Repeated games offer a playground for cooperation and strategy. The folk theorem shows that patient players can achieve any individually rational payoff in the long run. This opens up possibilities beyond one-shot game outcomes.
Feasible payoffs, minmax strategies, and punishment mechanisms are key concepts. Understanding these tools helps players navigate repeated interactions, balancing cooperation with self-interest to reach sustainable equilibria.
Feasible and Rational Payoffs
Feasible Payoff Set and Pareto Frontier
- Feasible payoff set represents all possible payoff combinations players can achieve in a repeated game
- Includes payoffs from pure strategies, mixed strategies, and correlated strategies
- Convex set due to the ability to randomize between strategies
- Pareto frontier consists of payoff combinations where no player can improve their payoff without reducing another player's payoff
- Points on the Pareto frontier are Pareto efficient outcomes
Individually Rational Payoffs and Minmax
- Individually rational payoffs are those that provide each player with at least their minmax payoff
- Minmax payoff is the lowest payoff a player can guarantee themselves, regardless of the other players' actions
- Calculated by finding the maximum of the minimum payoffs for each player
- Players will not accept payoffs below their minmax payoff in a repeated game
- Individually rational payoff set is a subset of the feasible payoff set
Equilibrium Payoffs
Nash Equilibrium Payoffs
- Nash equilibrium payoffs are those that result from a Nash equilibrium strategy profile
- In a Nash equilibrium, no player can unilaterally deviate and improve their payoff
- Repeated games may have multiple Nash equilibria, each with different payoff combinations
- Nash equilibrium payoffs in a repeated game can be outside the convex hull of stage game Nash equilibrium payoffs
Subgame Perfect Equilibrium Payoffs
- Subgame perfect equilibrium (SPE) payoffs result from strategies that form a Nash equilibrium in every subgame
- SPE ensures credibility of threats and promises in a repeated game
- SPE payoffs must be individually rational, as players can always revert to their minmax strategies
- Set of SPE payoffs is a subset of the Nash equilibrium payoff set
Folk Theorem and Strategies
Folk Theorem
- Folk theorem states that any individually rational payoff can be sustained as a subgame perfect equilibrium in an infinitely repeated game with sufficiently patient players
- Patience is measured by the discount factor $\delta$, which represents the weight players put on future payoffs
- As $\delta$ approaches 1, players become more patient and the set of SPE payoffs expands to include all individually rational payoffs
- Folk theorem highlights the potential for cooperation and efficiency in repeated interactions
Punishment Strategies
- Punishment strategies are used to sustain cooperation in repeated games
- Trigger strategies, such as grim trigger or tit-for-tat, involve players punishing deviations from the cooperative outcome
- Optimal punishments minimize the punisher's loss while providing sufficient deterrence
- Stick-and-carrot punishments combine initial punishment with a return to cooperation
- Punishment should be severe enough to deter deviations but not so harsh that it is not credible