6.2 Folk theorems and equilibrium payoffs
3 min read•august 7, 2024
offer a playground for cooperation and strategy. The 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
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- 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
- 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
- are those that provide each player with at least their
- 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 are those that result from a 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
- (SPE) payoffs result from strategies that form a Nash equilibrium in every subgame
- SPE ensures credibility of threats and promises in a repeated game
- 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 , which represents the weight players put on future payoffs
- As 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
- are used to sustain cooperation in repeated games
- , such as or , 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
Key Terms to Review (19)
Equilibrium Payoffs: Equilibrium payoffs refer to the outcomes that players receive in a strategic game when all players have chosen their strategies and no player has an incentive to deviate unilaterally. These payoffs are crucial in determining the stability of strategies within a game, as they reflect the best responses of players to one another's choices, leading to a situation where mutual best responses create an equilibrium.
Feasible Payoff Set: The feasible payoff set refers to the collection of all possible outcomes or payoffs that players in a game can achieve while adhering to the constraints imposed by the game's rules and strategies. This set is crucial in understanding how players can reach equilibrium payoffs, as it helps identify which outcomes are attainable based on the strategies they choose. The concept also plays a vital role in folk theorems, which assert that any payoff within this set can potentially be sustained as an equilibrium under certain conditions.
Folk Theorem: The folk theorem is a concept in game theory that suggests that, in repeated games, a wide variety of outcomes can be sustained as Nash equilibria under certain conditions. This means that if players interact multiple times, they can potentially achieve cooperative outcomes that would not be possible in a one-shot game, particularly when there is an infinite horizon and players care about their future payoffs. The folk theorem highlights the importance of establishing trust and reputation over time, as players may adjust their strategies based on past actions.
Grim trigger: A grim trigger is a strategy used in repeated games where a player threatens to revert to a non-cooperative strategy if the other player deviates from a cooperative agreement. This strategy can effectively sustain cooperation among players by imposing severe consequences for any betrayal. It highlights the importance of threat and punishment in maintaining equilibrium in strategic interactions, making it relevant to understanding cooperation and stability in game dynamics.
Individually Rational Payoffs: Individually rational payoffs refer to the minimum level of utility or benefit that players expect to receive in a game, ensuring that each player prefers participating in the game rather than opting out. This concept is crucial for understanding how players assess their strategies and outcomes in various scenarios, especially in cooperative games and negotiations. These payoffs help to establish the boundaries for acceptable agreements and outcomes that maintain player motivation and engagement.
John Nash: John Nash was an influential mathematician and economist best known for his groundbreaking work in game theory, particularly the concept of Nash equilibrium. His theories have fundamentally shaped our understanding of strategic interactions among rational decision-makers, making them essential for analyzing competitive behaviors in various fields, including economics, political science, and biology.
Minmax payoff: The minmax payoff is the smallest maximum loss that a player can guarantee themselves in a strategic game, even in the worst-case scenario. This concept helps players identify a strategy that minimizes their potential losses while taking into account the strategies of their opponents. In the context of equilibrium payoffs and folk theorems, minmax payoffs serve as a benchmark for evaluating strategies and outcomes in competitive situations.
Nash Equilibrium: Nash Equilibrium is a concept in game theory where no player can benefit by unilaterally changing their strategy if the strategies of the other players remain unchanged. This means that each player's strategy is optimal given the strategies of all other players, resulting in a stable outcome where players have no incentive to deviate from their chosen strategies.
Pareto Efficiency: Pareto efficiency is an economic state where resources are allocated in a way that no individual can be made better off without making someone else worse off. This concept emphasizes the idea of optimal distribution of resources among players in a game, relating closely to strategies, payoffs, and the rational behavior of individuals involved.
Pareto Frontier: The Pareto Frontier, also known as the Pareto Boundary, is a concept in economics and game theory that represents the set of all possible outcomes in which no individual can be made better off without making someone else worse off. This concept is crucial for understanding efficient allocations of resources in cooperative scenarios, where players seek to maximize their outcomes while considering the welfare of others. It helps identify equilibrium payoffs in games where multiple players interact and strive for mutually beneficial results.
Payoff Matrix: A payoff matrix is a table that shows the payoffs or outcomes for each player in a game, given all possible combinations of strategies chosen by the players. It visually represents the choices available to players and their potential results, making it essential for analyzing strategic interactions in various types of games.
Punishment strategies: Punishment strategies are methods used to discourage undesirable behavior in strategic interactions by imposing costs or negative consequences on the offending party. These strategies play a vital role in maintaining cooperation among self-interested individuals, as they can deter deviations from agreed-upon actions, thus stabilizing relationships and fostering equilibrium outcomes. By aligning individual incentives with collective goals, punishment strategies contribute significantly to concepts like folk theorems and collusion dynamics.
Repeated games: Repeated games are strategic situations where the same game is played multiple times by the same players, allowing for the possibility of strategies to evolve based on past interactions. This framework enables players to build reputations, establish trust, and potentially achieve cooperative outcomes that would not be attainable in a one-shot game. The dynamics of repeated interactions can lead to various equilibria, including the possibility of sustaining cooperation over time.
Robert Axelrod: Robert Axelrod is a political scientist and professor known for his pioneering work in game theory and its applications to social science, particularly in the context of cooperation and conflict. His most famous contribution is the 'Prisoner's Dilemma' simulations which demonstrate how cooperation can emerge among self-interested individuals, connecting to concepts such as credible threats and promises, folk theorems, evolutionary stable strategies, and applications in biology and social evolution.
SPE Payoffs: SPE payoffs, or Subgame Perfect Equilibrium payoffs, refer to the outcomes of a game that remain consistent and optimal in every subgame of the original game. This concept emphasizes that players' strategies should not only lead to a Nash Equilibrium but should also be robust at every point of the game, ensuring that each player’s strategy is optimal given the strategies of others, even in future moves. The importance of SPE payoffs lies in their ability to establish credibility and sustainability in strategic interactions.
Subgame Perfect Equilibrium: Subgame perfect equilibrium is a refinement of Nash equilibrium applicable to dynamic games where players make decisions at various stages. It requires that players' strategies constitute a Nash equilibrium in every subgame of the original game, ensuring that the strategies are credible and optimal, even if the game is played out from any point along the decision path.
Tit-for-tat: Tit-for-tat is a strategy in game theory where a player responds to another player's actions with the same actions, whether cooperative or uncooperative. This approach encourages cooperation by mirroring the opponent's previous move, thus fostering an environment of mutual benefit and trust. It is often used to promote cooperation in repeated games, highlighting the importance of reciprocity and the potential for long-term collaboration over short-term gains.
Trigger Strategies: Trigger strategies are contingent plans used in repeated games where a player responds to another player's actions with predetermined responses, often designed to enforce cooperation or deter defection. These strategies can be particularly effective in sustaining equilibrium outcomes by threatening punitive responses to uncooperative behavior. They are essential in understanding dynamics like collusion, as they help maintain cooperative agreements among players through the threat of reverting to a less favorable strategy if the agreement is violated.
Utility Function: A utility function is a mathematical representation that assigns a numerical value to the level of satisfaction or happiness an individual derives from consuming goods and services. This function helps in understanding consumer preferences and choices, making it crucial for analyzing decision-making in various economic scenarios, including repeated games and equilibrium payoffs, where players aim to maximize their utility over time.