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Maximal lotteries refers to a probabilistic voting system first considered by Germain Kreweras〔G. Kreweras. ''Aggregation of preference orderings''. In Mathematics and Social Sciences I: Proceedings of the seminars of Menthon-Saint-Bernard, France (1–27 July 1960) and of Gösing, Austria (3–27 July 1962), pages 73–79, 1965.〕 in 1965 and independently rediscovered and studied in more detail in 1984 by Peter C. Fishburn.〔P. C. Fishburn. ''Probabilistic social choice based on simple voting comparisons''. Review of Economic Studies, 51(4):683–692, 1984.〕 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,〔F. Brandl, F. Brandt, and H. G. Seedig. Consistent probabilistic social choice. Working paper. 2015.〕 participation,〔F. Brandl, F. Brandt, and J. Hofbauer. Welfare Maximization Entices Participation. Working paper. 2015.〕 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. In 2015, maximal lotteries were axiomatically characterized by showing that only maximal lotteries satisfy probabilisitic versions of population-consistency (a weakening of reinforcement), agenda-consistency, and composition-consistency (a strengthening of independence of clones).〔 It was also shown that maximal lotteries satisfy a strong notion of Pareto efficiency and a weak notion of strategyproofness.〔H. Aziz, F. Brandt, and M Brill. On the Tradeoff between Economic Efficiency and Strategyproofness in Randomized Social Choice. In Proceedings of the 12th International Conference on Autonomous Agents and Multi-Agent Systems (AAMAS), pages 455–462. 2013.〕 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,〔G. Laffond, J.-F. Laslier, and M. Le Breton. ''The bipartisan set of a tournament game''. Games and Economic Behavior, 5(1):182–201, 1993.〕 mathematicians,〔D. C. Fisher and J. Ryan. ''Tournament games and positive tournaments''. Journal of Graph Theory, 19(2):217–236, 1995.〕 political scientists, philosophers,〔D. S. Felsenthal and M. Machover. ''After two centuries should Condorcet’s voting procedure be implemented?'' Behavioral Science, 37(4):250–274, 1992.〕 and computer scientists.〔R. L. Rivest and E. Shen. An optimal single-winner preferential voting system based on game theory. In Proceedings of 3rd International Workshop on Computational Social Choice, pages 399–410, 2010.〕 In particular, the support of maximal lotteries, which is known as the ''essential set'' or the ''bipartisan set'', has been studied in detail.〔 == Preferences over lotteries == The input to this solution concept consists of the agents' ordinal preferences over outcomes (not lotteries over outcomes), but a relation on the set of lotteries is constructed in the following way: if and are different lotteries over outcomes, 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. While this relation is not necessarily transitive, it does always contain at least one maximal element. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Maximal lotteries」の詳細全文を読む スポンサード リンク
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