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