Suppose we have a simultaneous game, that has a strong Nash equilibrium (SNA), i.e. a weak Pareto efficient Nash equilibrium (no deviation of any subset of player brings a benefit to them).


Now suppose we play this game repeatedly. Does the repeated game has a strong Nash equilibrium, too? 


I would keep the question about the payoff function of the repeated game as open. Choose whatever adopted payoff works for the answer.


The idea behind the question is as follows: Games that don't have SNA, like prisoner's dilemma, might have those in the repeated scenario, since additional effects like long term strategies come into play.


Based on this, my guess is, that playing the SNA in every game, would also give a SNA in the repeated game.



Strong Nash Equilibria in repeated games
Let's first recall the definition of a strong nash equilibrium: A strong Nash equilibrium is a Nash equilibrium in which no coalition, taking the actions of its complements as given, can cooperatively deviate in a way that benefits all of its members. In particular, some of the deviating players may profit, but then others must lose.
Consider now the following two player game:


1,1 0,0 0,0


0,0 3,0 0,0


0,0 0,0 0,3


The top-left entry is a strong Nash equilibrium of the one stage game, but repeating it twice is not a SNE of the two-stage game. Indeed, in the two-stage game the players can get the payoff 1.5,1.5, which they cannot in the one-stage game.
Does this example answers the question?


