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

[Jobst Heitzig]

Dear Forest!

You wrote:
>   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 inter!
 esting?   Forest 

It is interesting indeed. It also holds when voters judge by median utility instead of mean utility (and then we don't need the uniform distribution assumption)...

Jobst
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