The Emergence of Chaos in Population Game Dynamics Induced by Comparisons

Precise description of population game dynamics introduced by revision protocols—an economic model describing the agent’s propensity to switch to a better-performing strategy—is of importance in economics and social sciences in general. In this setting imitation of others and innovation are forces which drive the evolution of the economic system. As the continuous-time game dynamics is relatively well understood, the same cannot be said about revision driven dynamics in the discrete time. We investigate the behavior of agents using revision protocols in a $$2\times 2$$ anti-coordination game with symmetric random matching and a unique mixed Nash equilibrium. We show that in discrete time one can construct a revision protocol (either innovative or imitative) such that if a large enough fraction of agents revise their choices the game dynamics becomes Li-Yorke chaotic, inducing complex and unpredictable behavior of the system, precluding stable predictions of equilibrium. This is in stark contrast to the continuous case. Moreover, we reveal that this unpredictability is encoded in any imitative revision protocol. Furthermore, we show that for any such game there exists a perturbed pairwise proportional imitation protocol introducing chaotic behavior of agents when a sufficiently large part of the population reconsiders their strategies.