Maximal lotteries
Maximal lotteries refers to a probabilistic voting system first considered by Germain Kreweras[1] in 1965 and independently rediscovered and studied in much more detail in 1984 by Peter C. Fishburn.[2]. 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. 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).[3] It was also shown that maximal lotteries satisfy a strong notion of Pareto efficiency and a weak notion of strategyproofness.[4]
Maximal lotteries or variants thereof have been rediscovered multiple times by economists,[5] mathematicians,[6] political scientists, philosophers,[7] and computer scientists.[8] In particular, the support of maximal lotteries, which is known as the essential set or the bipartisan set, has been studied in detail.
References
- ^ 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.
- ^ P. C. Fishburn. Probabilistic social choice based on simple voting comparisons. Review of Economic Studies, 51(4):683–692, 1984.
- ^ F. Brandl, F. Brandt, and H. G. Seedig. Consistent probabilistic social choice. Working paper. 2015.
- ^ 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.
- ^ 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.
- ^ D. C. Fisher and J. Ryan. Tournament games and positive tournaments. Journal of Graph Theory, 19(2):217–236, 1995.
- ^ D. S. Felsenthal and M. Machover. After two centuries should Condorcet’s voting procedure be implemented? Behavioral Science, 37(4):250–274, 1992.
- ^ 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.
External links
- Pnyx - Practical Preference Aggregation (voting website that, among other methods, offers maximal lotteries)
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