Exact Scale Invariance in Mixing of Binary Candidates in Voting Model

Abstract

We introduce a voting model and discuss the scale invariance in the mixing of candidates. The Candidates are classified into two categories $\mu \in \{0,1\}$ and are called as `binary' candidates. There are in total $N=N_0+N_1$ candidates, and voters vote for them one by one. The probability that a candidate gets a vote is proportional to the number of votes. The initial number of votes (`seed') of a candidate $\mu$ is set to be $s_{\mu }$. After infinite counts of voting, the probability function of the share of votes of the candidate $\mu$ obeys gamma distributions with the shape exponent $s_{\mu }$ in the thermodynamic limit $Z_0=N_1{s}_1+N_0{s}_0\to \infty$. Between the cumulative functions $\{x_{\mu }\}$ of binary candidates, the power-law relation $1-x_1\sim (1-x_0{)}^{\alpha }$ with the critical exponent $\alpha =s_1/s_0$ holds in the region $1-x_0,1-x_1<<1$. In the double scaling limit $(s_1,s_0)\to (0,0)$ and $Z_0\to \infty$ with $s_1/s_0=\alpha$ fixed, the relation $1-x_1=(1-x_0{)}^{\alpha }$ holds exactly over the entire range $0\le x_0,x_1\le 1$. We study the data on horse races obtained from the Japan Racing Association for the period 1986 to 2006 and confirm scale invariance.

19 pages, 8 figures, 2 tables

References in corpus (2)

• Power laws, Pareto distributions and Zipf's law
• Correlated Binomial Models and Correlation Structures

Cited by in corpus (10)

• Phase transition and information cascade in a voting model
• Digital herders and phase transition in a voting model
• Phase transition to two-peaks phase in an information cascade voting experiment
• Information cascade, Kirman's ant colony model, and kinetic Ising model
• A voter model on networks and multivariate beta distribution
• Finite-size scaling analysis of binary stochastic processes and universality classes of information cascade phase transition
• Two kinds of Phase transitions in a Voting model
• Mean Field Voter Model of Election to the House of Representatives in Japan
• Collective Adoption of Max-Min Strategy in an Information Cascade Voting Experiment
• From the multi-terms urn model to the self-exciting negative binomial distribution and Hawkes processes