Information Cascade and Phase Transition

Abstract

In this chapter, we discuss a voting model with two candidates. 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. 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. Here we obtain the exact solutions of the model. The main contents of this chapter are based on Hisakado and Mori (J Phys A 22:275204, 2011).

Appendices

Appendix A Catalan Number and Extended

Here, we consider the number of monotonic paths along the edges of a grid with square cells, which do not pass lower the diagonal. Let m and n be the horizontal axis and the vertical axis, respectively. The coordinates of the lower left corner are (0, 0). A monotonic path is one which starts in the lower left corner; finishes in the upper triangle (m, n), where 0 ≤ m ≤ n; and consists entirely of edges pointing rightward or upward.

The number of paths from (0, 0) to (m, n) can be calculated as

$$\displaystyle \begin{aligned} C_{m,n}=\frac{(n-m+1)(n+m)!}{m!(n+1)!} = \left( \begin{array}{cc} n+m\\ n \end{array} \right) - \left( \begin{array}{cc} n+m\\ n+1 \end{array} \right) . {} \end{aligned} $$

(22)

These numbers are known as generalized Catalan number.

If the finish point is (m, m), the number of paths becomes the Catalan number.

$$\displaystyle \begin{aligned} C_{m,m}=c_m = \frac{2m!}{m!(m+1)!} = \left( \begin{array}{cc} 2m\\ m \end{array} \right) - \left( \begin{array}{cc} 2m\\ m+1 \end{array} \right) . \end{aligned} $$

(23)

Next, we compute the distribution of the number of the paths that starts in the lower left corner, finishes in the upper triangle (m, n), and touches the diagonal k times (Di Francesco et al. 1997; Lang 2000). Let A _m,n,k denote the number of paths that touches the diagonal k times. We get a simple recursion relation about A _m,n,k.

$$\displaystyle \begin{aligned} A_{m,n,k}=\sum_{j=0}^{m-1}c_{j}A_{m-j-1,n-j-1,k-1}, {} \end{aligned} $$

(24)

for k ≥ 0, n, m ≥ 0, and m ≥ k, with the initial condition A _0,0,0 = 1. This defines the numbers A _m,n,k uniquely, and it is easy to prove that

$$\displaystyle \begin{aligned} \begin{array}{rcl} A_{m,n,k}&=&\frac{(n-m+k)(n+m-k-1)!}{n!(m-k)!} \\ &=& \left( \begin{array}{cc} n+m-k-1\\ n-1 \end{array} \right) - \left( \begin{array}{cc} n+m-k-1\\ n \end{array} \right) . {} \end{array} \end{aligned} $$

(25)

From (22) and (25), we can obtain the relation

$$\displaystyle \begin{aligned} A_{m,m,k}=C_{m-k,m-1}. {} \end{aligned} $$

(26)

The well-known generating function C ₀(x) of Catalan numbers is given by

$$\displaystyle \begin{aligned} \begin{array}{rcl} C_0(x)&\displaystyle =&\displaystyle \sum_{n=0}^{\infty}C_{m,m}x^{n} \\ &\displaystyle =&\displaystyle 1+x+2x^2+5x^3+14x^4+42x^5+132x^6+\cdots, \end{array} \end{aligned} $$

(27)

subject to the algebraic relation

$$\displaystyle \begin{aligned} xC_0(x)^2=C_0(x)-1, \end{aligned} $$

(28)

and we can obtain

$$\displaystyle \begin{aligned} C_0(x)=\frac{1-\sqrt{1-4x}}{2x}. {} \end{aligned} $$

(29)

Here, we obtain the generating function A _m,k(x) of A _m,m,k.

$$\displaystyle \begin{aligned} \begin{array}{rcl} A_{m,k}(x)&\displaystyle =&\displaystyle \sum_{m-k=0}^{\infty}A_{m,m,k}x^{m-k} =\sum_{m-k=0}^{\infty}C_{m-k,m-1}x^{m-k} \\ &\displaystyle =&\displaystyle \sum_{l=0}^{\infty}C_{l,l+k-1}x^{l} =C_{k-1}(x). {} \end{array} \end{aligned} $$

(30)

We use (26) for the second equality. $C_{j}(x)=\sum _{l=1}^{\infty }C_{l,l+j}x^{l}$ is the generating function of the generalized Catalan number (22). The generating function of the generalized Catalan number is given by

$$\displaystyle \begin{aligned} \begin{array}{rcl} C_1(x)&\displaystyle =&\displaystyle \sum_{n=0}^{\infty}C_{m,m+1}x^{n} \\ &\displaystyle =&\displaystyle 1+2x+5x^2+14x^3+42x^4+132x^5+429x^6+\cdots, \\ C_2(x)&\displaystyle =&\displaystyle \sum_{n=0}^{\infty}C_{m,m+2}x^{n} \\ &\displaystyle =&\displaystyle 1+3x+9x^2+28x^3+90x^4+297x^5+1001x^6+\cdots, \\ C_3(x)&\displaystyle =&\displaystyle \sum_{n=0}^{\infty}C_{m,m+3}x^{n} \\ &\displaystyle =&\displaystyle 1+4x+14x^2+48x^3+165x^4+572x^5\cdots. \end{array} \end{aligned} $$

From (24), we can obtain

$$\displaystyle \begin{aligned} C_{j}(x)=C_{j-1}(x)C_{0}(x). \end{aligned} $$

(31)

Thus, the simple relation between the generating functions is given by

$$\displaystyle \begin{aligned} C_{j}(x)=\{ C_{0}(x) \}^{j+1}. {} \end{aligned} $$

(32)

Appendix B Derivation of $\tilde {R_1}$, R ₁, and R ₂

$\tilde {R_1}$ is the probability that the path starts from (0, 0), goes across the diagonal only once, and reaches the wall n = m − 1 in II (m > n) (Fig. 5).

$$\displaystyle \begin{aligned} \begin{array}{rcl} \tilde{R_1}&\displaystyle =&\displaystyle (1-B) [1+y\gamma_1 C_0(y)+(y\gamma_1)^2C_1(y)+(y\gamma)^3 C_2(y)+\cdots ] \\ &\displaystyle =&\displaystyle (1-B) [1+y\gamma_1 C_0(y)+(y\gamma_1)^2\{C_0(y)\}^2+(y\gamma_1)^3\{ C_0(y)\}^{3}+\cdots ] \\ &\displaystyle =&\displaystyle (1-B)[\sum_{k=0}^{\infty} \{\gamma_1 yC_{0} (y)\}^{k}] = \frac{1-B}{1-\gamma_1 AC_0(A)} \\ &\displaystyle =&\displaystyle \frac{2(1-B)}{2-\gamma_1(1-\sqrt{1-4A(1-A)})}, {} \end{array} \end{aligned} $$

(33)

where A and B are given by (1), γ ₁ = B∕A, and y = A(1 − A). C _k(y) is the generation function of the generalized Catalan number (30). Here, we use the relations (32) and (29). When q = 1, (33) reduces to (8).

Fig. 5

$\tilde {R_1}$, R ₁, and R ₂. $\tilde {R_1}$ is the probability that the path starts from (0, 0), goes across the diagonal only once, and reaches the wall n = m − 1 in II (m > n). R ₁ is the probability that the path starts from the wall n = m + 1 in I (m < n), goes across the diagonal only once, and reaches the wall n = m − 1 in II (m > n). R ₂ is the probability that the path starts from the wall n = m − 1 in II (m > n), goes across the diagonal only once, and reaches the wall n = m + 1 in I (m < n)

R ₁ is the probability that the path starts from the wall n = m + 1 in I (m < n), goes across the diagonal only once, and reaches the wall n = m − 1 in II (m > n).

$$\displaystyle \begin{aligned} \begin{array}{rcl} R_1&\displaystyle =&\displaystyle \frac{1-B}{B} [y\gamma_1 C_0(y)+(y\gamma_1)^2C_1(y)+(y\gamma_1)^3 C_2(y)+\cdots ] \\ &\displaystyle =&\displaystyle \frac{1-B}{B}[y\gamma_1 C_0(y)+(y\gamma_1)^2\{C_0(y)\}^2+(y\gamma_1)^3\{ C_0(y)\}^{3}+\cdots ] \\ &\displaystyle =&\displaystyle \frac{1-B}{B}[\frac{\tilde{R_1}}{1-B}-1] = \frac{(1-B)\gamma_1(1-\sqrt{1-4A(1-A)})}{B\{2-\gamma_1(1-\sqrt{1-4A(1-A)})\}}. {} \end{array} \end{aligned} $$

(34)

R ₂ is the probability that the path starts from the wall n = m − 1 in II (m > n), goes across the diagonal only once, and reaches the wall n = m + 1 in I (m < n).

$$\displaystyle \begin{aligned} \begin{array}{rcl} R_2&\displaystyle =&\displaystyle \frac{B}{1-B} [z\gamma_2 C_0(z)+(z\gamma_2)^2C_1(z)+(z\gamma_2)^3 C_2(z)+\cdots ] \\ &\displaystyle =&\displaystyle \frac{B}{1-B}[z\gamma_2 C_0(z)+(z\gamma_2)^2\{C_0(z)\}^2+(z\gamma_2)^3\{ C_0(z)\}^{3}+\cdots ] \\ &\displaystyle =&\displaystyle \frac{B\gamma_2(1-\sqrt{1-4C(1-C)})}{(1-B)\{2-\gamma_2(1-\sqrt{1-4C(1-C)})\}}. {}, \end{array} \end{aligned} $$

(35)

where C is given by (1), γ ₂ = (1 − B)∕(1 − C), and z = C(1 − C).