The evolution of n-player cooperation-threshold games and ESS bifurcations

Area of Science:

• Evolutionary game theory
• Mathematical biology
• Behavioral economics

Background:

• Cooperation is crucial for collective benefits in biological systems like group foraging.
• Traditional models often use two-player games, which may not fully capture multi-player dynamics.

Purpose of the Study:

• To model cooperation in n-player Prisoner's Dilemma games with a threshold for collective benefit.
• To investigate the emergence and stability of cooperation in multi-player scenarios.
• To highlight differences between two-player and n-player game dynamics.

Main Methods:

• Modeling cooperation as an n-player Prisoner's Dilemma game.
• Introducing a threshold condition for the number of cooperators needed for collective benefit.
• Analyzing replicator dynamics and Evolutionarily Stable Strategies (ESS) bifurcations.

Main Results:

• A three-player game exhibits complex behavior, including a catastrophic event separating coexistence and pure defection.
• Cooperation emerges via an ESS bifurcation, thriving only beyond a critical cost-benefit point.
• A repelling fixed point acts as a barrier to introducing cooperation into defecting populations.

Conclusions:

• N-player games show qualitative differences from two-player games, limiting the generality of findings from simpler models.
• A procedure is presented for determining ESS in n-player games with variable costs and benefits.
• Threshold conditions significantly impact cooperation dynamics in multi-player evolutionary games.