The Chance of a Voting Cycle

Three voters each independently submit a strict ranking of candidates A, B and C, chosen uniformly at random from the 6 possible rankings. Majority preference between two candidates is decided by which one more voters rank higher (with three voters there are no ties). The majority relation is *cyclic* if it has the form X beats Y, Y beats Z, Z beats X for some ordering of the candidates (a Condorcet cycle). What is the probability that the majority relation is cyclic? Give a reduced fraction.

• distributions
• permutations

Source: Condorcet's paradox (1785); the 1/18 figure for three voters and three candidates is a standard impartial-culture computation. Statement written for AxiomIQ.