TRAJECTORIES OF DYNAMIC EQUILIBRIUM AND REPLICATOR DYNAMICS IN COORDINATION GAMES

Nikolay A. Krasovskii     (Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, 16 S. Kovalevskaya Str., Ekaterinburg, 620108, Russian Federation)
Alexander M. Tarasyev     (Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences,16 S. Kovalevskaya Str., Ekaterinburg, 620108, Russian Federation; Ural Federal University, 19 Mira Str., Ekaterinburg, 620002, Russian Federation)

Abstract


The paper analyzes average integral payoff indices for trajectories of the dynamic equilibrium and replicator dynamics in bimatrix coordination games. In such games, players receive large payoffs when choosing the same type of behavior. A special feature of a \(2\times2\) coordination game is the presence of three static Nash equilibria. In the dynamic formulation, the trajectories of coordination games are estimated by the average integral payoffs for a wide range of models arising in economics and biology. In optimal control problems and dynamic games, average integral payoffs are used to synthesize guaranteed strategies, which are involved, among other things, in the constructions of the dynamic Nash equilibrium. In addition, average integral payoffs are a natural tool for assessing the quality of trajectories of replicator dynamics. In the paper, we compare values of average integral indices for trajectories of replicator dynamics and trajectories generated by guaranteed strategies in constructing the dynamic Nash equilibrium. An analysis is provided for trajectories of mixed dynamics when the first player plays a guaranteed strategy, and the behavior of replicator dynamics guides the second player.


Keywords


Dynamic bimatrix games, Coordination games, Average integral payoffs, Guaranteed strategies, Replicator dynamics, Dynamic Nash equilibrium

Full Text:

PDF

References


  1. Arnold V.I. Optimization in mean and phase transitions in controlled dynamical systems. Funct. Anal. Appl., 2002. Vol. 36, No. 2. P. 83–92. DOI: 10.1023/A:1015655005114
  2. Bratus A.S., Novozhilov A.S., Platonov A.P. Dinamicheskiye systemy i modeli biologii [Dynamic Systems and Models of Biology]. Moscow: Fizmatlit, 2010. 400 p. (in Russian)
  3. Hofbauer J., Sigmund K. The Theory of Evolution and Dynamical Systems. Cambridge etc.: Cambridge Univ. Press, 1988. 341 p. DOI: 10.1002/zamm.19900700210
  4. Kleimenov A.F. Neantagonisticheskiye pozitsionniye differentsial’niye igry [Non-antagonistic Positional Differential Games]. Yekaterinburg: Nauka, 1993. 185 p. (in Russian)
  5. Krasovskii A.N., Krasovskii N.N. Control Under Lack of Information. Boston: Burkhäuser, 1995. 322 p. DOI: 10.1007/978-1-4612-2568-3
  6. Krasovskii N.A., Tarasyev A.M. Equilibrium trajectories in dynamical bimatrix games with average integral payoff functionals. Autom. Remote Control, 2018. Vol. 79, No. 6. P. 1148–1167. DOI: 10.1134/S0005117918060139
  7. Krasovskii N.N. Upravleniye dinamicheskoy sistemoy [Control of Dynamic System]. Moscow: Nauka, 1985. 520 p. (in Russian)
  8. Krasovskii N.N., Subbotin A.I. Game–Theoretical Control Problems. New-York: Springer–Verlag, 1988. 517 p.
  9. Mazalov V.V., Rettieva A.N. Application of bargaining schemes for equilibrium determination in dynamic games. Mat. Teor. Igr Pril., 2023. Vol. 15., No. 2. P. 75–88. (in Russian)
  10. Mertens J.-F., Sorin S., Zamir S. Repeated Games. Cambridge: Cambridge University Press, 2015. 567 p.
  11. Petrosjan L.A., Zenkevich N.A. Conditions for sustainable cooperation. Autom. Remote Control, 2015. Vol. 76. P. 1894–1904. DOI: 10.1134/S0005117915100148
  12. Pontryagin L.S., Boltyanskii V.G., Gamkrelidze R.V., Mischenko E.F. The Mathematical Theory of Optimal Processes. New-York: Wiley Interscience, 1962. 360 p.
  13. Sorin S. Replicator dynamics: old and new. J. Dynam. Games, 2020. Vol. 7, No. 4. P. 365–386. DOI: 10.3934/jdg.2020028
  14. Vinnikov E.V., Davydov A.A., Tunitskiy D.V. Existence of maximum of time averaged harvesting in the KPP-model on sphere with permanent and impulse harvesting. Dokl. Math., 2023. Vol. 108, No. 3. P. 472–476. DOI: 10.1134/S1064562423701387
  15. Vorobyev N.N. Teoriya igr dlya ekonomistov–kibernetikov [The Theory of Games for Economists–Cyberneticians]. Moscow: Nauka, 1985. 272 p. (in Russian)
  16. Yakushkina T.S. A distributed replicator system corresponding to a bimatrix game. Moscow Univ. Comput. Math. Cybernet., 2016. Vol. 40, No. 1. P. 19–27. DOI: 10.3103/S0278641916010064




DOI: http://dx.doi.org/10.15826/umj.2024.2.009

Article Metrics

Metrics Loading ...

Refbacks

  • There are currently no refbacks.