Level $$r$$ r consensus and stable social choice

Mahajne, Muhammad; Nitzan, Shmuel; Volij, Oscar

doi:10.1007/s00355-015-0882-7

Cited by 11 publications

(18 citation statements)

References 20 publications

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“…In social choice theory, it is common to study restricted domains of preference profiles, such as single-peaked, single-crossing or level-r-consensus [37,39]. Many problems are much easier to solve in such restricted domains than in the domain of all preferences [27,28].…”

Section: Related Workmentioning

confidence: 99%

Fair Allocation with Diminishing Differences

Segal-Halevi¹,

Hassidim

Aziz

2020

jair

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Ranking alternatives is a natural way for humans to explain their preferences. It is being used in many settings, such as school choice, course allocations and residency matches. In some cases, several "items" are given to each participant. Without having any information on the underlying cardinal utilities, arguing about fairness of allocation requires extending the ordinal item ranking to ordinal bundle ranking. The most commonly used such extension is stochastic dominance (SD), where a bundle X is preferred over a bundle Y if its score is better according to all additive score functions. SD is a very conservative extension, by which few allocations are necessarily fair while many allocations are possibly fair. We propose to make a natural assumption on the underlying cardinal utilities of the players, namely that the difference between two items at the top is larger than the difference between two items at the bottom. This assumption implies a preference extension which we call diminishing differences (DD), where X is preferred over Y if its score is better according to all additive score functions satisfying the DD assumption. We give a full characterization of allocations that are necessarily-proportional or possibly-proportional according to this assumption. Based on this characterization, we present a polynomial-time algorithm for finding a necessarily-DD-proportional allocation if it exists. Using simulations, we show that with high probability, a necessarily-proportional allocation does not exist but a necessarily-DD-proportional allocation exists, and moreover, that allocation is proportional according to the underlying cardinal utilities. We also consider chore allocation under the analogous conditionincreasing-differences. (Avinatan Hassidim), UNSW Sydney and Data61 CSIRO, Australia. haris.aziz@unsw.edu.au (Haris Aziz)A preliminary version of this paper was published in the proceedings of IJCAI 2017 [42]. In the present version, the proofs are substantially revised and simplified, omitted proofs are included, and there are two new sections: one on division of chores (section 9) and one on binary utilities (section 10).

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Section: Related Workmentioning

confidence: 99%

Fair Allocation with Diminishing Differences

Segal-Halevi¹,

Hassidim

Aziz

2020

jair

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“…In the second experiment we used preference-profiles that were generated according to Mallows' phi model [12], which was claimed to favor level-1 consensus [11]. Mallows' model assumes that there is a "correct" preference * , and the actual preferences of the voters are noisy variants of it.…”

Section: Probability Of Level-1 Consensusmentioning

confidence: 99%

“…Our proofs below closely follow the proofs of[11]. Their proofs are stated for level-r consensus for general r, and indeed Flexible Condition 1 can also be adapted to general r, but for the sake of simplicity we prefer to focus on the case r = 1.…”

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confidence: 96%

Flexible level-1 consensus ensuring stable social choice: analysis and algorithms

Nitzan

Segal-Halevi

2017

Soc Choice Welf

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Level-1 Consensus is a property of a preference-profile. Intuitively, it means that there exists a preference relation which induces an ordering of all other preferences such that frequent preferences are those that are more similar to it. This is a desirable property, since it enhances the stability of social choice by guaranteeing that there exists a Condorcet winner and it is elected by all scoring rules.In this paper, we present an algorithm for checking whether a given preference profile exhibits level-1 consensus. We apply this algorithm to a large number of preference profiles, both real and randomly-generated, and find that level-1 consensus is very improbable. We support these empirical findings theoretically, by showing that, under the impartial culture assumption, the probability of level-1 consensus approaches zero when the number of individuals approaches infinity.Motivated by these observations, we show that the level-1 consensus property can be weakened while retaining its stability implications. We call this weaker property Flexible Consensus. We show, both empirically and theoretically, that it is considerably more probable than the original level-1 consensus. In particular, under the impartial culture assumption, the probability for Flexible Consensus converges to a positive number when the number of individuals approaches infinity. JEL classification number: D71

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“…A well-known method of this type is that of Kemeny [11] , where the Kendall (tau) distance [16] between rankings and the unanimous consensus state are considered. Other relevant proposals are due to Bogart [17,18] or Cook and Seiford [19,20] , where different distance functions are considered; or due to Meskanen and Nurmi [10,21] , Rademaker and De Baets [22] or Pérez-Fernández et al [23] , where different consensus states are considered [10,15,[24][25][26] .…”

Section: Introductionmentioning

confidence: 99%