The evolution of n-player cooperation-threshold games and ESS bifurcations |
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Authors: | Bach L A Helvik T Christiansen F B |
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Institution: | Department of Theoretical Ecology, Ecology Building, Lund University, SE-223 62 Lund, Sweden. lars.bach@teorekol.lu.se |
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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. |
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Keywords: | _method=retrieve& _eid=1-s2 0-S0022519305002419& _mathId=si8 gif& _pii=S0022519305002419& _issn=00225193& _acct=C000054348& _version=1& _userid=3837164& md5=c78b13ac5c1ad1d3098640b34e370809')" style="cursor:pointer n-Player game" target="_blank">" alt="Click to view the MathML source" title="Click to view the MathML source">n-Player game Prisoner's Dilemma Cooperation Bifurcation Hysteresis |
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