We discussed the full unitary matrix models from the viewpoints of integrable equations and string equations. Coupling the Toda equations and the string equations, we derive a special case of the Painlevé III equation. From the Virasoro constraints, we can use the radial coordinate. The relation between t1 and t−1 is like the complex conjugate.

In this paper, we discuss a voting model with two candidates, C 1 and C 2 . We set two types of voters-herders and independents. The voting of independent voters is based on their fundamental values; on the other hand, the voting of herders is based on the number of votes. Herders always select the majority of the previous r votes, which is visible to them. We call them digital herders. We can accurately calculate the distribution of votes for special cases. When r ≥ 3, we find that a phase transition occurs at the upper limit of t, where t is the discrete time (or number of votes). As the fraction of herders increases, the model features a phase transition beyond which a state where most voters make the correct choice coexists with one where most of them are wrong. On the other hand, when r < 3, there is no * [1] masato hisakado@standardandpoors.com † [2] mori@sci.kitasato-u.ac.jp phase transition. In this case, the herders' performance is the same as that of the independent voters. Finally, we recognize the behavior of human beings by conducting simple experiments.

We introduce a voting model that is similar to a Keynesian beauty contest and analyze it from a mathematical point of view. There are two types of voters-copycat and independent-and two candidates. Our voting model is a binomial distribution (independent voters) doped in a beta binomial distribution (copycat voters). We find that the phase transition in this system is at the upper limit of t, where t is the time (or the number of the votes). Our model contains three phases. If copycats constitute a majority or even half of the total voters, the voting rate converges more slowly than it would in a binomial distribution. If independents constitute the majority of voters, the voting rate converges at the same rate as it would in a binomial distribution. We also study why it is difficult to estimate the conclusion of a Keynesian beauty contest when there is an information cascade. * [1] masato hisakado@standardandpoors.com

We discuss a general method to construct correlated binomial distributions by imposing several consistent relations on the joint probability function. We obtain self-consistency relations for the conditional correlations and conditional probabilities. The beta-binomial distribution is derived by a strong symmetric assumption on the conditional correlations. Our derivation clarifies the 'correlation' structure of the beta-binomial distribution. It is also possible to study the correlation structures of other probability distributions of exchangeable (homogeneous) correlated Bernoulli random variables. We study some distribution functions and discuss their behaviors in terms of their correlation structures.

Observational learning is an important information aggregation mechanism. However, it occasionally leads to a state in which an entire population chooses a suboptimal option. When this occurs and whether it is a phase transition remain unanswered. To address these questions we perform a voting experiment in which subjects answer a two-choice quiz sequentially with and without information about the prior subjects' choices. The subjects who could copy others are called herders. We obtain a microscopic rule regarding how herders copy others. Varying the ratio of herders leads to qualitative changes in the macroscopic behavior of about 50 subjects in the experiment. If the ratio is small, the sequence of choices rapidly converges to the correct one. As the ratio approaches 100%, convergence becomes extremely slow and information aggregation almost terminates. A simulation study of a stochastic model for 10(6) subjects based on the herder's microscopic rule shows a phase transition to the two-peak phase, where the convergence completely terminates as the ratio exceeds some critical value.