Consistency of scoring rules: a reinvestigation of composition-consistency
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Consistency of scoring rules: a reinvestigation of composition-consistency Z. Emel Öztürk1 Accepted: 16 January 2020 © The Author(s) 2020
Abstract We consider a collective choice problem in which the number of alternatives and the number of voters vary. Two fundamental axioms of consistency in such a setting, reinforcement and composition-consistency, are incompatible. We first observe that the latter implies four conditions each of which can be formulated as a consistency axiom on its own right. We find that two of these conditions are compatible with reinforcement. In fact, one of these, called composition-consistency with respect to non-clone winners, turns out to characterize a class of scoring rules which contains the Plurality rule. When combined with a requirement of monotonicity, compositionconsistency with respect to non-clone winners uniquely characterizes the Plurality rule. A second implication of composition-consistency leads to a class of scoring rules that always select a Plurality winner when combined with monotonicity. Keywords Plurality rule · Cloning-consistency · Composition-consistency · Reinforcement · Scoring rules · Monotonicity JEL Classification D70 · D71 · D72
1 Introduction The outcome of an election can be manipulated by an interested party to achieve strategic results. This paper studies two consistency conditions that rule out manipulations via partitioning the electorate into sub-electorates and manipulations via altering the set of alternatives running in the election.
Z. Emel Öztürk: I am grateful to Patrick Harless for helpful comments. I would also like to thank audiences in the 5th World Congress of the Game Theory Society and participants in seminars in Glasgow and Maastricht.
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Z. Emel Öztürk [email protected] Department of Economics, Tilburg University, 5000 LE Tilburg, The Netherlands
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Z. E. Öztürk
The framework includes a potential set of electorates and a potential set of alternatives. Each individual has preferences over the set of all possible alternatives. A social choice rule specifies the winning alternative(s) as a function of the set of entering alternatives (called agendas) and voters’ preferences over the entering alternatives. We focus only on social choice rules that are neutral and anonymous. Additionally, we often insist that the social choice rule satisfy faithfulness. The condition states that if there is only one voter, the most preferred alternative of the single voter should be uniquely chosen. Our first consistency condition, reinforcement due to Young (1974), says that if two disjoint electorates have some alternative in common in their choice set, then the choice set of their union should consist of the common choices of the separate groups.1 The second consistency axiom, composition-consistency due to Laffond et al. (1996), uses the notion of a set of clones. We say that a set of alternatives A is a set of clones at a preference profile if the alternatives in A perform identically in pairwise comparisons with alternatives
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