Maximizing cooperation in the prisoner's dilemma evolutionary game via optimal control

Area of Science:

• Evolutionary Game Theory
• Optimal Control Theory
• Dynamical Systems

Background:

• The prisoner's dilemma (PD) game models competition where defection is a stable but suboptimal outcome.
• Replicator dynamics in PD games lead to a Nash equilibrium favoring defection, resulting in lower average payoffs.
• Cooperation in PD games offers higher payoffs but is not an evolutionarily stable strategy without intervention.

Purpose of the Study:

• To develop an optimal control theory for the prisoner's dilemma evolutionary game.
• To maximize cooperation (minimize defector population) over a defined cycle time.
• To adaptively control strategies in dynamic populations using feedback.

Main Methods:

• Applied optimal control theory to the PD payoff matrix.
• Utilized time-dependent controllers in a bang-bang sequence to dynamically alter payoffs.
• Employed Pontryagin's maximum principle for optimization.
• Developed an adaptive method using end-of-cycle defector population for subsequent control.

Main Results:

• Demonstrated a method to dynamically adjust incentives and penalties to promote cooperation.
• Showcased an adaptive control strategy that optimizes cooperation over multiple cycles.
• Identified optimal timing for payoff adjustments based on initial population distributions.

Conclusions:

• Optimal control theory provides a framework to overcome suboptimal Nash equilibria in evolutionary games like the PD.
• The adaptive control method effectively maximizes cooperation in dynamic, evolving populations.
• This approach has broad applications in fields utilizing replicator dynamics, such as biology, economics, and social sciences.