Maximal lotteries

Maximal lotteries are a probabilistic voting rule that use ranked ballots and returns a lottery over candidates that a majority of voters will prefer, in expectation, to any other. More formally, the rule has the property that, when averaging over a series of repeated head-to-head matchups, at least half of all voters will prefer the result of a maximal lottery to the result produced by any other voting rule.

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

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. Maximal lotteries are also nonmonotonic in probabilities, meaning it is possible for the probability of an alternative to decrease if this alternative up. However, they satisfy relative monotonicity, the probability of ${\displaystyle x}$ relative to that of ${\displaystyle y}$ does not decrease when ${\displaystyle x}$ is improved over ${\displaystyle y}$.

The maximal lottery satisfies some generalizations of reinforcement and participation to the non-deterministic case: the probability of superiority exceeds 50%, meaning a voter will prefer the result they get by participating to the one they get by not participating more than half the time. However, a maximal lottery can still exhibit a no-show paradox in the expected utility sense, so a voter feels the lottery they get by voting is worse (has lower expected utility) than the lottery they would receive if they had not turned out.

The support of maximal lotteries, which is known as the essential set or the bipartisan set, has been studied in detail.