Strong and safe Nash equilibrium in some repeated 3-player games

We consider a 3-player game in the normal form, in which each player has two actions. We assume that the game is symmetric and repeated infinitely many times. At each stage players make their choices knowing only the average payoffs from previous stages of all the players. A strategy of a player in the repeated game is a function defined on the convex hull of the set of payoffs. Our aim is to construct a strong Nash equilibrium in the repeated game, i.e. a strategy profile being resistant to deviations by coalitions. Constructed equilibrium strategies are safe, i.e. the non-deviating player payoff is not smaller than the equilibrium payoff in the stage game, and deviating players' payoffs do not exceed the non-deviating player payoff more than a positive constant which can be arbitrary small and chosen by the non-deviating player. Our construction is inspired by Smale's good strategies described in \cite{smale}, where the repeated Prisoner's Dilemma was considered. In proofs we use arguments based on approachability and strong approachability type results.

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