[Paper Review] Time and the Prisoner's Dilemma

Yishay Mor, Jeffrey S. Rosenschein|Open Research Online (The Open University)|Jan 22, 2007

Game Theory and ApplicationsDecision Sciences15 references21 citations

TL;DR

This paper introduces a finite-time repeated Prisoner’s Dilemma model with computationally bounded players, showing that limited time for computation enables cooperative equilibria. By incorporating opting-out behavior and competitive analysis, it demonstrates that sub-optimal, satisfying strategies can become optimal under resource constraints, making cooperation rational even in theory-contradicting settings.

ABSTRACT

This paper examines the integration of computational complexity into game theoretic models. The example focused on is the Prisoner's Dilemma, repeated for a finite length of time. We show that a minimal bound on the players' computational ability is sufficient to enable cooperative behavior. In addition, a variant of the repeated Prisoner's Dilemma game is suggested, in which players have the choice of opting out. This modification enriches the game and suggests dominance of cooperative strategies. Competitive analysis is suggested as a tool for investigating sub-optimal (but computationally tractable) strategies and game theoretic models in general. Using competitive analysis, it is shown that for bounded players, a sub-optimal strategy might be the optimal choice, given resource limitations.

Motivation & Objective

• To address the paradox of non-cooperation in the repeated Prisoner’s Dilemma under standard rationality assumptions.
• To model real-world agents with limited computational power, challenging the assumption of unbounded rationality in game theory.
• To investigate how opting out as a strategic choice can stabilize cooperative equilibria in repeated games.
• To apply competitive analysis to evaluate sub-optimal strategies in resource-constrained environments.
• To provide design guidelines for systems where cooperation is desirable but not guaranteed by standard game-theoretic equilibria.

Proposed method

• Introduces a finite-time repeated Prisoner’s Dilemma model where players have limited computation time per round.
• Defines 'complexity-bounded' (CB) players whose rationality is constrained by time, preventing full backward induction.
• Proposes an 'opting out' variant of the repeated PD, where players can exit a match if the opponent fails to cooperate.
• Introduces the Opt-for-Tat (OFT) strategy: cooperate as long as the opponent cooperates, otherwise opt out and seek new partners.
• Applies competitive analysis to compare satisfying strategies (e.g., OFT) with maximizing strategies, defining the competitive ratio as SL(S)/h(S).
• Uses probabilistic reasoning to model the likelihood of rematching with cooperative agents, deriving bounds on expected payoffs.

Experimental results

Research questions

• RQ1 Can cooperative behavior emerge in the repeated Prisoner’s Dilemma when players are computationally bounded?
• RQ2 How does the introduction of an 'opting out' mechanism affect the stability and emergence of cooperative equilibria?
• RQ3 Under what conditions does a sub-optimal, satisfying strategy become optimal in resource-constrained environments?
• RQ4 What is the relationship between computational time and the emergence of cooperation in repeated games?
• RQ5 How can competitive analysis be used to evaluate and justify non-maximizing strategies in bounded rationality models?

Key findings

• A minimal bound on computational ability—specifically, limited time per round—is sufficient to enable cooperative equilibria in the finite-time repeated Prisoner’s Dilemma.
• The Opt-for-Tat (OFT) strategy ensures a security level of N*R - const, where const depends on the expected time to rematch and the payoff structure.
• When the probability q of encountering a cooperative player is bounded away from zero, the competitive ratio of satisfying strategies approaches 1 as N increases.
• A player using a satisfying strategy like OFT cannot be outperformed by any optimizing strategy in the long run if the opponent is also bounded and cooperative.
• The 'defect always' strategy is no longer an equilibrium when opting out is allowed and computational time is limited, due to the risk of permanent exclusion.
• Theoretical and practical guidelines are derived for system designers to promote cooperation via initial deployment of cooperative agents and adoption of OFT-like protocols.