Explicit equilibria in a kinetic model of gambling

TL;DR

This paper models wealth distribution in gambling using a nonlinear kinetic equation, deriving explicit steady states including Gibbs and Gamma distributions, and analyzing cases with conservative-in-the-mean dynamics.

Contribution

It introduces a kinetic model for gambling wealth exchange and explicitly characterizes steady states for various random sharing fractions, including heavy-tailed distributions.

Findings

Gibbs distribution as steady state for uniform sharing

Gamma distribution as steady state for Beta sharing

Heavy-tailed distribution in conservative-in-the-mean case

Abstract

We introduce and discuss a nonlinear kinetic equation of Boltzmann type which describes the evolution of wealth in a pure gambling process, where the entire sum of wealths of two agents is up for gambling, and randomly shared between the agents. For this equation the analytical form of the steady states is found for various realizations of the random fraction of the sum which is shared to the agents. Among others, Gibbs distribution appears as steady state in case of a uniformly distributed random fraction, while Gamma distribution appears for a random fraction which is Beta distributed. The case in which the gambling game is only conservative-in-the-mean is shown to lead to an explicit heavy tailed distribution.