Stochastic and Boltzmann-like models for behavioral changes, and their relation to game theory

TL;DR

This paper develops stochastic and Boltzmann-like models to describe social behavioral changes, revealing phase transitions and self-organization phenomena, and connects these models to game theory through approximation methods.

Contribution

It introduces a configurational master equation for social pair interactions and links Boltzmann-like equations to game dynamical equations, highlighting new insights into social phase transitions.

Findings

Identification of phase transition in behavioral conventions

Boltzmann-like equations can exhibit oscillatory or chaotic solutions

Game dynamical equations derived from approximations

Abstract

In the last decade, stochastic models have shown to be very useful for quantitative modelling of social processes. Here, a configurational master equation for the description of behavioral changes by pair interactions of individuals is developed. Three kinds of social pair interactions are distinguished: Avoidance processes, compromising processes, and imitative processes. Computational results are presented for a special case of imitative processes: the competition of two equivalent strategies. They show a phase transition that describes the selforganization of a behavioral convention. This phase transition is further analyzed by examining the equations for the most probable behavioral distribution, which are Boltzmann-like equations. Special cases of Boltzmann-like equations do not obey the H-theorem and have oscillatory or even chaotic solutions. A suitable Taylor approximation leads to the socalled game dynamical equations (also known as selection-mutation equations in the theory of evolution).