Abstract
Suppose the only observable characteristic of each of n random variables that is uniformly distributed on the six rankings of objects in a three-element set is its first-ranked object. Let ƒ( n 1, n 2, n 3) be the probability that one of the three objects has majorities over the other two within the rankings when n j of the n rankings have the jth object in first place. It is assumed that n is odd, so that ƒ( n 1, n 2, n 3)=1 only if n j ≥( n+1)/2 for some j. It is shown that ƒ(a+1,b,c)<ƒ(a,b+1,c) if a <b,a ≤ c ≤ b+1 and max { b, c}≤( n−1)/2. It follows from this that ƒ is minimized for fixed n if and only if n j − n k ≤1 for all j, kϵ {1,2,3}. However, ƒ does not necessarily increase when two of its arguments get farther apart. For example, ƒ( b, b,3)>ƒ( b−1, b+1,3) for b≥28, and ƒ( b, b,2 b−1)>ƒ( b−1, b+1,2 b−1) for b≥12.
Published Version
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