Abstract
Stability analysis is an important research direction in evolutionary game theory. Evolutionarily stable states have a close relationship with Nash equilibria of repeated games, which are characterized by the folk theorem. When applying the folk theorem, one needs to compute the minimax profile of the game in order to find Nash equilibria. Computing the minimax profile is an NP-hard problem. In this paper, we investigate a new methodology to compute evolutionary stable states based on the level- ${k}$ equilibrium, a new refinement of Nash equilibrium in repeated games. A level- ${k}$ equilibrium is implemented by a group of players who adopt reactive strategies and who have no incentive to deviate from their strategies simultaneously. Computing the level- ${k}$ equilibria is tractable because the minimax payoffs and strategies are not needed. As an application, this paper develops a tractable algorithm to compute the evolutionarily stable states and the Pareto front of ${n}$ -player symmetric games. Three games, including the iterated prisoner’s dilemma, are analyzed by means of the proposed methodology.
Highlights
Evolutionary game theory has been successful in helping to explain many complex and challenging aspects of biological and social phenomena in recent decades [1, 2]
We prove the existence of level-k equilibria, a subset of Nash equilibrium (NE) of repeated games, which are not characterized by the folk theorem
We propose a new stability analysis of repeated games and evolutionary games based on a subset of NE, which we call them the level-k equilibrium
Summary
Evolutionary game theory has been successful in helping to explain many complex and challenging aspects of biological and social phenomena in recent decades [1, 2]. We propose a new refinement of NE in repeated games, namely the level-k equilibrium, in order to develop tractable and general algorithms to compute NE of repeated games. Based on this concept, we show that specific Pareto optimums of the convex hull of the feasible payoff profiles are NE payoff profiles. We propose a tractable algorithm based on the level-k equilibrium to compute the evolutionarily stable states of n-player symmetric repeated games.
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