[EM] A class of ballot set with "unbeaten in mean lotteries."

[Simmons, Forest]

Let x, y, and z be positive integers such that x+y+z=N, and max(x,y,z)<N/2, where N is the number of some large population of voters, and the ordinal preferences are divided into three factions:
 
x: A>B>C
y: B>C>A
z: C>A>B
 
Further assume that the cardinal ratings of the middle candidate within each faction are distributed uniformly, so that in the first faction the cardinal ratings of B are distributed evenly between zero and 100%.
 
Let (alpha, beta, gamma) be a "lottery" for this election.
 
Then the number of voters that prefer A to this lottery is given by the expression
 
       p(A) = x + beta*z/(gamma+beta)
 
Corresponding expressions for B and C are
 
      p(B) = y + gamma*x/(alpha+gamma)  and
      p(C) = z + alpha*y/(beta+alpha)
 
If we set (alpha, beta, gamma) equal to
 
     (x+y-z, y+z-x, z+x-y)/(2*N) ,
 
then p(A)=p(B)=p(C)=N/2 , which means that none of the candidates is preferred over the lottery by more than half of the population.
 
Isn't that interesting?
 
Forest
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