this repeated game, a strategy for one of the players is a deterministic rule that specifies the player's choice in each iteration of the stage game, based on all other player's choices in the prior iterations. A choice of strategy for each of the players is a strategy profile, and it leads to a payout profile for the repeated game. There are a number of different ways such a strategy profile can be translated into a payout profile, outlined below. Any Nash equilibrium payoff profile of a repeated game must satisfy two properties: Individual rationality: the payoff must weakly dominate the minmax payoff profile of the constituent stage game. That is, the equilibrium payoff of each player must be at least as large as the minmax payoff of that player. This is because a player achieving less than his minmax payoff always has incentive to deviate by simply playing his minmax strategy at every history. Feasibility: the payoff must be a convex combination of possible payoff profiles of the stage game. This is because the payoff in a repeated game is just a weighted average of payoffs in the basic games. Folk theorems are partially converse claims: they say that, under certain conditions (which are different in each folk theorem), every payoff profile that is both individually rational and feasible can be realized as a Nash equilibrium payoff profile of the repeated game. There are various folk theorems; some relate to finitely-repeated games while others relate to infinitely-repeated games. Infinitely-repeated games without discounting In the undiscounted model, the players are patient. They don't differentiate between utilities in different time periods. Hence, their utility in the repeated game is represented by the sum of utilities in the basic games. When the game is infinite, a common model for the utility in the infinitely-repeated game is the limit inferior of mean utility: If the game results in a path of outcomes , where denotes the collective choices of the players at iteration t (t=0,1,2,...), player is utility is defined as where is the basic-game utility function of player i. An infinitely-repeated game without discounting is often called a "supergame". The folk theorem in this case is very simple and contains no pre-conditions: every individually rational and feasible payoff profile in the basic game is a Nash equilibrium payoff profile in the repeated game. The proof employs what is called a grim or