Repeated Games and Folk Theorem Definition: Repeated games are a sequence of independent games played by a finite number of players, each with perfect k...
Repeated Games and Folk Theorem
Definition:
Repeated games are a sequence of independent games played by a finite number of players, each with perfect knowledge of the other players' strategies. Each player's strategy in a repeated game is independent of all other players' strategies, but it is influenced by the strategies of all other players in past rounds.
Folk Theorem:
The Nash equilibrium is the unique stationary strategy profile for a repeated game. A Nash equilibrium is a set of strategies that, for every player, is a best response to the strategies of all other players in the game.
Formal Definition:
Let a repeated game with N players be represented by the tuple (N, s, R), where:
N is the number of players.
s is the strategy space, which is a finite set of actions.
R is the set of all possible game histories.
A Nash equilibrium for this game is a strategy profile p that, for every player i, is a best response to the strategies of all other players in the game. A best response is a strategy that is a best response to all possible other players' strategies, given the history of the game up to that point.
Example:
Consider a repeated game of rock, paper, scissors between two players. The strategy space for this game would be {rock, paper, scissors}. A Nash equilibrium for this game would be the strategy profile in which player 1 always chooses rock, and player 2 always chooses paper.
Implications of the Folk Theorem:
The Folk theorem states that repeated games converge to a Nash equilibrium as the number of players increases. In other words, there is a positive probability that the game will converge to a Nash equilibrium, given that the number of players is sufficiently large.
Applications:
The Folk theorem has a wide range of applications in economics and other fields. For example, it can be used to analyze the behavior of strategic agents in games, such as economic agents, political parties, and organizations