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 mu is an element of{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) -> infinity. Between the cumulative functions {x(mu)} of binary candidates, the power-law relation 1 - x(1) similar to (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))-> (0,0) and Z(0) -> infinity with s(1)/s(0) = alpha fixed, the relation 1 - x(1) = (1 - x(0))(alpha) holds exactly over the entire range 0 <= x(0),x(1) <= 1. We study the data on horse races obtained from the Japan Racing Association for the period 1986 to 2006 and confirm scale invariance.