title: The evolution of n-player cooperation-threshold games and ESS bifurcations.
abstract: An evolutionary game of individuals cooperating to obtain a collective benefit is here modelled as an n-player Prisoner's Dilemma game. With reference to biological situations, such as group foraging, we introduce a threshold condition in the number of cooperators required to obtain the collective benefit. In the simplest version, a three-player game, complex behaviour appears as the replicator dynamics exhibits a catastrophic event separating a parameter region allowing for coexistence of cooperators and defectors and a region of pure defection. Cooperation emerges through an ESS bifurcation, and cooperators only thrive beyond a critical point in cost-benefit space. Moreover, a repelling fixed point of the dynamics acts as a barrier to the introduction of cooperation in defecting populations. The results illustrate the qualitative difference between two-player games and multiple player games and thus the limitations to the generality of conclusions from two-player games. We present a procedure to find the evolutionarily stable strategies in any n-player game with cost and benefit depending on the number of cooperators. This was previously done by Motro [1991. Co-operation and defection: playing the field and the ESS. J. Theor. Biol. 151, 145-154] in the special cases of convex and concave benefit functions and constant cost.
authors.affiliation: Department of Theoretical Ecology, Ecology Building, Lund University, SE-223 62 Lund, Sweden. lars.bach@teorekol.lu.se
authors.forename: L A
authors.initials: LA
authors.lastname: Bach
authors.forename: T
authors.initials: T
authors.lastname: Helvik
authors.forename: F B
authors.initials: FB
authors.lastname: Christiansen
mesh_terms: D000818:Animals; D005075:Biological Evolution; D003299:Cooperative Behavior; D005716:Game Theory; D008954:Models, Biological; D011157:Population Dynamics