Detection of phase transition in generalized P\'olya urn in information cascade experiment

Abstract

We propose a method of detecting a phase transition in a generalized P\'olya 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 P\'olya 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.