The dynamics of fitness and wealth distributions — a stochastic game-theoretic model

Sylvain Gibaud; Jörgen Weibull

doi:10.3934/jdg.2022016

Article Contents

2022, Volume 9, Issue 4: 405-432. Doi: 10.3934/jdg.2022016

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The dynamics of fitness and wealth distributions — a stochastic game-theoretic model

Sylvain Gibaud^1, and
Jörgen Weibull^2,

1.
Institut de Mathématique de Toulouse, Université Paul Sabatier Toulouse, France
2.
Department of Economics, Stockholm School of Economics, Sweden

Received: October 2021

Revised: April 2022

Early access: July 2022

Published: October 2022

Sylvain Gibaud (gibaudsylvain@gmail.com) thanks Unité Mixte de Recherche 5219 for its financial support. Jörgen Weibull (jorgen.weibull@hhs.se) thanks the Knut and Alice Wallenberg Research Foundation, the Agence Nationale de la Recherche, Chaire IDEX ANR-11-IDEX-0002-02, and the Tore Browald and Tom Hedelius Foundation for financial support.

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Abstract

A model of the dynamics of distributions of individual wealth, or of individual Darwinian fitness, is here developed. Pairs of individuals are recurrently and randomly matched to play a game over a resource. In addition, all individuals have random access to a constant background source, and their fitness or wealth depreciates over time. For brevity, we focus on the well-known Hawk-Dove game. In the base-line model, the probability of winning a fight over a resource is the same for both parties. In an extended version, the individual with higher current fitness or wealth has a higher probability of winning. Analytical results are given for the fitness/wealth distribution at any given time, for the evolution of average fitness/wealth over time, and for the asymptotics with respect to both time and population size. Long-run average fitness/wealth is non-monotonic in the value of the resource, thus providing a potential explanation of the so-called curse of the riches.

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Mathematics Subject Classification: Primary: 60J28, 91A22, 91B68; Secondary: 92D25.

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Figure 1. The extensive-form representation of the Hawk-Dove game $ G\left( v,c\right) $

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Figure 2. Empirical long-run wealth distribution under partly binomial depreciation

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Figure 3. Steady-state average wealth as a function of the value $ v $ in game $ G\left( v,c\right) $

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Figure 4. Steady-state average wealth as a function of the cost $ c $ in game $ G\left( v,c\right) $

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Figure 5. The generalized Hawk-Dove game $ G^{\ast }(v,c,p) $

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Figure 6. The long-run distribution of individual wealth when "wealth is strength", compared with the distribution in the base-line model

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