# American Institute of Mathematical Sciences

doi: 10.3934/jdg.2022016
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## The dynamics of fitness and wealth distributions — a stochastic game-theoretic model

 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

Fund Project: 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

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.

Citation: Sylvain Gibaud, Jörgen Weibull. The dynamics of fitness and wealth distributions — a stochastic game-theoretic model. Journal of Dynamics and Games, doi: 10.3934/jdg.2022016
##### References:
 [1] P. Barbe and M. Ledoux, Probabilité, Enseignement SUP-Maths, EDP Sciences. De Gruyter: Berlin, 2007. [2] P. Bardhan, S. Bowles and H. Gintis, Wealth inequality, wealth constraints, and economic performance, Handbook of Income Distribution, Elsevier Science: Amsterdam, 1 (1999). [3] C. Bordenave, D. McDonald and A. Proutière, A particle system in interaction with a rapidly varying environment: Mean field limits and applications, Netw. Heterog. Media, 5 (2010), 31-62.  doi: 10.3934/nhm.2010.5.31. [4] P. Brémaud, Probability Theory and Stochastic Processes, Universitext, Springer, Cham, 2020. doi: 10.1007/978-3-030-40183-2. [5] L. A. Childs, A Concrete Introduction to Higher Algebra, 3$^rd$ edition, Undergraduate Texts in Mathematics, Springer, New York, 2009. doi: 10.1007/978-0-387-74725-5. [6] P. Crowley, Hawks, doves, and mixed-symmetry games, Journal of Theoretical Biology, 204 (2000), 543-563. [7] J. Davies and A. Shorrock, The distribution of wealth, Chapter 11 in A. Atkinson and F. Bourguignon (eds. ), Handbook of Income Distribution, Amsterdam: Elsevier, 1 (1999) [8] M. Enquist and O. Leimar, Evolution of fighting behavior: Decision rules and assessment of relative strength,, Journal of Theoretical Biology, 102 (1983), 387-410. [9] M. Enquist and O. Leimar, Effects of asymmetries in owner-intruder conflicts, Journal of Theoretical Biology, 111 (1984), 475-491. [10] M. Enquist and O. Leimar, Evolution of fighting behavior: The effect of variation in resource value, J. Theoret. Biol., 127 (1987), 187-205.  doi: 10.1016/S0022-5193(87)80130-3. [11] M. Enquist and O. Leimar, The evolution of fatal fighting, Animal Behavior, 39 (1990), 1-9. [12] S. N. Ethier and T. G. Kurtz, Markov processes: Characterization and convergence, John Wiley & Sons., New York, 1986. doi: 10.1002/9780470316658. [13] M. Fischer, Lectures on Markov Processes and Martingales Problems, Department of Mathematics, University of Padua, 2012. [14] J. Franke, C. Kanzow and W. Leininger, Effort maximization in asymmetric contest games with heterogeneous contestants, Economic Theory, 52 (2013), 589-630.  doi: 10.1007/s00199-011-0657-z. [15] S. Gibaud, Evolution Markovienne de Systèmes Multi-Joueurs Accumulant Leurs Gains, Chapter 3 of Thèse de doctorat, Université de Toulouse, Université Toulouse Ⅲ-Paul Sabatier, (2017). [16] C. Graham and S. Méléard, Stochastic particle approximations for generalized Boltzmann models and convergence estimates, Ann. Probab., 25 (1997), 115-132.  doi: 10.1214/aop/1024404281. [17] P. Hammerstein and O. Leimar, Evolutionary game theory and biology, Chapter 11 in P. Young and S. Zamir (eds. ), Handbook of Game Theory with Economic Applications, Amsterdam: Elsevier, 4 (2015). [18] A. Houston and J. McNamara, Fighting for food: A dynamic version of the Hawk-Dove game, Evolutionary Ecology, 2 (1988), 51-64. [19] T. G. Kurtz, Lectures on Stochastic Analysis, Department of Mathematics and Statistics, University of Wisconsin, Madison, updated version 2007. [20] J. Maynard Smith, Evolution and the theory of games, American Scientist, 64 (1976), 41-45. [21] J. Maynard Smith and G. R. Price, The logic of animal conflict, Nature, 246 (1973), 15-18. [22] J. McNamara and A. Houston, If animals know their own fighting ability, the evolutionarily stable level of fighting is reduced, J. Theoret. Biol., 232 (2005), 1-6.  doi: 10.1016/j.jtbi.2004.07.024. [23] P. Molander, Ojämlikhetens Anatomi, ("The Anatomy of Inequality", in Swedish), Weylers förlag: Stockholm. (2014). [24] J. R. Norris, Markov Chains, Cambridge University Press: Cambridge, 1998. [25] T. Piketty, Capital in the 21st Century, Belknap Press. Cambridge, MA. 201. [26] A. Rubinstein and A. Wolinsky, Decentralized trading, strategic behaviour and the Walrasian outcome, Rev. Econom. Stud., 57 (1990), 63-78.  doi: 10.2307/2297543. [27] R. Selten, A note on evolutionarily stable strategies in asymmetric animal conflicts, J. Theoret. Biol., 84 (1980), 93-101.  doi: 10.1016/S0022-5193(80)81038-1. [28] D. W. Stroock, An Introduction to Markov Processes, 2$^nd$ ection, Springer, Heidelberg, 2014. doi: 10.1007/978-3-642-40523-5. [29] A. -S. Sznitman, Topics in propagation of chaos, In Hennequin, P-L. (ed. ), Ecole d'Eté de Probabilités de Saint-Flour XIX–-1989, Lecture Notes in Mathematics, Springer Verlag: Berlin, 1464 (1991). [30] B. Torres, O. Alfonso and I. Soares, A Survey of Literature On the resource curse: critical Analysis of the Main Explanations, Empirical Tests and Resource Proxies, CEF. UP Working Paper 2013-02, 2013. [31] G. Tullock, Toward a Theory of the Rent-Seeking Society, Efficient rent seeking, in Buchanan, J. R. Tollison and G. Tullock, G. (eds. ), Texas A & M University Press: College Station, 1980. [32] J. Weibull, National Wealth Accumulation as A Recurrent Game Between Its Citizens, Mimeo., Stockholm School of Economics and the I. U. I. Institute, Stockholm, 1999.

show all references

##### References:
 [1] P. Barbe and M. Ledoux, Probabilité, Enseignement SUP-Maths, EDP Sciences. De Gruyter: Berlin, 2007. [2] P. Bardhan, S. Bowles and H. Gintis, Wealth inequality, wealth constraints, and economic performance, Handbook of Income Distribution, Elsevier Science: Amsterdam, 1 (1999). [3] C. Bordenave, D. McDonald and A. Proutière, A particle system in interaction with a rapidly varying environment: Mean field limits and applications, Netw. Heterog. Media, 5 (2010), 31-62.  doi: 10.3934/nhm.2010.5.31. [4] P. Brémaud, Probability Theory and Stochastic Processes, Universitext, Springer, Cham, 2020. doi: 10.1007/978-3-030-40183-2. [5] L. A. Childs, A Concrete Introduction to Higher Algebra, 3$^rd$ edition, Undergraduate Texts in Mathematics, Springer, New York, 2009. doi: 10.1007/978-0-387-74725-5. [6] P. Crowley, Hawks, doves, and mixed-symmetry games, Journal of Theoretical Biology, 204 (2000), 543-563. [7] J. Davies and A. Shorrock, The distribution of wealth, Chapter 11 in A. Atkinson and F. Bourguignon (eds. ), Handbook of Income Distribution, Amsterdam: Elsevier, 1 (1999) [8] M. Enquist and O. Leimar, Evolution of fighting behavior: Decision rules and assessment of relative strength,, Journal of Theoretical Biology, 102 (1983), 387-410. [9] M. Enquist and O. Leimar, Effects of asymmetries in owner-intruder conflicts, Journal of Theoretical Biology, 111 (1984), 475-491. [10] M. Enquist and O. Leimar, Evolution of fighting behavior: The effect of variation in resource value, J. Theoret. Biol., 127 (1987), 187-205.  doi: 10.1016/S0022-5193(87)80130-3. [11] M. Enquist and O. Leimar, The evolution of fatal fighting, Animal Behavior, 39 (1990), 1-9. [12] S. N. Ethier and T. G. Kurtz, Markov processes: Characterization and convergence, John Wiley & Sons., New York, 1986. doi: 10.1002/9780470316658. [13] M. Fischer, Lectures on Markov Processes and Martingales Problems, Department of Mathematics, University of Padua, 2012. [14] J. Franke, C. Kanzow and W. Leininger, Effort maximization in asymmetric contest games with heterogeneous contestants, Economic Theory, 52 (2013), 589-630.  doi: 10.1007/s00199-011-0657-z. [15] S. Gibaud, Evolution Markovienne de Systèmes Multi-Joueurs Accumulant Leurs Gains, Chapter 3 of Thèse de doctorat, Université de Toulouse, Université Toulouse Ⅲ-Paul Sabatier, (2017). [16] C. Graham and S. Méléard, Stochastic particle approximations for generalized Boltzmann models and convergence estimates, Ann. Probab., 25 (1997), 115-132.  doi: 10.1214/aop/1024404281. [17] P. Hammerstein and O. Leimar, Evolutionary game theory and biology, Chapter 11 in P. Young and S. Zamir (eds. ), Handbook of Game Theory with Economic Applications, Amsterdam: Elsevier, 4 (2015). [18] A. Houston and J. McNamara, Fighting for food: A dynamic version of the Hawk-Dove game, Evolutionary Ecology, 2 (1988), 51-64. [19] T. G. Kurtz, Lectures on Stochastic Analysis, Department of Mathematics and Statistics, University of Wisconsin, Madison, updated version 2007. [20] J. Maynard Smith, Evolution and the theory of games, American Scientist, 64 (1976), 41-45. [21] J. Maynard Smith and G. R. Price, The logic of animal conflict, Nature, 246 (1973), 15-18. [22] J. McNamara and A. Houston, If animals know their own fighting ability, the evolutionarily stable level of fighting is reduced, J. Theoret. Biol., 232 (2005), 1-6.  doi: 10.1016/j.jtbi.2004.07.024. [23] P. Molander, Ojämlikhetens Anatomi, ("The Anatomy of Inequality", in Swedish), Weylers förlag: Stockholm. (2014). [24] J. R. Norris, Markov Chains, Cambridge University Press: Cambridge, 1998. [25] T. Piketty, Capital in the 21st Century, Belknap Press. Cambridge, MA. 201. [26] A. Rubinstein and A. Wolinsky, Decentralized trading, strategic behaviour and the Walrasian outcome, Rev. Econom. Stud., 57 (1990), 63-78.  doi: 10.2307/2297543. [27] R. Selten, A note on evolutionarily stable strategies in asymmetric animal conflicts, J. Theoret. Biol., 84 (1980), 93-101.  doi: 10.1016/S0022-5193(80)81038-1. [28] D. W. Stroock, An Introduction to Markov Processes, 2$^nd$ ection, Springer, Heidelberg, 2014. doi: 10.1007/978-3-642-40523-5. [29] A. -S. Sznitman, Topics in propagation of chaos, In Hennequin, P-L. (ed. ), Ecole d'Eté de Probabilités de Saint-Flour XIX–-1989, Lecture Notes in Mathematics, Springer Verlag: Berlin, 1464 (1991). [30] B. Torres, O. Alfonso and I. Soares, A Survey of Literature On the resource curse: critical Analysis of the Main Explanations, Empirical Tests and Resource Proxies, CEF. UP Working Paper 2013-02, 2013. [31] G. Tullock, Toward a Theory of the Rent-Seeking Society, Efficient rent seeking, in Buchanan, J. R. Tollison and G. Tullock, G. (eds. ), Texas A & M University Press: College Station, 1980. [32] J. Weibull, National Wealth Accumulation as A Recurrent Game Between Its Citizens, Mimeo., Stockholm School of Economics and the I. U. I. Institute, Stockholm, 1999.
The extensive-form representation of the Hawk-Dove game $G\left( v,c\right)$
Empirical long-run wealth distribution under partly binomial depreciation
Steady-state average wealth as a function of the value $v$ in game $G\left( v,c\right)$
Steady-state average wealth as a function of the cost $c$ in game $G\left( v,c\right)$
The generalized Hawk-Dove game $G^{\ast }(v,c,p)$
The long-run distribution of individual wealth when "wealth is strength", compared with the distribution in the base-line model
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