The Pitman-Yor Process and Choice Behavior

Abstract

In this chapter, 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. 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 (J Phys Soc Jpn 87(2):024002-2419, 2018).

Appendices

Appendix A Fixed Number of Candidates Case

We model the voting of K candidates, C ₁⋯C _K. At time t, candidate C _j has c _j(t) votes. In this appendix, we consider the case in which the number of candidates K is fixed, that is, no new entry is allowed. In each time step, one voter votes for one candidate; the voting is sequential. Hence, at time t, the tth voter votes, after which the total number of votes is t. Voters are allowed to see r previous votes for each candidate and, thus, are aware of public perception. r is a constant number. We consider the case in which all voters vote for the candidate with a probability proportional to the previous votes ratio, which is visible to the voters.

The transition is

$$\displaystyle \begin{aligned} c_j(t)=k \rightarrow k+1: P_{j,k,t:l,t-r}=\frac{\frac{q_j(1-\rho)}{\rho}+ (k-l)}{\frac{1-\rho}{\rho}+ r}=\frac{\beta_j+(k-l)}{\theta+ r}, {} \end{aligned} $$

(20)

where c _j(t − r) = l, ρ is the correlation coefficient and q _j is the initial constant of the jth candidate (Hisakado et al. 2006). ρ is the correlation of the beta-binomial model. The constraint $\sum _{j=1}^{K}q_j=1$ exists. We define θ = (1 − ρ)∕ρ and β _j = q _j(1 − ρ)∕ρ. P _{j,k,t:l,t−r} denotes the probabilities of the process. The voting ratio for C _j at t − r is c _j(t − r) = l. We consider the case β _j ≥ 0 from P _{j,k,t:l,t−r} > 0 and the constraint ∑_j β _j = θ. When β _j = β, the constraint becomes βK = θ.

We consider the hopping rate among (r + 1) states $\hat {k}_j=k-l$, $\hat {k}_j=0,1,\cdots , r$. In each step of t, the vote at time (t − r) is deleted, and a new vote is obtained. $\hat {k}$ is the number of votes candidate C _j obtained in the latest r votes. In case K = 2, the model becomes Kirman’s ant colony model (Kirman 1993). The dynamic evolution of the process is given by

$$\displaystyle \begin{aligned} \begin{array}{rcl} \hat{k}_j&\displaystyle \rightarrow&\displaystyle \hat{k}_j+1: P_{\hat{k}_j,\hat{k}_j+1,t}=\frac{r-\hat{k}_j}{r}\frac{\beta_j+ \hat{k}_j}{\theta+ r-1}, \\ \hat{k}_j &\displaystyle \rightarrow&\displaystyle \hat{k}_j-1: P_{\hat{k}_j,\hat{k_j}-1,t}=\frac{\hat{k}_j}{r}\frac{(\theta-\beta_j)+ (r-1-\hat{k}_j)}{\theta+ r-1}, \\ \hat{k}_j &\displaystyle \rightarrow&\displaystyle \hat{k}_j: P_{\hat{k}_j,\hat{k}_j,t}=1-P_{\hat{k},\hat{k}-1,t}-P_{\hat{k},\hat{k}+1,t}. \end{array} \end{aligned} $$

$P_{\hat {k}_j,\hat {k}_j\pm 1,t}$ are the probabilities of the process and the products of exit votes and new entry votes.

We consider hopping from candidate C _i to C _j.

$$\displaystyle \begin{aligned} \begin{array}{rcl} \hat{k}_i&\displaystyle \rightarrow&\displaystyle \hat{k}_i-1,\hat{k}_j \rightarrow \hat{k}_j+1: P_{\hat{k}_i\rightarrow\hat{k}_i-1,\hat{k}_j\rightarrow\hat{k}_j+1,t}=\frac{\hat{k}_i}{r}\frac{\beta_j+ \hat{k}_j}{\theta+ r-1}, \\ \hat{k}_i&\displaystyle -&\displaystyle 1\rightarrow \hat{k}_i,\hat{k}_j +1\rightarrow \hat{k}_j: P_{\hat{k}_i-1\rightarrow\hat{k}_i,\hat{k}_j+1\rightarrow\hat{k}_j,t}=\frac{\hat{k}_j+1}{r}\frac{\beta_i+ \hat{k}_i-1}{\theta+ r-1}. \end{array} \end{aligned} $$

Here, we define $\mu _{r}(\hat {k},t)$ as a distribution function of the state $\hat {k}$ at time t. The number of all states is (r + 1). Given that the process is reversible, we have

$$\displaystyle \begin{aligned} \frac{\mu_{r}(\hat{k}_i,\hat{k}_j, t)}{\mu_{r}(\hat{k}_i-1,\hat{k}_j+1, t)} =\frac{\hat{k}_j+1}{\hat{k}_i}\frac{\beta_i+\hat{k}_i-1}{\beta_j+\hat{k}_j}. \end{aligned} $$

(21)

We can separate indexes i and j and obtain

$$\displaystyle \begin{aligned} \begin{array}{rcl} \frac{\mu_{r}^i(\hat{k}_i, t)}{\mu_{r}^i(\hat{k}_i-1, t)} &\displaystyle =&\displaystyle \frac{\beta_i+\hat{k}_i-1}{\hat{k}_i}c \\ \frac{\mu_{r}^j(\hat{k}_j+1, t)}{\mu_{r}^j(\hat{k}_j, t)} &\displaystyle =&\displaystyle \frac{\beta_j+\hat{k}_j}{\hat{k}_j+1}c, {} \end{array} \end{aligned} $$

(22)

where c is a constant. Using (22) sequentially, in the limit t →∞, we can obtain the equilibrium distribution, which can be written as

$$\displaystyle \begin{aligned} \mu_r(\boldsymbol{a} ,\infty)= \left( \begin{array}{r} \theta+r-1\\ r \end{array} \right)^{-1} \prod_{j=1}^{K} \left( \begin{array}{r} \beta_j +\hat{k}_j-1\\ \hat{k}_j \end{array} \right), {} \end{aligned} $$

(23)

where $\boldsymbol {a}=(\hat {k}_1,\hat {k}_2,\cdots , \hat {k}_K)$. This distribution is written as

$$\displaystyle \begin{aligned} \mu_r(\boldsymbol{a},\infty)=\frac{r !}{\theta^{[r]}}\prod_{i=1}^{K} \frac{\beta_i^{[\hat{k}_i]}}{\hat{k}_i !}, {} \end{aligned} $$

(24)

where x ^[n] = x(x + 1)⋯(x + n − 1). This is the Dirichlet-multinomial distribution.

Here, we set β _j = β. The relation β = −α exists, where α is the parameter used in the main text. We write (23) as

$$\displaystyle \begin{aligned} \mu_r(\hat{\boldsymbol{a}},\infty)= \left( \begin{array}{r} \theta+r-1 \\ r \end{array} \right)^{-1} \prod_{j=1}^{r} \left( \begin{array}{r} \beta+j-1 \\ j \end{array} \right)^{a_{j }}, \end{aligned} $$

(25)

where a _j is the number of candidates for whom j voters voted and $\hat {\boldsymbol {a}}=(a_1, \cdots , a_{r})$. Hence, the relations $\sum _{i=1}^{r}a_i=K_r<K$ and $\sum _{i=1}^{r}i a_i=r$ exist. Here, we define K _r as the number of candidates who have more than one vote. (K − K _r) candidates have no vote.

We consider the partitions of integer K _r. To normalize, we add the term of combination: K!∕a ₁!⋯a _r!(K − K _r)!. We obtain

$$\displaystyle \begin{aligned} \begin{array}{rcl} \mu_r(\hat{\boldsymbol{a}},\infty)&=& \frac{K!}{a_1 !\cdots a_r !(K-K_r )!} \left( \begin{array}{r} \theta+r-1 \\ r \end{array} \right)^{-1} \prod_{j=1}^{r} \left( \begin{array}{r} \beta+j-1 \\ j \end{array} \right)^{a_{j }} \\ &=& \frac{r! \theta^{[K_r:-\beta]}}{\theta^{[r]}} \prod_{j=1}^{r}(\frac{(1+\beta)^{[j-1]}}{j!})^{a_j}\frac{1}{a_j!}, {} \end{array} \end{aligned} $$

(26)

where x ^[n:−β] = x(x − β)⋯(x − (n − 1)β). We use the relation θ = Kβ. Equation (26) is simply the Pitman sampling formula (Pitman 2006).

In the limit β → 0 and K →∞, subject to a fixed θ = βK, we can obtain the Ewens sampling formula. In this case, the sum of the probabilities that a candidate who has zero votes can obtain one vote is θ∕(θ + r). This case is the same as α = 0 in Sect. 2.

Appendix B Partition Number and Pitman Distribution

A partition of positive integer is called integer partition problem. It is the number of the representations that positive integer n as the sum of the positive integer. Here we write the number p(n). For example, 3 is presented as 3, 2+1, 1+1+1, and p(3) = 3.

The generating function of p(n) is known as

$$\displaystyle \begin{aligned} \begin{array}{rcl} \sum_{n=0}^{\infty}p(n)x^n&\displaystyle =&\displaystyle 1+x+2x^2+3x^3+5x^4+7x^5+11x^6+\dots \\ &\displaystyle =&\displaystyle \prod_{k=1}^{\infty}\left(\frac{1}{1-x^k}\right)=(1+x+x^2\cdots)(1+x^2+x^4 +\cdots)\\ &\displaystyle &\displaystyle (1+x^3+\cdots). \end{array} \end{aligned} $$

(27)

We can confirm the combinatorics of n as the coefficients of x ⁿ. The Pitman distribution represents the distribution for each representation. We show the Pitman distribution and the votes for 2ch in Fig. 10.

Fig. 10

The black bar is the Pitman distribution. The red line is the empirical data of 2ch. Here we show n = 3, 5, 20, 50 cases

The red line is the business data which fits well to the Pitman distribution. The parameter is calibrated by the historical data, when n is less than the range of the auto correlation which we discussed in the Sect. 4.1. We can confirm 2ch data is represented by the Pitman distribution.