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Social Choice and Welfare

Erschienen in:

23.08.2019 | Original Paper

Single-crossing choice correspondences

verfasst von: Matheus Costa, Paulo Henrique Ramos, Gil Riella

Erschienen in: Social Choice and Welfare | Ausgabe 1/2020

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Abstract

We characterize choice correspondences that can be justified by a collection of preferences that satisfy the single-crossing and/or single-peak properties. These are properties vastly used in the applied literature, often in the context of parametric families of objective (utility) functions that satisfy them. Apesteguia et al. (Econometrica 85(2):661–674, 2017) characterize random utility models that satisfy those properties, and we provide similar results in the context of choice correspondences that admit a pseudo-rational or justifiable choice representation. These results might be useful for researchers in situations where it is not possible to work with random utility models or parametric families of objective (utility) functions. In addition, we use them to discuss some aspects of the connection between deterministic and stochastic choice.

Vorheriger Artikel Identifying changing taste from demand data via golden eggs

Nächster Artikel Robust bounds on choosing from large tournaments

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We discuss this small literature in Sect. 4.

Martin and Merlin define a social choice correspondence (SCC) as a mapping from collections of individual preferences (strict linear orders) to the set of possible choices. In the context of majority voting, if, for each possible profile of individual preferences, the set of alternatives that are not dominated in pairwise majority voting by any other alternative is not empty, then the mapping from the preference profiles to these winner sets can be seen as a SCC. This winner set may, though, be empty for a given profile of preferences. In this sense, as is well known, single-peakedness and single-crossingness are sufficient conditions over the profile of preferences to ensure that this will not happen (i.e. there will always be a Condorcet winner) and majority voting can be represented as a SCC.

Investigating the implications of single-crossingness on social choice, Saporiti and Tohmé (2006) explore the impact of this property in a strategic voting environment, showing that it is capable of ensuring not only the existence of majority voting equilibria but also of non-manipulable choice rules.

Bredereck et al. (2013) explore the characterization of a single-crossing profile of preferences, exposing some forbidden structures and allowing a computationally viable way to test if a preference profile satisfies single-crossingness when the ordering of voters is given, but not necessarily the exogenous ordering of the alternatives. This result relates to our work in the sense that we explore the characterization of choice correspondences that derive from a single-crossing profile, though here we specify the ordering of the alternatives but not the ordering of voters or, in our case, individual preferences.

That is, \(x\succsim y\) and \(y\succsim x\) imply that \(x=y\), for every \(x,y\in X\).

For any binary relation \(\succsim \subseteq X\times X\) and choice problem \(A\in \Omega _{X}\), the set of maximum elements of \(\succsim \) in A is defined as \( max (A,\succsim ):=\{x\in A:x\succsim y,\forall y\in A\}\)

We are making a slight abuse of notation here. Since \(\succcurlyeq _{i}\) is a linear order, we are writing \(\max (X,\succcurlyeq _{i})\) to represent the unique element \(x^{*}\in X\) such that \(x^{*}\succcurlyeq _{i}z\) for every \(z\in X\).

We again make an abuse of notation here. We are writing \(\min (X,\succcurlyeq _{i})\) to represent the unique element \(x^{*}\in X\) such that \(z\succcurlyeq _{i}x^{*}\) for every \(z\in X\).

By \(\hat{w}\) being the successor of w with respect to \(\succcurlyeq _{n}\) we mean that \(\hat{w}\succ _{n}w\) and for no \(\tilde{w}\in X\) it is true that \(\hat{w}\succ _{n}\tilde{w}\succ _{n}w\).

Notice that if \(x\succ ^{*}y\) and \(y\succ x\), then \(y\succ _{j}x\) for every \(j\in \{1,\dots ,n\}\).

Since \(x\notin \max (A,\succcurlyeq _{i^{*}-1})\), such z must exist.

Otherwise we would have \(\max (A,\succcurlyeq _{i^{*}-1})\succcurlyeq _{i^{*}-1}z\succ _{i^{*}-1}w\) for every \(w\in X{\setminus } A\), contradicting Claim 1.

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Titel: Single-crossing choice correspondences
verfasst von: Matheus Costa
Paulo Henrique Ramos
Gil Riella
Publikationsdatum: 23.08.2019
Verlag: Springer Berlin Heidelberg
Erschienen in: Social Choice and Welfare / Ausgabe 1/2020
Print ISSN: 0176-1714
Elektronische ISSN: 1432-217X
DOI: https://doi.org/10.1007/s00355-019-01212-7

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