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Phase transition in the Kolkata Paise Restaurant problem.

A novel phase transition behavior is observed in the Kolkata Paise Restaurant problem where a large number (N) of agents or customers collectively (and iteratively) learn to choose among the N restaurants where she would expect to be alone that evening and would get the only dish available there (or may get randomly picked up if more than one agent arrive there that evening). The players are expected to evolve their strategy such that the publicly available information about past crowds in different restaurants can be utilized and each of them is able to make the best minority choice. For equally ranked restaurants, we follow two crowd-avoiding strategies: strategy I, where each of the ni(t) number of agents arriving at the ith restaurant on the tth evening goes back to the same restaurant the next evening with probability [ni(t)]-α, and strategy II, with probability p, when ni(t)>1. We study the steady state (t-independent) utilization fraction f:(1-f) giving the steady state (wastage) fraction of restaurants going without any customer at any particular evening. With both strategies, we find, near αc=0+ (in strategy I) or p=1- (in strategy II), the steady state wastage fraction (1-f)∝(α-αc)β or (pc-p)β with β≃0.8,0.87,1.0, and the convergence time τ [for f(t) becoming independent of t] varies as τ∝(α-αc)-γ or (pc-p)-γ, with γ≃1.18,1.11,1.05 in infinite-dimensions (rest of the N-1 neighboring restaurants), three dimensions (six neighbors), and two dimensions (four neighbors), respectively.