A game theoretic model of wealth distribution

TL;DR

This paper introduces an agent-based game theoretic model to analyze wealth distribution, demonstrating how different game equilibria influence wealth dynamics and distribution shapes through simulations and theoretical analysis.

Contribution

It develops a novel model combining learning dynamics with wealth exchange, deriving equations similar to replicator dynamics, and characterizing wealth distributions under different game equilibria.

Findings

Pure strategy equilibrium leads to fixed wealth distribution with wealthier agents near optimal strategies.

Mixed strategy equilibrium results in wealth distributions approximating a Gamma distribution.

Theoretical computation of the second moment of wealth distribution in mixed strategy cases.

Abstract

In this work we consider an agent based model in order to study the wealth distribution problem where the interchange is determined with a symmetric zero sum game. Simultaneously, the agents update their way of play trying to learn the optimal one. Here, the agents use mixed strategies. We study this model using both simulations and theoretical tools. We derive the equations for the learning mechanism, and we show that the mean strategy of the population satisfies an equation close to the classical replicator equation. Concerning the wealth distribution, there are two interesting situations depending on the equilibrium of the game. If the equilibrium is a pure strategy, the wealth distribution is fixed after some transient time, and those players which are close to optimal strategy are richer. When the game has an equilibrium in mixed strategies, the stationary wealth distribution is close to a Gamma distribution with variance depending on the coefficients of the game matrix. We compute theoretically their second moment in this case.