Adiabatic theory for the population distribution in the evolutionary minority game

We study the evolutionary minority game (EMG) using a statistical mechanics approach. We derive a theory for the steady-state population distribution of the agents. The theory is based on an "adiabatic approximation" in which short time fluctuations in the population distribution are integrated out to obtain an effective equation governing the steady-state distribution. We discover the mechanism for the transition from segregation (into opposing groups) to clustering (towards cautious behaviors). The transition is determined by two generic factors: the market impact (of the agents' own actions) and the short time market inefficiency (arbitrage opportunities) due to fluctuations in the numbers of agents using opposite strategies. A large market impact favors "extreme" players who choose fixed opposite strategies, while large market inefficiency favors cautious players. The transition depends on the number of agents (N) and the effective rate of strategy switching. When N is small, the market impact is relatively large; this favors the extreme behaviors. Frequent strategy switching, on the other hand, leads to a clustering of the cautious agents.