Information Cascade, Kirman’s Ant Colony Model, and Kinetic Ising Model

Abstract

We discuss a voting model in which voters can obtain information from a finite number of previous voters. It is the equilibrium process. There exist three groups of voters: (i) digital herders and independent voters, (ii) analog herders and independent voters, and (iii) $\tanh$-type herders. In the case (i), we show that the solution oscillates between the two states. A good (bad) equilibrium is where a majority of r select the correct (wrong) candidate. We show that there is no phase transition when r is finite. If the annealing schedule is adequately slow from finite r to infinite r, the voting rate converges only to the good equilibrium. In case (ii), the state of reference votes is equivalent to that of Kirman’s ant colony model, and it follows beta-binomial distribution. In case (iii), we show that the model is equivalent to the finite-size kinetic Ising model. If the voters are rational, a simple herding experiment of information cascade is conducted. Information cascade results from the quenching of the kinetic Ising model. As case (i) is the limit of case (iii) when $\tanh$ function becomes a step function, the phase transition can be observed in infinite-size limit. We can confirm that there is no phase transition when the reference number r is finite. This chapter is based on Hisakado and Mori (Physica A 417:63–75, 2015).

Appendices

Appendix A Ising Model

Here we introduce the infinite-range model. It is one of the most popular model in statistical physics which explains the phase transition. In the model spins interact all other spins. Hamiltonian is

$$\displaystyle \begin{aligned} H(\sigma)=-\frac{J}{N}\sum_{i>j} \sigma_i \sigma_j -h\sum_{i=1}^N \sigma_i, \end{aligned} $$

(29)

where N is the number of spins, σ is the spin that has the value ± 1, J is the parameter of interaction, and h is the outer field. Here we define average of spins, as order parameter, $m=1/N\sum \sigma _i$.

In large N →∞ limit, the self-consistent equation is

$$\displaystyle \begin{aligned} m=\tanh \beta(Jm+h), \end{aligned} $$

(30)

where β = 1∕k _B T. k _B is Boltzmann constant and T is temperature. When infinite-range model, we can obtain the strict solution from the self-consistent equation. When the symmetric case, under the transition temperature T _C, there are two solutions. One of the solution is selected, and it is called spontaneous symmetry braking. On the other hand, above T _c, there is only one solution. It is the simple model which represents the phase transition. When there is the outer field h, the model becomes the asymmetric model, and there is no phase transition.

In this chapter we discussed the equilibrium voting model. The model which we discussed in Hisakado and Mori (2012) is the non-equilibrium model. The large r limit corresponds to the large N limit of the infinite-range model. We can observe the phase transition in both models.

In the case of non-equilibrium, we can obtain the same self-consistent equation. In this case the solution which is against the outer field may be selected. It is the characteristic point of the non-equilibrium model which we discuss in other chapters.

Appendix B Response Function of Animals

In this chapter we use three kinds of herders, digital, analog, and tanh type. In Sect. 3, we studied the Kirman’s ant colony model. In this model the response function is analog. Here we have the question whether the real ant is analog herder. In this appendix we review the several experiments of animals about decision of the two choices. In Table 1 we show the list of the response function of the animals. These are the decision of two selections of animals. For the nonlinear response function, there is the spontaneous symmetry braking which corresponds to the phase transition. On the other hand, for linear response function, there is no phase transition. From the experiments some ants are analog herders, and some are tanh-type herders. About humans we presented in Chaps. 8 and 9. In the case of animals, the nonlinear response function is sometimes called quorum response.

Table 1 Response functions of animalsl