{"type":"ScholarlyArticle","title":"An Analytical and Experimental Comparison of Maximal Lottery Schemes","authors":[{"name":"Florian Brandl","slug":"florian-brandl","given":"Florian","family":"Brandl","affiliation":"Technische Universität München"},{"name":"Felix Brandt","slug":"felix-brandt","given":"Felix","family":"Brandt","affiliation":"Technische Universität München"},{"name":"Christian Stricker","slug":"christian-stricker","given":"Christian","family":"Stricker","affiliation":"Technische Universität München"}],"abstract":"Randomized voting rules are gaining increasing attention in computational and non-computational social choice. A particularly interesting class of such rules are maximal lottery (ML) schemes, which were proposed by Peter Fishburn in 1984 and have been repeatedly recommended for practical use. However, the subtle differences between different ML schemes are often ignored. Two canonical subsets of ML schemes are C1-ML schemes (which only depend on unweighted majority comparisons) and C2-ML schemes (which only depend on weighted majority comparisons). We prove that C2-ML schemes are the only Pareto efficient---but also among the most manipulable---ML schemes. Furthermore, we evaluate the frequency of manipulable preference profiles and the degree of randomization of ML schemes via extensive computer simulations. In general, ML schemes are rarely manipulable and often do not randomize at all, especially when there are only few alternatives. For up to 21 alternatives, the average support size of ML schemes lies below 4 under reasonable assumptions. The average degree of randomization (in terms of Shannon entropy) of C2-ML schemes is significantly lower than that of C1-ML schemes.","doi":"10.24963/ijcai.2018/16","url":"https://ijcai.org/proceedings/2018/16","pdf":"https://ijcai.org/proceedings/2018/0016.pdf","markdown":"https://ijcai.org/proceedings/2018/16.md","bibtex":"https://ijcai.org/proceedings/2018/bibtex/16","volume":{"slug":"2018","title":"Proceedings of the Twenty-Seventh International Joint Conference on Artificial Intelligence","short_title":"IJCAI 2018","url":"https://ijcai.org/proceedings/2018/","year":2018,"location":"Stockholm","dates":"13-19 July 2018","editors":"Jérôme Lang","isbn":"978-0-9992411-2-7","publisher":"International Joint Conferences on Artificial Intelligence Organization","doi_prefix":"10.24963/ijcai.2018"},"section":"Agent-based and Multi-agent Systems","track":"Main track","number":16,"pages":{"first":114,"last":120},"keywords":["Agent-based and Multi-agent Systems: Voting","Agent-based and Multi-agent Systems: Computational Social Choice"],"videos":[],"references":[{"position":1,"raw":"[Aziz et al., 2013] H. Aziz, F. Brandt, and M. Brill. On the tradeoff between economic efficiency and strategyproof- ness in randomized social choice. In Proc. of 12th AAMAS Conference, pages 455-462. IFAAMAS, 2013.","title":"On the tradeoff between economic efficiency and strategyproofness in randomized social choice","authors":"H Aziz, F Brandt, M Brill","year":"2013","venue":"Proc. of 12th AAMAS Conference","doi":null,"url":null,"internal_url":null},{"position":2,"raw":"[Brandl and Brandt, 2018] F. Brandl and F. Brandt. Arrovian aggregation of convex preferences. 2018. Working Paper.","title":"Arrovian aggregation of convex preferences","authors":"F Brandl, F Brandt","year":"2018","venue":null,"doi":null,"url":null,"internal_url":null},{"position":3,"raw":"[Brandl et al., 2016] F. Brandl, F. Brandt, and H. G. Seedig. Consistent probabilistic social choice. Econometrica, 84(5):1839-1880, 2016.","title":"Consistent probabilistic social choice","authors":"F Brandl, F Brandt, H G Seedig","year":"2016","venue":"Econometrica","doi":null,"url":null,"internal_url":null},{"position":4,"raw":"[Brandl et al., 2018a] F. Brandl, F. Brandt, M. Eberl, and C. Geist. Proving the incompatibility of efficiency and strategyproofness via SMT solving. Journal of the ACM, 65(2), 2018.","title":"Proving the incompatibility of efficiency and strategyproofness via SMT solving","authors":"Brandl","year":"2018","venue":"Journal of the ACM","doi":null,"url":null,"internal_url":null},{"position":5,"raw":"[Brandl et al., 2018b] F. Brandl, F. Brandt, and J. Hofbauer. Welfare maximization entices participation. Games and Economic Behavior, 2018. Forthcoming.","title":"Welfare maximization entices participation","authors":"Brandl","year":"2018","venue":"Games and Economic Behavior","doi":null,"url":null,"internal_url":null},{"position":6,"raw":"[Brandt and Seedig, 2014] F. Brandt and H. G. Seedig. On the discriminative power of tournament solutions. In Proc. of 1st EXPLORE Workshop, 2014.","title":"On the discriminative power of tournament solutions","authors":"F Brandt, H G Seedig","year":"2014","venue":"Proc. of 1st EXPLORE Workshop","doi":null,"url":null,"internal_url":null},{"position":7,"raw":"[Brandt et al., 2016] F. Brandt, C. Geist, and P. Harrenstein. A note on the McKelvey uncovered set and Pareto opti- mality. Social Choice and Welfare, 46(1):81-91, 2016.","title":"A note on the McKelvey uncovered set and Pareto optimality","authors":"F Brandt, C Geist, P Harrenstein","year":"2016","venue":"Social Choice and Welfare","doi":null,"url":null,"internal_url":null},{"position":8,"raw":"[Brandt et al., 2018] F. Brandt, M. Brill, H. G. Seedig, and W. Suksompong. On the structure of stable tournament solutions. Economic Theory, 65(2):483-507, 2018.","title":"On the structure of stable tournament solutions","authors":"F Brandt, M Brill, H G Seedig, W Suksompong","year":"2018","venue":"Economic Theory","doi":null,"url":null,"internal_url":null},{"position":9,"raw":"[Campbell and Kelly, 2003] D. E. Campbell and J. S. Kelly. A strategy-proofness characterization of majority rule. Economic Theory, 22(3):557-568, 2003.","title":"A strategy-proofness characterization of majority rule","authors":"D E Campbell, J S Kelly","year":"2003","venue":"Economic Theory","doi":null,"url":null,"internal_url":null},{"position":10,"raw":"[Dutta and Laslier, 1999] B. Dutta and J.-F. Laslier. Compar- ison functions and choice correspondences. Social Choice and Welfare, 16(4):513-532, 1999.","title":"Comparison functions and choice correspondences","authors":"B Dutta, J.-F Laslier","year":"1999","venue":"Social Choice and Welfare","doi":null,"url":null,"internal_url":null},{"position":11,"raw":"[Felsenthal and Machover, 1992] D. S. Felsenthal and M. Machover. After two centuries should Condorcet's voting procedure be implemented? Behavioral Science, 37(4):250-274, 1992.","title":"After two centuries should Condorcet's voting procedure be implemented?","authors":"D S Felsenthal, M Machover","year":"1992","venue":"Behavioral Science","doi":null,"url":null,"internal_url":null},{"position":12,"raw":"[Fishburn, 1977] P. C. Fishburn. Condorcet social choice functions. SIAM Journal on Applied Mathematics, 33(3):469-489, 1977.","title":"Condorcet social choice functions","authors":"P C Fishburn","year":"1977","venue":"SIAM Journal on Applied Mathematics","doi":null,"url":null,"internal_url":null},{"position":13,"raw":"[Fishburn, 1984a] P. C. Fishburn. Dominance in SSB util- ity theory. Journal of Economic Theory, 34(1):130-148, 1984.","title":"Dominance in SSB utility theory","authors":"P C Fishburn, Fishburn","year":"1984","venue":"Journal of Economic Theory","doi":null,"url":null,"internal_url":null},{"position":14,"raw":"[Fishburn, 1984b] P. C. Fishburn. Probabilistic social choice based on simple voting comparisons. Review of Economic Studies, 51(4):683-692, 1984.","title":"Probabilistic social choice based on simple voting comparisons","authors":"P C Fishburn, Fishburn","year":"1984","venue":"Review of Economic Studies","doi":null,"url":null,"internal_url":null},{"position":15,"raw":"[Fisher and Ryan, 1995] D. C. Fisher and J. Ryan. Tourna- ment games and positive tournaments. Journal of Graph Theory, 19(2):217-236, 1995.","title":"Tournament games and positive tournaments","authors":"D C Fisher, J Ryan","year":"1995","venue":"Journal of Graph Theory","doi":null,"url":null,"internal_url":null},{"position":16,"raw":"[Gehrlein and Lepelley, 2011] W. V. Gehrlein and D. Lepel- ley. Voting Paradoxes and Group Coherence. Studies in Choice and Welfare. Springer-Verlag, 2011.","title":"Voting Paradoxes and Group Coherence","authors":"W V Gehrlein, D Lepelley","year":"2011","venue":null,"doi":null,"url":null,"internal_url":null},{"position":17,"raw":"[Gehrlein, 2002] W. V. Gehrlein. Condorcet's paradox and the likelihood of its occurrence: Different perspectives on balanced preferences. Theory and Decision, 52(2):171- 199, 2002.","title":"Condorcet's paradox and the likelihood of its occurrence: Different perspectives on balanced preferences","authors":"W V Gehrlein","year":"2002","venue":"Theory and Decision","doi":null,"url":null,"internal_url":null},{"position":18,"raw":"[Gibbard, 1977] A. Gibbard. Manipulation of schemes that mix voting with chance. Econometrica, 45(3):665-681, 1977.","title":"Manipulation of schemes that mix voting with chance","authors":"A Gibbard","year":"1977","venue":"Econometrica","doi":null,"url":null,"internal_url":null},{"position":19,"raw":"[Hoang, 2017] L. N. Hoang. Strategy-proofness of the ran- domized Condorcet voting system. Social Choice and Wel- fare, 48(3):679-701, 2017.","title":"Strategy-proofness of the randomized Condorcet voting system","authors":"L N Hoang","year":"2017","venue":"Social Choice and Welfare","doi":null,"url":null,"internal_url":null},{"position":20,"raw":"[Kreweras, 1965] G. Kreweras. Aggregation of preference orderings. In Mathematics and Social Sciences I, pages 73-79, 1965.","title":"Aggregation of preference orderings","authors":"G Kreweras","year":"1965","venue":"Mathematics and Social Sciences I","doi":null,"url":null,"internal_url":null},{"position":21,"raw":"[Laffond et al., 1993] 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.","title":"The bipartisan set of a tournament game","authors":"G Laffond, J.-F Laslier, M Le Breton","year":"1993","venue":"Games and Economic Behavior","doi":null,"url":null,"internal_url":null},{"position":22,"raw":"[Laffond et al., 1997] G. Laffond, J.-F. Laslier, and M. Le Breton. A theorem on symmetric two-player zero-sum games. Journal of Economic Theory, 72(2):426-431, 1997.","title":"A theorem on symmetric two-player zero-sum games","authors":"G Laffond, J.-F Laslier, M Le Breton","year":"1997","venue":"Journal of Economic Theory","doi":null,"url":null,"internal_url":null},{"position":23,"raw":"[Laslier, 2000] J.-F. Laslier. Aggregation of preferences with a variable set of alternatives. Social Choice and Welfare, 17(2):269-282, 2000.","title":"Aggregation of preferences with a variable set of alternatives","authors":"J.-F Laslier","year":"2000","venue":"Social Choice and Welfare","doi":null,"url":null,"internal_url":null},{"position":24,"raw":"[Laslier, 2010] J.-F. Laslier. In silico voting experiments. In Handbook on Approval Voting, chapter 13, pages 311-335. Springer-Verlag, 2010.","title":"In silico voting experiments","authors":"J.-F Laslier","year":"2010","venue":"Handbook on Approval Voting","doi":null,"url":null,"internal_url":null},{"position":25,"raw":"[Mattei et al., 2012] N. Mattei, J. Forshee, and J. Goldsmith. An empirical study of voting rules and manipulation with large datasets. In Proc. of 4th COMSOC, 2012.","title":"An empirical study of voting rules and manipulation with large datasets","authors":"N Mattei, J Forshee, J Goldsmith","year":"2012","venue":"Proc. of 4th COMSOC","doi":null,"url":null,"internal_url":null},{"position":26,"raw":"[Moulin, 1988] H. Moulin. Axioms of Cooperative Decision Making. Cambridge University Press, 1988.","title":"Axioms of Cooperative Decision Making","authors":"H Moulin","year":"1988","venue":null,"doi":null,"url":null,"internal_url":null},{"position":27,"raw":"[Peyre, 2013] R. Peyre. La quête du graal électoral. Images des Mathématiques, CNRS, 2013.","title":"La quête du graal électoral","authors":"R Peyre","year":"2013","venue":null,"doi":null,"url":null,"internal_url":null},{"position":28,"raw":"[Regenwetter et al., 2006] M. Regenwetter, B. Grofman, A. A. J. Marley, and I. M. Tsetlin. Behavioral Social Choice: Probabilistic Models, Statistical Inference, and Applications. Cambridge University Press, 2006.","title":"Behavioral Social Choice: Probabilistic Models, Statistical Inference, and Applications","authors":"M Regenwetter, B Grofman, A A J Marley, I M Tsetlin","year":"2006","venue":null,"doi":null,"url":null,"internal_url":null},{"position":29,"raw":"[Rivest and Shen, 2010] R. L. Rivest and E. Shen. An op- timal single-winner preferential voting system based on game theory. In Proc. of 3rd COMSOC, pages 399-410, 2010.","title":"An optimal single-winner preferential voting system based on game theory","authors":"R L Rivest, E Shen","year":"2010","venue":"Proc. of 3rd COMSOC","doi":null,"url":null,"internal_url":null},{"position":30,"raw":"[Tideman and Plassmann, 2012] N. Tideman and F. Plass- mann. Modeling the outcomes of vote-casting in actual elections. In Electoral Systems: Paradoxes, Assumptions, and Procedures. Springer, 2012.","title":"Modeling the outcomes of vote-casting in actual elections","authors":"N Tideman, F Plassmann","year":"2012","venue":"Electoral Systems: Paradoxes, Assumptions, and Procedures","doi":null,"url":null,"internal_url":null}],"license_note":"Copyright International Joint Conferences on Artificial Intelligence."}