Abstract
Voting rules that are based on the majority graph typically output large sets of winners. In this full original paper our goal is to investigate a general method which leads to randomized version of such rules. We use the idea of parallel universes, where each universe is connected with a permutation over alternatives. The permutation allows us to construct resolute voting rules (i.e. rules that always choose unique winners). Such resolute rules can be constructed in a variety of ways: we consider using binary voting trees to select a single alternative. In turn this permits the construction of neutral rules that output the set the possible winners of every parallel universe. The question of which rules can be constructed in this way has already been partially studied under the heading of agenda implementability. We further propose a randomised version in which the probability of being the winner is the ratio of universes in which the alternative wins. We also briefly consider (typically novel) rules that elect the alternatives that have maximal winning probability. These rules typically output small sets of winners, thus provide refinements of known tournament solutions.
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Notes
- 1.
A binary relation \(R\subseteq X\times X\) trichotomous if for all \(a,b\in X\), either aTb, bTa or \(a=b\).
- 2.
More precisely, this is one instance of the simple tree. We give a full recursive definition later in this section.
- 3.
By permutation here do not mean all possible assignments of alternatives to the leaves: indeed, this would imply that all alternatives are trivially returned, as the tree where all leaves are the same alternative must return that alternative. We are not aware of any work that considers possible winners under some generalised “multiple assignment procedure” of the leaves of a binary tree.
- 4.
The question of what tournament rules are implementable by parallel universes in this manner has been studied by Horan [13], who gives necessary and sufficient conditions.
- 5.
The top cycle of a tournament is the maximal set of alternatives such that the restriction of the tournament to these alternatives contains a cycle.
- 6.
An alternative is undominated if it there is some alternative that does not dominate it.
References
Altman, A., Kleinberg, R.: Nonmanipulable randomized tournament selections. In: AAAI (2010)
Aziz, H.: Maximal recursive rule: a new social decision scheme. In Proceedings of IJCAI 2013, pp. 34–40. AAAI Press (2013)
Aziz, H., Stursberg, P.: A generalization of probabilistic serial to randomized social choice. In: Proceedings of the Twenty-Eighth AAAI Conference on Artificial Intelligence, 27–31 July 2014, Québec City, Québec, Canada, pp. 559–565 (2014)
Banks, S.J.: Sophisticated voting outcomes and agenda control. Soc. Choice Welfare 1(4), 295–306 (1985)
Barberà, S.: Majority and positional voting in a probabilistic framework. Rev. Econ. Stud. 46(2), 379–389 (1979)
Bogomolnaia, A., Moulin, H., Stong, R.: Collective choice under dichotomous preferences. J. Econ. Theory 122(2), 165–184 (2005)
Brandt, F., Brill, M., Harrenstein, P.: Tournament solutions. In: Handbook of Computational Social Choice, chap. 3. Cambridge University Press, Cambridge (2016)
Brill, M., Fischer, F.: The price of neutrality for the ranked pairs method. In: Proceedings of AAAI-2012, pp. 1299–1305 (2012)
Conitzer, V., Rognlie, M., Xia, L.: Preference functions that score rankings and maximum likelihood estimation. In: Proceedings of IJCAI-2009, pp. 109–115 (2009)
Fischer, F., Procaccia, A.D., Samorodnitsky, A.: A new perspective on implementation by voting trees. In: Proceedings of the 10th ACM Conference on Electronic Commerce, pp. 31–40. ACM (2009)
Freeman, R., Brill, M., Conitzer, V.: General tiebreaking schemes for computational social choice. In: Proceedings of AAMAS-2015 (2015)
Gibbard, A.: Manipulation of schemes that mix voting with chance. Econometrica 45, 665–681 (1977)
Horan, S.: Implementation of majority voting rules. Preprint (2013)
Hudry, O.: A note on “Banks winners in tournaments are difficult to recognize” by G. J. Woeginger. Soc. Choice Welfare 23(1), 113–114 (2004)
Kreweras, G.: Aggregation of preference orderings. In: Mathematics and Social Sciences I: Proceedings of the Seminars of Menthon-Saint-Bernard, France, 1–27 July 1960, Gösing, Austria, 3–27 July 1962, pp. 73–79 (1965)
Lang, J., Pini, M.S., Rossi, F., Venable, K.B., Walsh, T.: Winner determination in sequential majority voting. In: IJCAI 2007, vol. 7, pp. 1372–1377 (2007)
Laslier, J.-F.: Tournament Solutions and Majority Voting. Springer, Heidelberg (1997)
Moon, J.W.: Topics on Tournaments in Graph Theory. Holt, Rinehart and Winston (1968)
Moulin, H.: Choosing from a tournament. Soc. Choice Welfare 3(4), 271–291 (1986)
Nicolaus, T.: Independence of clones as a criterion for voting rules. Soc. Choice Welfare 4(3), 185–206 (1987)
Woeginger, G.J.: Banks winners in tournaments are difficult to recognize. Soc. Choice Welfare 20(3), 523–528 (2003)
Acknowledgement
Justin Kruger and Stéphane Airiau are supported by the ANR project CoCoRICo-CoDec.
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A Tournament for which \(\mathsf {st}^\mathsf {AM}\) is distinct from \(\mathsf {ic}^\mathsf {AM}\)
A Tournament for which \(\mathsf {st}^\mathsf {AM}\) is distinct from \(\mathsf {ic}^\mathsf {AM}\)
We note that this tournament was found by computer; the actual counting of the different universes for which a particular alternative wins is slightly tedious. The tournament itself is shown in Fig. 6.
A six alternative tournament T for which \(\mathsf {st}^\mathsf {AM}= \{4\}\) and \(\mathsf {ic}^\mathsf {AM}=\{0\}\).
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Kruger, J., Airiau, S. (2018). Permutation-Based Randomised Tournament Solutions. In: Belardinelli, F., Argente, E. (eds) Multi-Agent Systems and Agreement Technologies. EUMAS AT 2017 2017. Lecture Notes in Computer Science(), vol 10767. Springer, Cham. https://doi.org/10.1007/978-3-030-01713-2_17
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