Breaking ties in collective decision-making

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.

Most Equitable Voting Rules

“…Our characterization of ANR k-list rules (Proposition 2) is new. Our MFP tie-breaking mechanisms, despite being simple and natural, address the computational challenge pointed out in previous work [Bubboloni and Gori, 2021]. Therefore, we view the surprising simplicity in Lemma 1 and MFP breaking an advantage.…”

“…For choosing a k-committee, Bubboloni and Gori [2016] proved that an ANR rule that satisfies Pareto efficiency exists if and only if gcd(m!, n) = 1. Bubboloni and Gori [2021] provided sufficient conditions for the existence of ANR rules that satisfy weaker notions of anonymity and neutrality. However, the fundamental question of the existence of ANR k-committee rules (without being required to satisfy any other properties) remained open, which is resolved by our Proposition 1.…”

“…Dogan and Giritligil [2015] proposed a mechanism to reduce the number of winners to minimize the "irresoluteness outlook", i.e., the set of possible sizes of the winners under all profiles. Bubboloni and Gori [2021] proposed tie-breaking mechanisms for k-committees and rankings, that are based on defining a system of "representatives" for each equivalent class of profiles (see the proof of our Lemma 1 for a similar high-level idea). They also highlighted the computational chal-lenges in their approach: (using our notation) "finding explicitly a system of representatives is in general a complex problem.…”

Characterization of tie-breaking plurality rules

“…Both studies analyzed binary social choice functions satisfying anonymity and some weakenings of neutrality. Recently, Bubboloni and Gori (2021) and Ozkes and Sanver (2021) treated the same setting as ours. Bubboloni and Gori (2021) employed two axioms: one is a weakening of anonymity and the other is a weakening of neutrality.…”

“…Recently, Bubboloni and Gori (2021) and Ozkes and Sanver (2021) treated the same setting as ours. Bubboloni and Gori (2021) employed two axioms: one is a weakening of anonymity and the other is a weakening of neutrality. They gave a condition for which those two axioms are compatible.…”