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 μ∈ \0,1\ and are called as `binary' candidates. There are in total N=N0+N1 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 μ is set to be sμ. After infinite counts of voting, the probability function of the share of votes of the candidate μ obeys gamma distributions with the shape exponent sμ in the thermodynamic limit Z0=N1s1+N0s0 ∞. Between the cumulative functions \xμ\ of binary candidates, the power-law relation 1-x1 (1-x0)α with the critical exponent α=s1/s0 holds in the region 1-x0,1-x1<<1. In the double scaling limit (s1,s0) (0,0) and Z0 ∞ with s1/s0=α fixed, the relation 1-x1=(1-x0)α holds exactly over the entire range 0 x0,x1 1. We study the data on horse races obtained from the Japan Racing Association for the period 1986 to 2006 and confirm scale invariance.