Evol Ecol Res 16: 293-314 (2014)     Full PDF if your library subscribes.

ESS versus Nash: solving evolutionary games

Joe Apaloo1, Joel S. Brown2, Gordon G. McNickle3, Tania L.S. Vincent4 and Thomas L. Vincent5

1Department of Mathematics, Statistics and Computer Science, St. Francis Xavier University, Antigonish, Nova Scotia, Canada,  2Department of Biological Sciences, University of Illinois at Chicago, Chicago, Illinois, USA,  3Biology Department, Wilfrid Laurier University, Waterloo, Ontario, Canada,  4Alaska Pacific University, Anchorage, Alaska, USA and  5Department of Aerospace and Mechanical Engineering, University of Arizona, Tucson, Arizona, USA

Correspondence: J. Apaloo, Department of Mathematics, Statistics and Computer Science, St. Francis Xavier University, PO Box 5000, Antigonish, Nova Scotia B2G 2W5, Canada.
e-mail: japaloo@stfx.ca

ABSTRACT

Question: Can a Nash solution concept be used for the analysis of evolutionary games? What are the advantages of ESS solutions over Nash solutions in evolutionary games?

Mathematical methods: ESS and Nash equilibrium solution concepts and Darwinian dynamics for evolutionary games.

Conclusions: An ESS contains the properties of Nash but not the other way around. The properties of evolutionary games make them fit poorly with the classical notion of a Nash solution. These properties include six key differences between the ESS and Nash concepts: (1) in the evolutionary game, players inherit rather than choose their strategies; (2) the focus of evolutionary games is on the strategies and not on the actual players who come and go via births and deaths; (3) the payoffs in the evolutionary game represent fitness, creating a dynamical link between payoffs and changes in the frequency of strategies; (4) in state-dependent games, players in a classical game may possess forethought and anticipate the consequences of their actions, whereas in the evolutionary game organisms do not; (5) unlike classical games, the actual number of players in the game can expand and contract via changes in population sizes; and (6) evolutionary games have a particular kind of symmetry where collections of individuals may have different strategies, yet their strategies all arise from the same set of evolutionarily feasible strategies (pure or mixed) and each experiences the same fitness consequence of possessing a particular strategy. For these reasons, we argue that the use of Nash as a solution concept in evolutionary games will be misleading, and that the ESS deserves primacy over Nash as the solution concept for evolutionary games.

Keywords: Nash equilibrium, ESS, Darwinian dynamics.