Maximal lotteries refers to a probabilistic voting system first considered by Germain Kreweras in 1965. The method uses preferential ballots and returns so-called maximal lotteries, i.e., probability distributions over the alternatives that are weakly preferred to any other probability distribution. Maximal lotteries satisfy the Condorcet criterion, the Smith criterion, reversal symmetry, polynomial runtime, and probabilistic versions of reinforcement, participation, and independence of clones.
Maximal lotteries are equivalent to mixed maximin strategies (or Nash equilibria) of the symmetric zero-sum game given by the pairwise majority margins. As such, they can be computed using linear programming. The voting system that returns all maximal lotteries is axiomatically characterized as the only one satisfying probabilistic versions of population-consistency (a weakening of reinforcement) and composition-consistency (a strengthening of independence of clones). Further, maximal lotteries satisfy a strong notion of Pareto efficiency and a weak notion of strategyproofness. In contrast to random dictatorship, maximal lotteries do not satisfy the standard notion of strategyproofness. Also, maximal lotteries are not monotonic in probabilities, i.e., it is possible that the probability of an alternative decreases when this alternative is ranked up. However, the probability of the alternative will remain positive.
Maximal lotteries or variants thereof have been rediscovered multiple times by economists, mathematicians, political scientists, philosophers, and computer scientists. In particular, the support of maximal lotteries, which is known as the essential set or the bipartisan set, has been studied in detail.
Example
Suppose there are five voters who have the following preferences over three alternatives:
The pairwise preferences of the voters can be represented in the following table or skew-symmetric matrix, where the entry for row
This matrix can be interpreted as a zero-sum game and admits a unique Nash equilibrium (or minimax strategy)