doi: 10.3934/jdg.2021030
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Using chemical reaction network theory to show stability of distributional dynamics in game theory

1. 

Department of Mathematics, Wilfrid Laurier University, Waterloo, Ontario, Canada

2. 

Department of Mathematics, Faculty of Sciences, University of South Bohemia, Branišovská 1760,370 05 České Budějovice, Czech Republic

3. 

Czech Academy of Sciences, Biology Centre, Branišovská 31,370 05 České Budějovice, Czech Republic

* Corresponding author: Vlastimil Křivan

Received  February 2021 Revised  October 2021 Early access November 2021

This article shows how to apply results of chemical reaction network theory (CRNT) to prove uniqueness and stability of a positive equilibrium for pairs/groups distributional dynamics that arise in game theoretic models. Evolutionary game theory assumes that individuals accrue their fitness through interactions with other individuals. When there are two or more different strategies in the population, this theory assumes that pairs (groups) are formed instantaneously and randomly so that the corresponding pairs (groups) distribution is described by the Hardy–Weinberg (binomial) distribution. If interactions times are phenotype dependent the Hardy-Weinberg distribution does not apply. Even if it becomes impossible to calculate the pairs/groups distribution analytically we show that CRNT is a general tool that is very useful to prove not only existence of the equilibrium, but also its stability. In this article, we apply CRNT to pair formation model that arises in two player games (e.g., Hawk-Dove, Prisoner's Dilemma game), to group formation that arises, e.g., in Public Goods Game, and to distribution of a single population in patchy environments. We also show by generalizing the Battle of the Sexes game that the methodology does not always apply.

Citation: Ross Cressman, Vlastimil Křivan. Using chemical reaction network theory to show stability of distributional dynamics in game theory. Journal of Dynamics & Games, doi: 10.3934/jdg.2021030
References:
[1]

M. Broom, R. Cressman and V. Křivan, Revisiting the "fallacy of averages" in ecology: Expected gain per unit time equals expected gain divided by expected time, J. Theor. Biol., 483 (2019), 109993, 6pp. doi: 10.1016/j.jtbi.2019.109993.  Google Scholar

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V. Křivan and R. Cressman, Interaction times change evolutionary outcomes: Two player matrix games, J. Theor. Biol., 416 (2017), 199–207. doi: 10.1016/j.jtbi.2017.01.010.  Google Scholar

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V. Křivan, T. Galanthay and R. Cressman, Beyond replicator dynamics: From frequency to density dependent models of evolutionary games, J. Theor. Biol., 455 (2018), 232–248. doi: 10.1016/j.jtbi.2018.07.003.  Google Scholar

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J. O. Ledyard, Public goods: A survey of experimental research, in The Handbook of Experimental Economics (ed. J. H. Kagel), Princeton University Press, 1995. Google Scholar

[22] W. H. Sandholm, Population Games and Evolutionary Dynamics, The MIT Press, Cambridge, MA, 2010.   Google Scholar
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L. WardilI. R. Silva and J. K. L. da Silva, Positive interactions may decrease cooperation in social dilemma experiments, Sci. Rep., 9 (2019), 1017.  doi: 10.1038/s41598-018-37674-5.  Google Scholar

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Wikipedia contributors, Gershgorin circle theorem — Wikipedia, the free encyclopedia, https://en.wikipedia.org/w/index.php?title=Gershgorin_circle_theorem&oldid=1003302183, 2021, [Online; accessed 6-February-2021]. Google Scholar

show all references

References:
[1]

M. Broom, R. Cressman and V. Křivan, Revisiting the "fallacy of averages" in ecology: Expected gain per unit time equals expected gain divided by expected time, J. Theor. Biol., 483 (2019), 109993, 6pp. doi: 10.1016/j.jtbi.2019.109993.  Google Scholar

[2]

M. Broom and V. Křivan, Two-strategy games with time constraints on regular graphs, J. Theor. Biol., 506 (2020), 110426, 13pp. doi: 10.1016/j.jtbi.2020.110426.  Google Scholar

[3]

A. Chaudhuri, Recent advances in experimental studies of social dilemma games, Games, 7 (2016), Paper No. 7, 11 pp. doi: 10.3390/g7010007.  Google Scholar

[4]

G. Craciun, Toric differential inclusions and the proof of the global attractor conjucture, arXiv: 1501.02860, 2015. Google Scholar

[5]

G. CraciunA. DickensteinA. Shiu and B. Sturmfels, Toric dynamical systems, J. Symb. Comp., 44 (2009), 1551-1565.  doi: 10.1016/j.jsc.2008.08.006.  Google Scholar

[6]

R. Cressman and V. Křivan, Bimatrix games that include interaction times alter the evolutionary outcome: The Owner–Intruder game, J. Theor. Biol., 460 (2019), 262–273. doi: 10.1016/j.jtbi.2018.10.033.  Google Scholar

[7]

R. Cressman and V. Křivan, Reducing courtship time promotes marital bliss: The battle of the sexes game revisited with costs measured as time lost, J. Theor. Biol., 503 (2020), 110382, 14pp. doi: 10.1016/j.jtbi.2020.110382.  Google Scholar

[8]

R. M. Dawes, Social dilemmas, Ann. Rev. Psychol., 31 (1980), 169-193.   Google Scholar

[9] R. Dawkins, The Selfish Gene, Oxford University Press, Oxford, 1976.   Google Scholar
[10]

M. Feinberg, Foundations of Chemical Reaction Network Theory, Springer, 2019.  Google Scholar

[11]

J. Garay, V. Csiszár and T. F. Móri, Evolutionary stability for matrix games under time constraints, J. Theor. Biol., 415 (2017), 1–12. doi: 10.1016/j.jtbi.2016.11.029.  Google Scholar

[12]

J. A. P. Heesterbeek and J. A. J. Metz, The saturating contact rate in marriage- and epidemic models, J. Math. Biol., 31 (1993), 529-539.  doi: 10.1007/BF00173891.  Google Scholar

[13] J. Hofbauer and K. Sigmund, Evolutionary Games and Population Dynamics, Cambridge University Press, Cambridge, UK, 1998.  doi: 10.1017/CBO9781139173179.  Google Scholar
[14]

R. D. Holt and M. Barfield, On the relationship between the ideal-free distribution and the evolution of dispersal, in Dispersal (eds. J. C. E. Danchin, A. Dhondt and J. Nichols), Oxford University Press, 2001, 83–95. Google Scholar

[15]

F. Horn, Necessary and sufficient conditions for complex balancing in chemical kinetics, Arch. Ration. Mech. Anal., 49 (1972), 172-186.  doi: 10.1007/BF00255664.  Google Scholar

[16]

F. Horn, The dynamics of open reaction systems, in Mathematical Aspects of Chemical and Biochemical Problems Amd Quantum Chemistry, vol. 8 of Proceedings of the SIAM-AMS Symposium on Applied Mathematics, 1974,125–137.  Google Scholar

[17]

F. Horn and R. Jackson, General mass action kinetics, Arch. Ration. Mech. Anal., 47 (1972), 81-116.  doi: 10.1007/BF00251225.  Google Scholar

[18]

V. Křivan and R. Cressman, Defectors' intolerance of others promotes cooperation in the repeated public goods game with opting out, Sci. Rep., 10 (2020), 19511.  doi: 10.1038/s41598-020-76506-3.  Google Scholar

[19]

V. Křivan and R. Cressman, Interaction times change evolutionary outcomes: Two player matrix games, J. Theor. Biol., 416 (2017), 199–207. doi: 10.1016/j.jtbi.2017.01.010.  Google Scholar

[20]

V. Křivan, T. Galanthay and R. Cressman, Beyond replicator dynamics: From frequency to density dependent models of evolutionary games, J. Theor. Biol., 455 (2018), 232–248. doi: 10.1016/j.jtbi.2018.07.003.  Google Scholar

[21]

J. O. Ledyard, Public goods: A survey of experimental research, in The Handbook of Experimental Economics (ed. J. H. Kagel), Princeton University Press, 1995. Google Scholar

[22] W. H. Sandholm, Population Games and Evolutionary Dynamics, The MIT Press, Cambridge, MA, 2010.   Google Scholar
[23] K. Sigmund, The Calculus of Selfishness, Princeton University Press, Princeton, NJ, USA, 2010.  doi: 10.1515/9781400832255.  Google Scholar
[24]

L. WardilI. R. Silva and J. K. L. da Silva, Positive interactions may decrease cooperation in social dilemma experiments, Sci. Rep., 9 (2019), 1017.  doi: 10.1038/s41598-018-37674-5.  Google Scholar

[25]

Wikipedia contributors, Gershgorin circle theorem — Wikipedia, the free encyclopedia, https://en.wikipedia.org/w/index.php?title=Gershgorin_circle_theorem&oldid=1003302183, 2021, [Online; accessed 6-February-2021]. Google Scholar

Figure 1.  Distributional dynamics (10) of pairs (panel A) and groups of size four (panel B) for PGG (20). Both panels assume that initially there are only singles and initial conditions are $ n_1(0) = n_2(0) = 5 $ ($ n_C(0) = n_D(0) = 5 $) in panel A (panel B). Parameters used in Panel A: $ \lambda_{11} = \lambda_{12} = \lambda_{21} = \lambda_{22} = \lambda = 0.1 $, $ \tau_{11} = 5 $, $ \tau_{12} = \tau_{21} = 3 $, $ \tau_{22} = 1 $. Parameters used in Panel B: $ \lambda_i = \frac{\lambda}{4}\binom{4}{i} $ with $ \lambda = 0.05 $, $ \tau_{0} = 1 $, $ \tau_{1} = 2 $, $ \tau_{2} = 4 $, $ \tau_{3} = 6 $, $ \tau_4 = 15 $
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