Breaking ties in collective decision-making
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Breaking ties in collective decision-making Daniela Bubboloni1
· Michele Gori2
Received: 5 February 2020 / Accepted: 13 July 2020 © The Author(s) 2020
Abstract Many classic social preference (multiwinner social choice) correspondences are resolute only when two alternatives and an odd number of individuals are considered. Thus, they generally admit several resolute refinements, each of them naturally interpreted as a tie-breaking rule. A tie-breaking rule is compulsory every time a single final decision is needed. Unfortunately, using a tie-breaking rule on some social preference (multiwinner social choice) correspondence can dramatically compromise its properties. In particular, very often, the arithmetic relation between the number of alternatives and the number of voters does not allow to maintain both anonymity and neutrality. In those cases, the only possibility is to look at suitable different forms of symmetry that are coherent with the decision context. We find out conditions which make a social preference (multiwinner social choice) correspondence admit a resolute refinement fulfilling some weak versions of the anonymity and neutrality principles. We also clear when it is possible to obtain, for those resolute refinements, the reversal symmetry (immunity to the reversal bias). The theory we develop turns out to be useful in many common applicative contexts and allows to explicitly construct those refinements. Keywords Social preference correspondence · Multiwinner social choice correspondence · Resoluteness · Anonymity · Neutrality · Tie-breaking rule JEL Classification D71
Daniela Bubboloni was supported by GNSAGA of INdAM.
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Michele Gori [email protected] Daniela Bubboloni [email protected]
1
Dipartimento di Matematica e Informatica “Ulisse Dini”, Università degli Studi di Firenze, viale Morgagni 67/a, 50134 Florence, Italy
2
Dipartimento di Scienze per l’Economia e l’Impresa, Università degli Studi di Firenze, via delle Pandette 9, 50127 Florence, Italy
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D. Bubboloni, M. Gori
1 Introduction Consider a committee having h ≥ 2 members whose purpose is to determine a ranking of n ≥ 2 alternatives and assume that committee members are supposed to express their preferences via a ranking of the alternatives. A preference profile is a list of individual preferences, one for each committee member. A social preference correspondence (spc) is a procedure which associates with any preference profiles a family of rankings of the alternatives to be interpreted as the family of the social preferences which better fit the individual preferences. A k-multiwinner social choice correspondence (k-scc) is instead a procedure which associates with any preference profile a family of sets of k alternatives to be interpreted as the family of all the sets of k alternatives that can be considered the best k alternatives for the society. While the concept of spc is classic, the one of k-scc, which extends the well-established single-winner framework (corresponding to the case k = 1), is more recent. I
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