Kernkonzepte
Evolutionarily stable strategies (ESS) in asymmetric games, where players have different roles, can be defined in various ways. The dynamical stability of these ESS definitions under the replicator equation is analyzed, and connections between game theory, dynamical systems, and information theory are established.
Zusammenfassung
The paper discusses the concept of evolutionarily stable strategy (ESS) in asymmetric games, where players are categorized into different roles and may have different strategy sets. It presents and analyzes various definitions of ESS in such games, including:
Definitions 1a, 1b, and 1c: These definitions require the resident strategy to outperform the mutant strategy in both subpopulations simultaneously.
Definitions 2a, 2b, and 2c: These definitions compare the combined fitness of the resident and mutant strategies across both subpopulations.
Definitions 3a, 3b, and 3c: These definitions relax the requirement for the resident to outperform the mutant in both subpopulations simultaneously, and only require one subpopulation to have a higher fitness.
The paper establishes the equivalence between these different definitions of ESS. It then examines the dynamical stability of these ESS under the replicator equation, showing that the ESS defined by Definitions 1c and 2c correspond to asymptotically stable fixed points, while the 2ESS defined by Definition 3c may not be asymptotically stable.
The paper further investigates the effect of introducing intraspecific interactions, which transforms the bimatrix game into an asymmetric game. It shows that for pure ESS, the definitions requiring the resident strategy to outperform the mutant in both subpopulations (Definitions 1d and 2d) correspond to asymptotically stable fixed points of the replicator equation, while the weaker 2ESS definition (Definition 3d) may not.
The paper also explores the connections between game theory, dynamical systems, and information theory by invoking the concept of relative entropy to gain insights into the ESS in asymmetric games. Finally, the results are generalized to the case of multiplayer (hypermatrix) games.