# American Institute of Mathematical Sciences

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 and 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. [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. [3] A. Chaudhuri, Recent advances in experimental studies of social dilemma games, Games, 7 (2016), Paper No. 7, 11 pp. doi: 10.3390/g7010007. [4] G. Craciun, Toric differential inclusions and the proof of the global attractor conjucture, arXiv: 1501.02860, 2015. [5] G. Craciun, A. Dickenstein, A. Shiu and B. Sturmfels, Toric dynamical systems, J. Symb. Comp., 44 (2009), 1551-1565.  doi: 10.1016/j.jsc.2008.08.006. [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. [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. [8] R. M. Dawes, Social dilemmas, Ann. Rev. Psychol., 31 (1980), 169-193. [9] R. Dawkins, The Selfish Gene, Oxford University Press, Oxford, 1976. [10] M. Feinberg, Foundations of Chemical Reaction Network Theory, Springer, 2019. [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. [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. [13] J. Hofbauer and K. Sigmund, Evolutionary Games and Population Dynamics, Cambridge University Press, Cambridge, UK, 1998.  doi: 10.1017/CBO9781139173179. [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. [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. [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. [17] F. Horn and R. Jackson, General mass action kinetics, Arch. Ration. Mech. Anal., 47 (1972), 81-116.  doi: 10.1007/BF00251225. [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. [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. [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. [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. [22] W. H. Sandholm, Population Games and Evolutionary Dynamics, The MIT Press, Cambridge, MA, 2010. [23] K. Sigmund, The Calculus of Selfishness, Princeton University Press, Princeton, NJ, USA, 2010.  doi: 10.1515/9781400832255. [24] L. Wardil, I. 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. [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].

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. [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. [3] A. Chaudhuri, Recent advances in experimental studies of social dilemma games, Games, 7 (2016), Paper No. 7, 11 pp. doi: 10.3390/g7010007. [4] G. Craciun, Toric differential inclusions and the proof of the global attractor conjucture, arXiv: 1501.02860, 2015. [5] G. Craciun, A. Dickenstein, A. Shiu and B. Sturmfels, Toric dynamical systems, J. Symb. Comp., 44 (2009), 1551-1565.  doi: 10.1016/j.jsc.2008.08.006. [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. [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. [8] R. M. Dawes, Social dilemmas, Ann. Rev. Psychol., 31 (1980), 169-193. [9] R. Dawkins, The Selfish Gene, Oxford University Press, Oxford, 1976. [10] M. Feinberg, Foundations of Chemical Reaction Network Theory, Springer, 2019. [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. [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. [13] J. Hofbauer and K. Sigmund, Evolutionary Games and Population Dynamics, Cambridge University Press, Cambridge, UK, 1998.  doi: 10.1017/CBO9781139173179. [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. [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. [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. [17] F. Horn and R. Jackson, General mass action kinetics, Arch. Ration. Mech. Anal., 47 (1972), 81-116.  doi: 10.1007/BF00251225. [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. [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. [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. [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. [22] W. H. Sandholm, Population Games and Evolutionary Dynamics, The MIT Press, Cambridge, MA, 2010. [23] K. Sigmund, The Calculus of Selfishness, Princeton University Press, Princeton, NJ, USA, 2010.  doi: 10.1515/9781400832255. [24] L. Wardil, I. 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. [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].
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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