Heterogeneous Mixed Populations of Best-Responders and Imitators: Equilibrium Convergence and Stability

• Published in: IEEE transactions on automatic control Vol. 66; no. 8; pp. 3475 - 3488
• Main Authors: Le, Hien, Ramazi, Pouria
• Format: Journal Article
• Language: English
• Published: New York IEEE 01.08.2021; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Anticoordination game; best-response; Convergence; Decision making; Equilibrium; evolutionary game theory; Game theory; Games; heterogeneous population; imitation; Population; Populations; Resource allocation; Silicon; Sociology; Stability analysis; Statistics; Strategy
• ISSN: 0018-9286; 1558-2523
• DOI: 10.1109/TAC.2020.3016903

In anticoordination social contexts such as stock selection, resource allocation, and crowd dispersion, an individual earns more if the opponents adopt her opposite strategy. Based on their experience and available information, individuals may either evaluate all available options and decide on the most profitable one, or simply mimic successful others. These two types of decision-makers are known as best-responders and imitators , respectively. Previous studies have shown that in anticoordination social contexts, a population of best-responders reaches an equilibrium state, where every individual is satisfied with her decision, but a population of imitators is quite likely to never settle and undergo perpetual fluctuations. Most real-world populations, however, consist of both types of individuals, and it remains an open problem whether such mixed-populations eventually reach an equilibrium state. We provide a sharp, yet simple answer to this question: the population almost surely reaches an equilibrium if and only if it admits one. More specifically, we study a well-mixed population of both best-responders and imitators playing anticoordination games with two available strategies, cooperation and defection , and earning according to payoff matrices that can be unique to each player, resulting in a heterogeneous population. The individuals update their strategies asynchronously accordingly to their types: best-responders choose the strategy that maximizes their payoffs against the population and imitators copy the strategy of the individual earning the highest payoff. We find the necessary and sufficient condition for the population dynamics to admit an equilibrium, identify all possible equilibria, investigate their stability and perform convergence analysis.