Max-Min Probabilities in the Voting Paradox

Abstract

The voting paradox, that it is possible among three candidates to have $A$ more popular than $B, B$ more popular than $C$, and $C$ more popular than $A$ (e.g., let $\frac{1}{3}$ of a population prefer $A$ to $B, B$ to $C$; another $\frac{1}{3}$ prefer $B$ to $C, C$ to $A$; and the remaining $\frac{1}{3}$ prefer $C$ to $A, A$ to $B$) naturally raises the question of how much more popular they can be, and what results can be obtained with more than three candidates. The question corresponds to the mathematical problem of choosing the joint distribution of $n$ real-valued random variables so as to maximize $\min\{P(X_1 > X_2), \cdots, P(X_{n-1} > X_n), P(X_n > X_1)\}$. The fact that all these probabilities can exceed $\frac{1}{2}$ is well known, (see [1]), but the question of max-min does not appear to have been considered. This note considers this problem (a) with unrestricted $X_1, \cdots, X_n$ and (b) with $X_1, \cdots, X_n$ restricted to be independent. In Case (a), it is very easy to show that the largest possible minimum is $(n - 1)/n$, easily achievable. In Case (b), which is more interesting, there is also an achievable largest minimum $b(n)$, which can be found by solving a degree $\lbrack\frac{1}{2}(n + 1)\rbrack$ equation, and has $b(n + 1) > b(n), \lim_{n\rightarrow\infty} b(n) = \frac{3}{4}$, and $b(3) = .61803, b(4) = \frac{2}{3}, \cdots, b(10) = .73205$.