title: Gains from switching and evolutionary stability in multi-player matrix games.
abstract: In this paper we unify, simplify, and extend previous work on the evolutionary dynamics of symmetric N-player matrix games with two pure strategies. In such games, gains from switching strategies depend, in general, on how many other individuals in the group play a given strategy. As a consequence, the gain function determining the gradient of selection can be a polynomial of degree N-1. In order to deal with the intricacy of the resulting evolutionary dynamics, we make use of the theory of polynomials in Bernstein form. This theory implies a tight link between the sign pattern of the gains from switching on the one hand and the number and stability of the rest points of the replicator dynamics on the other hand. While this relationship is a general one, it is most informative if gains from switching have at most two sign changes, as is the case for most multi-player matrix games considered in the literature. We demonstrate that previous results for public goods games are easily recovered and extended using this observation. Further examples illustrate how focusing on the sign pattern of the gains from switching obviates the need for a more involved analysis.
authors.affiliation: Faculty of Business and Economics, University of Basel, Peter Merian-Weg 6, CH-4002 Basel, Switzerland. Electronic address: pena@evolbio.mpg.de.
authors.forename: Jorge
authors.initials: J
authors.lastname: Peña
authors.affiliation: Department of Ecology and Evolution, University of Lausanne, Le Biophore, CH-1015 Lausanne, Switzerland. Electronic address: laurent.lehmann@unil.ch.
authors.forename: Laurent
authors.initials: L
authors.lastname: Lehmann
authors.affiliation: Faculty of Business and Economics, University of Basel, Peter Merian-Weg 6, CH-4002 Basel, Switzerland. Electronic address: georg.noeldeke@unibas.ch.
authors.forename: Georg
authors.initials: G
authors.lastname: Nöldeke
mesh_terms: D005075:Biological Evolution; D005716:Game Theory; D006801:Humans; D016014:Linear Models; D008954:Models, Biological
keywords: Evolutionary game theory; Polynomials in Bernstein form; Public goods games; Replicator dynamics