Behavioral equilibrium for infinitely repeated games

Artem Baklanov of the Advanced Systems Analysis Program is analyzing iterated social dilemmas that will help reveal features of stability of interactions, thereby helping individuals learn, though interaction, how to cope with behavioral uncertainty, understand the interests of other individuals, and better adapt to changing social environments.

Artem Baklanov

Artem Baklanov

Introduction

Models of repeated games have utilized the concept of strategic equilibrium to help analyze global economic and political problems. As, in this connection, the stability of existing equilibrium solutions needs to be considered, my research focuses on the stability of behavioral strategies in this setting. The main goal of the research is the analysis of iterated social dilemmas that will help reveal features of stability of interactions, allowing individuals to cope with behavioral uncertainty, understand the interests of other individuals, and to adapt to changing social environments.

Methodology

In analysis of the Nash equilibrium in the class of behavior strategies, the mathematical expectations of the players’ current benefits (averaged over all game rounds) will play the key role. To define those expectations, we construct corresponding probability spaces and show that the game trajectories form an irreducible and aperiodic finite-order Markov chain with finite state spaces. The corresponding stationary distribution is then used to define the mathematical expectations of the players’ average benefits. To prove that a perturbed behavior strategy is an equilibrium we analyze the derivative of the expected average benefit with respect to a parameter responsible for perturbation. In more complex cases, the existence of Nash equilibria needs to be proved in a multi-step behavioral “meta-game” using the Kakutani fixed-point theorem. In this context, we expect the players’ sets of admissible behavior strategies to sometimes require further transformations—convexification or compactification. Such transformations may require extensions of the behavioral strategies originally provided into classes of countably additive or finitely additive probability measures. To carry out and evaluate numerical experiments we use statistical analysis.

Results

Recently begun research reveals the effects of the structural stability of Nash equilibria with respect to uncertainty in the complexity of the behavioral strategies of opponents. This uncertainty is restricted to some range of underlying beliefs reflecting the ability of opponents to perform (observe) specific actions or states in infinitely repeated bimatrix games. Moreover, the general characterization of Nash equilibrium pairs was obtained in the class of reactive stochastic strategies. The proposed approach will allow results to be further generalized. The outcome provides tools to allow policymakers to make the correct choice of Nash equilibria with respect to possible change and uncertainty in the structure of behavioral strategies of players.

Note

Artem Baklanov is a Russian citizen, and an IIASA-funded Postdoctoral Scholar (Aug 2014 – Aug 2016).


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Last edited: 19 February 2015

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