Maximal lotteries

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Maximal lotteries are a tournament voting rule that elects the majority-preferred candidate if one exists,^[1] and otherwise elects a candidate from the majority-preferred set by a randomized voting procedure.^[1] The method selects the probability distribution (or linear combination) of candidates that a majority of voters would prefer to any other.^[1]

Maximal lotteries satisfy a wide range of desirable properties, including satisfying all axioms of majority rule in the strongest sense: they elect the Condorcet winner with probability 1^[1] and never elect candidates outside the Smith set.^[1] Moreover, they satisfy reinforcement,^[2] participation,^[3] and independence of clones,^[2] and are weakly group-strategyproof (see below). The social welfare function that top-ranks maximal lotteries is uniquely characterized by Arrow's independence of irrelevant alternatives and Pareto efficiency.^[4]

Maximal lotteries do not satisfy the standard notion of strategyproofness, as shown by Gibbard's theorem. Maximal lotteries are also nonmonotonic in probabilities, i.e. it is possible for increasing the rank of a candidate to decrease their probability of winning.^[5]

The support of maximal lotteries, which is known as the essential set^[6] or the bipartisan set, has been studied in detail.^[7][8]