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Exact Scale Invariance in Mixing of Binary Candidates in Voting Model

We introduce a voting model and discuss the scale invariance in the mixing of candidates. The Candidates are classified into two categories $μ\in \{0,1\}$ and are called as `binary&#39; 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&#39;) of a candidate $μ$ is set to be $s_μ$. After infinite counts of voting, the probability function of the share of votes of the candidate $μ$ obeys gamma distributions with the shape exponent $s_μ$ in the thermodynamic limit $Z_{0}=N_{1}s_{1}+N_{0}s_{0}\to \infty$. Between the cumulative functions $\{x_μ\}$ of binary candidates, the power-law relation $1-x_{1} \sim (1-x_{0})^α$ with the critical exponent $α=s_{1}/s_{0}$ 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 $s_{1}/s_{0}=α$ fixed, the relation $1-x_{1}=(1-x_{0})^α$ 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.