Fault Tolerant Equilibria in Anonymous Games: best response correspondences and fixed points

The notion of fault tolerant Nash equilibria has been introduced as a way of studying the robustness of Nash equilibria. Under this notion, a fixed number of players are allowed to exhibit faulty behavior in which they may deviate arbitrarily from an equilibrium strategy. A Nash equilibrium in a game with $N$ players is said to be $\alpha$-tolerant if no non-faulty user wants to deviate from an equilibrium strategy as long as $N-\alpha-1$ other players are playing the equilibrium strategies, i.e., it is robust to deviations from rationality by $\alpha$ faulty players. In prior work, $\alpha$-tolerance has been largely viewed as a property of a given Nash equilibria. Here, instead we consider following Nash's approach for showing the existence of equilibria, namely, through the use of best response correspondences and fixed-point arguments. In this manner, we provide sufficient conditions for the existence an $\alpha$-tolerant equilibrium. This involves first defining an $\alpha$-tolerant best response correspondence. Given a strategy profile of non-faulty agents, this correspondence contains strategies for a non-faulty player that are a best response given any strategy profile of the faulty players. We prove that if this correspondence is non-empty, then it is upper-hemi-continuous. This enables us to apply Kakutani's fixed-point theorem and argue that if this correspondence is non-empty for every strategy profile of the non-faulty players then there exists an $\alpha$-tolerant equilibrium. However, we also illustrate by examples, that in many games this best response correspondence will be empty for some strategy profiles even though $\alpha$-tolerant equilibira still exist.