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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