Collective behaviour in binary games with memory

Cyclic dynamics are displayed in many social, economic, physical and biological systems governed by negative frequency-dependent selection and delayed feedback. Here, I study the emergence and collapse of cyclic dynamics in an anti-coordination binary game with memory. In this model, individuals are faced with two options and aim to choose the option adopted by a minority in the group. Between adaptation moves, individuals learn about the behaviour of others by attending to and recollecting their choices. I introduce a novel double-fold definition of individual memory, incorporating a rate of observation (how often one collects new information) and duration (how long one remembers information). In the context of bounded rationality, these parameters mirror limitations due to cognitive and environmental constraints. I show that finite and infinite observation rates generate different collective dynamics. In the limit of an infinite rate, the population exhibits deterministic and thus cyclic dynamics. In contrast, finite rates generate noise, enriching the game dynamics with different equilibria. The time evolution of the frequency of play of the two options in the group is investigated theoretically and computationally, and characterized as a function of the system parameters, particularly in terms of amplitude and period. The onset, decay and robustness of cyclic behaviour is discussed. To make this more concrete, the dynamics at play are illustrated with a specific example from fashion, intended as a sociological subject with collective and individual dynamics and imitation and distinction motifs. Applications to a number of other social, economic and biological contexts are also discussed.