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We introduce a voting model and discuss the scale invariance in the mixing ofcandidates. 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 acandidate gets a vote is proportional to the number of votes. The initialnumber of votes (`seed') of a candidate $\mu$ is set to be $s_{\mu}$. Afterinfinite counts of voting, the probability function of the share of votes ofthe 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 thecumulative 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 doublescaling limit $(s_{1},s_{0})\to (0,0)$ and $Z_{0} \to \infty$ with$s_{1}/s_{0}=\alpha$ fixed, the relation $1-x_{1}=(1-x_{0})^{\alpha}$ holdsexactly over the entire range $0\le x_{0},x_{1} \le 1$. We study the data onhorse races obtained from the Japan Racing Association for the period 1986 to2006 and confirm scale invariance.

Exact Scale Invariance in Mixing of Binary Candidates in Voting Model