Choosers as extension axioms

What is a monotonicity property? How should such a property be recast, so as to apply to voting rules that allow ties in the outcome? Our original interest was in the second question, as applied to six related properties for voting rules: monotonicity, participation, one-way monotonicity, half-way monotonicity, Maskin monotonicity, and strategy-proofness. This question has been considered for some of these properties: by Peleg and Barberà for monotonicity, by Moulin and Pérez et al, for participation, and by many authors for strategy-proofness. Our approach, however, is comparative; we examine the behavior of all six properties, under three general methods for handling ties: applying a set extension principle (in particular, Gärdenfors' sure-thing principle), using a tie-breaking agenda to break ties, and rephrasing properties via the "t-a-t" approach, so that only two alternatives are considered at a time. In attempting to explain the patterns of similarities and differences we discovered, we found ourselves obliged to confront the issue of what it is, exactly, that identifies these properties as a class. We propose a distinction between two such classes: the "tame" monotonicity properties (which include participation, half-way monotonicity, and strategy proofness) and the strictly broader class of "normal" monotonicity properties (which include monotonicity and one-way monotonicity, but not Maskin monotonicity). We explain why the tie-breaking agenda, t-a-t, and Gärdenfors methods are equivalent for tame monotonicities, and how, for properties that are normal but not tame, set-extension methods can fail to be equivalent to the other two (and may fail to make sense at all).

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“…Erdamar and Sanver (2009) characterize set extension principles in terms of the class of choice rules used to break ties.…”

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confidence: 99%

Monotonicity properties and their adaptation to irresolute social choice rules

Sanver

Zwicker

2012

Soc Choice Welf

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“…(Kelly, 1977) One interpretation of this extension is that voters are completely unaware of the mechanism (e.g., a lottery) that will be used to pick the winning alternative (Gärdenfors, 1979;Erdamar & Sanver, 2009). In other words, it contains exactly the pairwise comparisons which voters can make without knowledge of the mechanism (e.g., {a, b} R K i {c} if a P i b P i c).…”

Section: Strategyproofnessmentioning

confidence: 99%

“…Alternatively, the existence of a chairman who breaks ties according to a linear, but unknown, preference relation also rationalizes this preference extension (Erdamar & Sanver, 2009). For both of these interpretations, the extension describes exactly the conclusions a voter who is aware of the tie-breaking method can draw (e.g., {a, b} R F i {b, c} if a P i b P i c, which does not hold for Kelly's extension R K i ).…”

Section: Strategyproofnessmentioning

confidence: 99%

“…A function that yields a preference relation over subsets of alternatives when given a preference relation over single alternatives is called a set extension or preference extension. In this paper, we focus on two set extensions attributed to Kelly (1977) and Fishburn (1972), 1 which have been shown to arise uniquely under very natural assumptions (Gärdenfors, 1979;Erdamar & Sanver, 2009; see also Section 2.2 of this paper).…”

Section: Introductionmentioning

confidence: 99%

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Finding Strategyproof Social Choice Functions via SAT Solving

Brandt¹,

Geist²

2016

jair

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A promising direction in computational social choice is to address research problems using computer-aided proving techniques. In particular with SAT solvers, this approach has been shown to be viable not only for proving classic impossibility theorems such as Arrow's Theorem but also for finding new impossibilities in the context of preference extensions. In this paper, we demonstrate that these computer-aided techniques can also be applied to improve our understanding of strategyproof irresolute social choice functions. These functions, however, require a more evolved encoding as otherwise the search space rapidly becomes much too large. Our contribution is two-fold: We present an efficient encoding for translating such problems to SAT and leverage this encoding to prove new results about strategyproofness with respect to Kelly's and Fishburn's preference extensions. For example, we show that no Pareto-optimal majoritarian social choice function satisfies Fishburn-strategyproofness. Furthermore, we explain how human-readable proofs of such results can be extracted from minimal unsatisfiable cores of the corresponding SAT formulas.

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“…The assumptions we shall work with concern the principles by which a voter will extend her preferences over individual candidates to a preference order over (nonempty) sets of candidates when considering what ballot to submit. This is a well-studied problem in social choice theory, often referred to as ranking sets of objects (see, e.g., Barberà et al 2004;Gärdenfors 1976;Kelly 1977;Kannai and Peleg 1984;Nitzan and Pattanaik 1984;Puppe 1995;Can et al 2009;Erdamar and Sanver 2009;Geist and Endriss 2011) and we shall be examining incentives to vote sincerely under AV for a range of different such principles proposed in the literature.…”

Section: Introductionmentioning

confidence: 99%

Sincerity and manipulation under approval voting

Endriss

2012

Theory Decis

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Under approval voting, each voter can nominate as many candidates as she wishes and the election winners are those candidates that are nominated most often. A voter is said to have voted sincerely if she prefers all those candidates she nominated to all other candidates. As there can be a set of winning candidates rather than just a single winner, a voter's incentives to vote sincerely will depend on what assumptions we are willing to make regarding the principles by which voters extend their preferences over individual candidates to preferences over sets of candidates. We formulate two such principles, replacement and deletion, and we show that, under approval voting, a voter who accepts those two principles and who knows how the other voters will vote will never have an incentive to vote insincerely. We then discuss the consequences of this result for a number of standard principles of preference extension in view of sincere voting under approval voting. The insightful feedback received from the participants of these meetings is gratefully acknowledged. Thanks are also due to Steve Brams and an anonymous reviewer for comments on an earlier version of this paper, and to Remzi Sanver and Bill Zwicker for illuminating discussions on preference extensions. Keywords U. Endriss (B)

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Choosers as extension axioms

Abstract: Preferences over sets, Non-resolute outcomes,

Cited by 7 publications

References 30 publications

Monotonicity properties and their adaptation to irresolute social choice rules

Monotonicity properties and their adaptation to irresolute social choice rules

Finding Strategyproof Social Choice Functions via SAT Solving

Sincerity and manipulation under approval voting

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