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This study generalizes evolutionary stable strategy (ESS) to games with variable player numbers, showing mixed equilibria exist and are stable. Network simulations reveal phase transitions depend on interaction probability and network structure.

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Area of Science:

  • Evolutionary Game Theory
  • Complex Systems
  • Network Science

Background:

  • Evolutionary games typically analyze pairwise interactions.
  • Existing models often assume a fixed number of players per interaction.
  • Real-world scenarios frequently involve interactions with varying numbers of participants.

Purpose of the Study:

  • To generalize the concept of evolutionary stable strategy (ESS) for games with a variable number of players.
  • To analyze the "duel-truel" game, where interactions can involve two or three players with a given probability.
  • To investigate the dynamics of these games on complex networks using an agent-based model.

Main Methods:

  • Formalizing and generalizing the ESS definition for variable player interactions.
  • Analyzing the duel-truel game for pure and mixed strategy equilibria.
  • Developing an agent-based model on networks to simulate evolutionary dynamics.
  • Deriving mean-field and pair approximation equations for network dynamics.

Main Results:

  • Existence of pure and mixed strategy equilibria is shown, dependent on the interaction probability (p).
  • A specific range of p values leads to mixed equilibria where strategy proportions vary with p.
  • All identified mixed equilibria are proven to be ESS.
  • Network simulations demonstrate that phase transitions between equilibria are influenced by p and the network's mean degree.

Conclusions:

  • The generalized ESS framework accommodates variable player interactions effectively.
  • Mixed equilibria in the duel-truel game are robust and depend on interaction probabilities.
  • Network structure significantly impacts evolutionary dynamics in games with higher-order interactions.