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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]
Maximal lotteries were first proposed by the French mathematician and social scientist Germain Kreweras in 1965, [9] and popularized by Peter Fishburn. [1]
Since then, they have been rediscovered multiple times by economists, [7] mathematicians, [1] [10] political scientists, philosophers, [11] and computer scientists. [12]
Similar ideas have also appeared in the study of reinforcement learning [13] and in evolutionary biology to explain the multiplicity of co-existing species. [14]
The input to this voting system consists of the agents' ordinal preferences over outcomes (not lotteries over alternatives), but a relation on the set of lotteries can be constructed in the following way: if and are lotteries over alternatives, if the expected value of the margin of victory of an outcome selected with distribution in a head-to-head vote against an outcome selected with distribution is positive. In other words, if it is more likely that a randomly selected voter will prefer the alternatives sampled from to the alternative sampled from than vice versa. [4] While this relation is not necessarily transitive, it does always contain at least one maximal element.
It is possible that several such maximal lotteries exist, as a result of ties. However, the maximal lottery is unique whenever there an odd number of voters express strict preferences. [15] By the same argument, the bipartisan set is uniquely-defined by taking the support of all maximal lotteries that solve a tournament game. [5]
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 have a natural interpretation in terms of electoral competition between two political parties. [16]
Maximal lotteries satisfy several notable strategy-resistance properties, such as eliminating the possibility that a voter can, by misreporting their preferences, obtain a lottery that stochastically dominates another. [17] [18]
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 skew-symmetric matrix, where the entry for row and column denotes the number of voters who prefer to minus the number of voters who prefer to .
This matrix can be interpreted as a zero-sum game and admits a unique Nash equilibrium (or minimax strategy) where , , . By definition, this is also the unique maximal lottery of the preference profile above. The example was carefully chosen not to have a Condorcet winner. Many preference profiles admit a Condorcet winner, in which case the unique maximal lottery will assign probability 1 to the Condorcet winner.