally predicting the best response of its opponent.