What Is Folk Theorem?

The Folk Theorem explains that in an infinitely repeated game, any outcome in which each player receives at least their security pay-off can be an equilibrium. It is a fundamental result in game theory that was accepted informally before a formal proof was established.

Introduction

The Folk Theorem is a pivotal result in the field of game theory, a branch of mathematics that studies strategic interactions among rational decision-makers. The theorem addresses the outcomes of infinitely repeated games and provides deep insights into the concept of Nash Equilibrium, particularly how cooperative outcomes can be sustained without external enforcement. The term “Folk Theorem” was coined because the concept was widely accepted by game theorists before a formal proof was constructed, akin to how folk tales are passed down through generations based on tradition rather than empirical evidence.

Historical Context

The Folk Theorem emerged in the 1970s when game theory gained prominence as a tool to analyze strategic interactions in economics, political science, and evolutionary biology. It helped explain how cooperative behaviors could persist among self-interested agents in situations such as oligopolies, international negotiations, and social dilemmas.

Types and Categories

There are several variations of the Folk Theorem depending on the specifics of the game being analyzed:

1. Perfect Monitoring Folk Theorem: Assumes that players can perfectly observe each other’s actions.
2. Imperfect Monitoring Folk Theorem: Assumes that players have incomplete or noisy observations of each other’s actions.
3. Public Monitoring Folk Theorem: Assumes that all players observe the same public signal about the actions taken.

Key Events

• 1970s: The Folk Theorem is widely discussed and accepted in informal circles.
• 1980s: Formal proofs and generalizations are developed by economists such as Robert Aumann, Michael Maschler, and others.

Detailed Explanations

Nash Equilibrium in Repeated Games: In a repeated game, players interact over multiple periods, and their decisions in one period can affect future outcomes. The Folk Theorem posits that in such settings, a multitude of equilibria are possible, including those where players receive more than their security (minimax) pay-off.

Mathematical Formulation

For a repeated game \( G \), let \( v_i \) be the minimax pay-off for player \( i \). The Folk Theorem asserts that for any feasible and individually rational pay-off vector \( (u_1, u_2, \ldots, u_n) \) where \( u_i \geq v_i \), there exists a strategy profile that can sustain this vector as a Nash equilibrium of the infinitely repeated game.

Mermaid Diagram for Visualization:

    graph LR
	    A(Player 1 Strategy) -->|Interaction| B(Player 2 Strategy)
	    B -->|Future Impact| C(Player 3 Strategy)
	    C -->|Observation| A

Importance and Applicability

The Folk Theorem is critical for understanding:

• Collusive Behavior: How firms might sustain higher prices in oligopolies.
• International Relations: How countries can maintain cooperative agreements.
• Social Sciences: How cooperation can emerge in communities without external enforcement.

Examples

• Price Fixing in Oligopolies: Competing firms may sustain high prices through repeated interactions where defection from agreed prices leads to punitive price wars.
• International Treaties: Countries may adhere to treaties due to the threat of future retaliation.

Considerations

While the Folk Theorem provides a robust framework, its application hinges on the assumptions of infinite repetition and rationality. Real-world constraints like finite lifespans, imperfect information, and bounded rationality can limit its practical applicability.

Related Terms

• Nash Equilibrium: A situation where no player can benefit by unilaterally changing their strategy.
• Minimax Pay-off: The maximum loss a player can ensure in a game, assuming the worst-case scenario.

Comparisons

• Finite Repeated Games vs. Infinite Repeated Games: In finite games, cooperation is harder to sustain due to backward induction, unlike in infinite games as posited by the Folk Theorem.
• Folk Theorem vs. Subgame Perfect Equilibrium: The Folk Theorem often results in subgame-perfect equilibria due to the threat of future punishments.

Interesting Facts

• The Folk Theorem helps explain how cooperation can evolve even in the absence of central authority, mirroring mechanisms in biological evolution.

Famous Quotes

• “In game theory, as in life, the future impacts the present.” – John Nash

Proverbs and Clichés

• “Tit for tat.”
• “You scratch my back, and I’ll scratch yours.”

Jargon and Slang

• Grim Trigger Strategy: A strategy in repeated games where a player punishes defection by reverting to the worst possible outcome forever.
• Folk Wisdom: Collective knowledge accepted without formal evidence, akin to the Folk Theorem’s early acceptance.

FAQs

What is the Folk Theorem?

The Folk Theorem states that in an infinitely repeated game, any outcome where each player gets at least their security pay-off can be a Nash equilibrium.

Why is it called the Folk Theorem?

It was accepted informally by game theorists like folk wisdom before any formal proof.

What are the implications of the Folk Theorem?

It shows that cooperation can be sustained in strategic interactions through repeated interactions and the threat of future punishment.

References

1. Fudenberg, D., & Maskin, E. (1986). “The Folk Theorem in Repeated Games with Discounting or with Incomplete Information.” Econometrica.
2. Aumann, R. J., & Maschler, M. (1995). “Repeated Games with Incomplete Information.”

Summary

The Folk Theorem is a cornerstone of game theory that explains how a wide range of outcomes can be sustained as equilibria in infinitely repeated games. By highlighting the role of future interactions and punishment strategies, it provides a deep understanding of cooperation and strategic behavior among rational agents. Its applications span across economics, political science, and social dynamics, demonstrating the profound impact of repeated interactions on decision-making processes.

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