Understanding Nash Equilibrium in Infinitely Repeated Games

The Nash equilibrium in infinitely repeated games is a fundamental concept in game theory, particularly in the analysis of strategic interactions that occur over an indefinite number of periods. This equilibrium concept helps us understand how players can reach cooperative agreements and sustain them over time despite the potential for short-term gains through defection.

Definition of Nash Equilibrium in Infinitely Repeated Games

A Nash equilibrium in an infinitely repeated game is a strategy profile in which each player's strategy is a best response to the strategies of the other players, and no player has an incentive to unilaterally deviate from their strategy given the strategies of the others.

Key Features of Nash Equilibrium in Infinitely Repeated Games

Strategy Profile

A Nash equilibrium is characterized by a set of strategies, one for each player, that are played throughout the infinite sequence of games. These strategies are interdependent, and each player's choice depends on the actions of the other players.

Best Response

Each player's strategy must be the best response to the strategies chosen by the other players. This means that, given the other players' strategies, no player can improve their payoff by unilaterally changing their own strategy.

Payoff Considerations

In the context of infinitely repeated games, players take into account the future consequences of their current actions. This is crucial because cooperation can be sustained through the threat of future punishment for deviation from cooperative behavior. The threat of punishment serves as a deterrent for defection, ensuring that cooperative behavior is maintained.

The Folk Theorem in Infinitely Repeated Games

The Folk Theorem is a significant aspect of infinite games, particularly when players are patient and value future payoffs highly. It suggests that a wide range of outcomes can be sustained as Nash equilibria. This includes cooperative outcomes that would not be possible in a one-shot, non-cooperative game.

Subgame Perfect Equilibrium

In many cases, especially those involving punishment strategies for deviations, the equilibria in infinitely repeated games are also subgame perfect. This means that the strategies constitute a Nash equilibrium in every subgame of the infinite game. Subgame perfection ensures that the strategy profile is not only a Nash equilibrium at the start but also at every point in the game.

Example: Infinitely Repeated Cournot Competition

Consider two firms engaged in an infinitely repeated Cournot competition. If both firms choose to cooperate by setting higher prices, they can achieve higher profits over time. If one firm deviates and lowers its price, it may gain a short-term advantage but could face retaliation in the form of price wars, leading to lower long-term profits. The threat of future punishment ensures that cooperation continues as a Nash equilibrium.

Conclusion

The Nash equilibrium in infinitely repeated games incorporates the strategic dimension of future interactions, allowing for cooperation and complex strategies that evolve over time. This concept provides insights into how long-term relationships and repeated interactions can lead to cooperative outcomes in economic and social contexts.