Maximal lotteries

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Maximal lotteries refers to a probabilistic voting rule. The method uses preferential ballots and returns a probability distribution (or linear combination) of candidates that a majority of voters would weakly prefer to any other.^[1]

Maximal lotteries satisfy a wide range of desirable properties: they elect the Condorcet winner with probability 1 if it exists^[1] and never elect candidates outside the Smith set.^[1] Moreover, they satisfy reinforcement,^[2] participation,^[3] and independence of clones.^[2] The probabilistic voting rule that returns all maximal lotteries is the only rule satisfying reinforcement, Condorcet-consistency, and independence of clones.^[2] The social welfare function that top-ranks maximal lotteries has been uniquely characterized using Arrow's independence of irrelevant alternatives and Pareto efficiency.^[4]

Maximal lotteries do not satisfy the standard notion of strategyproofness, as Allan Gibbard has shown that only random dictatorships can satisfy strategyproofness and ex post efficiency.^[5] Maximal lotteries are also nonmonotonic in probabilities, i.e. it is possible that the probability of an alternative decreases when a voter ranks this alternative up.^[1] However, they satisfy relative monotonicity, i.e., the probability of ${\displaystyle x}$ relative to that of ${\displaystyle y}$ does not decrease when ${\displaystyle x}$ is improved over ${\displaystyle y}$.^[6]

The support of maximal lotteries, which is known as the essential set or the bipartisan set, has been studied in detail.^{[7][8][9][10]}