Microscopic Foundation of Stochastic Game Dynamical Equations

TL;DR

This paper derives stochastic game dynamical equations from microscopic pair interactions, discusses their limitations during phase transitions, and extends the model to include various behavioral and interaction complexities.

Contribution

It provides a microscopic foundation for stochastic game dynamics and explores extensions to incorporate expectations, multiple subpopulations, and memory effects.

Findings

Mean value equations differ from game dynamical equations during phase transitions.

Derived stochastic equations from Boltzmann-like pair interactions.

Extended models include effects of expectations, multiple subpopulations, and memory.

Abstract

The game dynamical equations are derived from Boltzmann-like equations for individual pair interactions by assuming a certain kind of imitation behavior, the so-called proportional imitation rule. They can be extended to a stochastic formulation of evolutionary game theory which allows the derivation of approximate and corrected mean value and covariance equations. It is shown that, in the case of phase transitions (i.e. multi-modal probability distributions), the mean value equations do not agree with the game dynamical equations. Therefore, their exact meaning is carefully discussed. Finally, some generalizations of the behavioral model are presented, including effects of expectations, other kinds of interactions, several subpopulations, or memory effects.