Detection of Phase Transition in Generalized Pólya Urn in Information Cascade Experiment

Abstract

We propose a method of detecting a phase transition in a generalized Pólya urn in an information cascade experiment. The method is based on the asymptotic behavior of the correlation C(t) between the first subject’s choice and the t + 1-th subject’s choice, the limit value of which, \(c \equiv \lim_{t \to \infty }C(t)\), is the order parameter of the phase transition. To verify the method, we perform a voting experiment using two-choice questions. An urn X is chosen at random from two urns A and B, which contain red and blue balls in different configurations. Subjects sequentially guess whether X is A or B using information about the prior subjects’ choices and the color of a ball randomly drawn from X. The color tells the subject which is X with probability q. We set \(q \in \{ 5/9,6/9,7/9,8/9\} \) by controlling the configurations of red and blue balls in A and B. The (average) lengths of the sequence of the subjects are 63, 63, 54.0, and 60.5 for \(q \in \{ 5/9,6/9,7/9,8/9\} \), respectively. We describe the sequential voting process by a nonlinear Pólya urn model. The model suggests the possibility of a phase transition when q changes. We show that c > 0 (= 0) for \(q = 5/9,6/9\) \((7/9,8/9)\) and detect the phase transition using the proposed method.