Best Reply Player Against Mixed Evolutionarily Stable Strategy User.

Area of Science:

• Evolutionary Game Theory
• Mathematical Biology
• Behavioral Economics

Background:

• Classical evolutionary matrix game theory assumes well-mixed interactions.
• Repetition in games allows players to react to opponent's past behavior.
• Mixed evolutionarily stable strategies (ESS) are key concepts in understanding population dynamics.

Purpose of the Study:

• To investigate the stability of mixed ESS in iterated matrix games with repetition.
• To determine conditions under which a best reply (BR) player can invade a population of ESS players.
• To analyze the impact of reactive strategies on evolutionary stability.

Main Methods:

• Analysis of matrix games with two player types: mixed ESS and best reply (BR).
• Focus on iterated games and well-mixed populations with a finite, positive number of repetitions.
• Mathematical modeling to identify conditions for BR player replacement of ESS players.

Main Results:

• Conditions exist on the payoff matrix for the BR player to replace the mixed ESS player.
• BR players, when paired, can develop cyclical dynamics that offset their disadvantage against ESS players.
• The classical mixed ESS loses general stability in repeated well-mixed population games, being vulnerable to BR invasion.

Conclusions:

• The assumption of non-reactive strategies in classical evolutionary game theory may limit its applicability.
• Reactive strategies, like the best reply, can destabilize established evolutionary equilibria.
• Findings highlight the importance of considering player reactivity and game repetition in evolutionary dynamics.