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. BordenaveD. 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. FrankeC. 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. BordenaveD. 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. FrankeC. 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.

Figure 1.  The extensive-form representation of the Hawk-Dove game $ G\left( v,c\right) $
Figure 2.  Empirical long-run wealth distribution under partly binomial depreciation
Figure 3.  Steady-state average wealth as a function of the value $ v $ in game $ G\left( v,c\right) $
Figure 4.  Steady-state average wealth as a function of the cost $ c $ in game $ G\left( v,c\right) $
Figure 5.  The generalized Hawk-Dove game $ G^{\ast }(v,c,p) $
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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