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Social Choice and Welfare

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01.01.2014 | Original Paper

Path independent choice and the ranking of opportunity sets

verfasst von: Matthew Ryan

Erschienen in: Social Choice and Welfare | Ausgabe 1/2014

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Abstract

The indirect utility principle provides an instrumentalist basis for ranking opportunity sets, given an underlying preference ranking on alternatives. Opportunity set A is weakly preferred to B if A includes at least one preference-maximising element from $A\cup B$. We introduce the Plott consistency principle as a natural extension of this logic to decision-makers who choose amongst alternatives according to a path independent choice function. Such choice functions need not be rationalisable by a preference order. Plott consistency requires that A is an acceptable choice from $\left\{ A, B\right\} $ if A includes at least one element from the set of acceptable choices from $A\cup B$. We explore necessary and sufficient conditions (imposed on a choice function defined on collections of opportunity sets) for Plott consistency.

Vorheriger Artikel A multidimensional Lorenz dominance relation

Nächster Artikel Sequential contests with synergy and budget constraints

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There is also a large literature on non-instrumentalist aspects of opportunity. Enhanced freedom of choice may have intrinsic value, over and above its instrumental utility. A useful survey of this literature is provided by Dowding and van Hees (2009).

See, for example, Nehring (1997).

As usual, we define $\succ $ to be the asymmetric part of $\succsim $ (so $ \left( x, y\right) \in \succ $ iff $\left( x, y\right) \in \succsim $ and $ \left( y, x\right) \notin \succsim $) and $\sim $ to be its symmetric part (so $\left( x, y\right) \in \sim $ iff $\left( x, y\right) \in \succsim $ and $ \left( y, x\right) \in \succsim $). We also follow convention by writing $ x\succsim y$ for $\left( x, y\right) \in \succsim $, $x\succ y$ for $\left( x, y\right) \in \succ $ and $x\sim y$ for $\left( x, y\right) \in \sim $ whenever convenient. Recall that $\succsim $ is complete if

$$\begin{aligned} \left\{ \left( x, y\right) ,\ \left( y, x\right) \right\} \cap \succsim \ \ne \ \emptyset \end{aligned}$$

for any $x, y\in X$. The binary relation $\succsim $ is acyclic if, for any $\left\{ x_{1}, x_{2},..., x_{n}\right\} \subseteq X$, $\left( x_{i}, x_{i+1}\right) \in \succ $ for each $i\in \left\{ 1, 2,..., n-1\right\} $ implies $x_{n}\ne x_{1}$. Analogous notation and definitions apply to binary relations on $\left[ X\right] $.

See, for example, Moulin (1985), Lemma 1.

Recall that $\succsim $ is transitive if, for any $x, y, z\in X$, $ x\succsim y$ and $y\succsim z$ implies $x\succsim z$. We say that $\succsim $ is quasi-transitive if $\succ $ is transitive. Once again, similar definitions apply to binary relations on $\left[ X\right] $.

If $\succsim ^{*}\subseteq \left[ X\right] \times \left[ X\right] $ satisfies (IU) for some complete and acyclic $\succsim \subseteq X\times X$, then

$$\begin{aligned} A\succsim ^{*}B\quad \quad&\Leftrightarrow \quad \quad \left[ \max _{\succsim }A\cup B \right] \cap A\ne \emptyset \\&\Leftrightarrow \quad \quad A\succsim ^{*}A\cup B \end{aligned}$$

Moreover, $A\cup B\succsim ^{*}A$ is immediate from (IU).

Proofs of all results may be found in the Appendix.

The mapping defined by (PC) is a choice function, since $\emptyset \ne c^{*}\left( \mathcal A \right) \subseteq \mathcal A $ for any $\mathcal A \in \Sigma $.

Our terminology is chosen to emphasise the relationship with Lahiri’s axioms, but versions of the SIC condition have appeared elsewhere in the literature under different names. Pattanaik and Peleg (1984) refer to a version of SIC in which $B$ is restricted to singletons as the union property. Danilov and Koshevoy (2006) likewise refer to SIC as the union property (with $\succ ^{*}$ interpreted as a binary dependence relation). Other authors (e.g., Arlegi 2003; Barberà et al. 2004) have referred to SIC with $\succsim ^{*}$ in place of $\succ ^{*}$ as the robustness property.

Condition (K) implies monotonicity with respect to set inclusion: for any $ A, B\in \left[ X\right] $

$$\begin{aligned} B\subseteq A\Rightarrow A\succsim ^{*}B \end{aligned}$$

(M)

To see why, note that if $B\subseteq A$ and $B\succ ^{*}A$ then (K) would imply the contradiction $B\sim ^{*}A$. We could therefore replace “$\subseteq $” in (SgM1) and (SgM2) with “$\precsim ^{*}$”. Hence the “monotone transitivity” interpretation of Strong Monotonicity.

The 1981 paper by Aizerman and Malishevski does not actually include a proof of this result. According to Kukushkin (2004), the original proof may be found dispersed through three other papers, all in Russian. Moulin (1985) provides a self-contained proof in English. The Aizerman-Malishevski result also follows from Theorems 5.1 and 5.2 in Edelman and Jamison (1985).

A binary relation is antisymmetric if its symmetric part contains only elements of the form $\left( x, x\right) $.

Property (M).

Aizerman MA, Malishevski AV (1981) General theory of best variants choice: some aspects. IEEE Trans Autom Control 26:1030–1041CrossRef

Arlegi R (2003) A note on Bossert, Pattanaik and Xu’s ‘Choice under complete uncertainty: axiomatic characterization of some decision rules’. Econ Theory 22:219–225CrossRef

Barberà S, Bossert W, Pattanaik PK (2004) Ranking sets of objects. In: Barberà P, Hammond J, Seidl C (eds) Handbook of Utility Theory, vol 2. Kluwer Academic, DordrechtCrossRef

Danilov V, Koshevoy G (2006) A new characterisation of the path independent choice functions. Math Soc Sci 51:238–245CrossRef

Dowding K, van Hees M (2009) Freedom of choice. In: Anand P, Pattanaik PK, Puppe C (eds) Handbook of Rational and Social Choice. Oxford University Press, Oxford

Edelman PH, Jamison RE (1985) The theory of convex geometries. Geom Dedic 19:247–270CrossRef

Koshevoy G (1999) Choice functions and abstract convex geometries. Math Soc Sci 38:35–44CrossRef

Kreps DM (1979) Preference for flexibility. Econometrica 47(3):565–577CrossRef

Kukushkin NS (2004) Path independence and Pareto dominance. Econ Bull 4(3):1–3

Lahiri S (2003) Justifiable preferences over opportunity sets. Soc Choice Welf 21:117–129CrossRef

Moulin H (1985) Choice functions over a finite set: a summary. Soc Choice Welf 2:147–160CrossRef

Nehring K (1997) Rational choice and revealed preference without binariness. Soc Choice Welf 14:403–425CrossRef

Pattanaik PK, Peleg B (1984) An axiomatic characterization of the lexicographic maximin extension of an ordering over a subset to the power set. Soc Choice Welf 1:113–122CrossRef

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Plott CR (1973) Path independence, rationality and social choice. Econometrica 41:1075–1091CrossRef

Puppe C, Xu Y (2010) Essential alternatives and freedom rankings. Social Choice and Welfare 35:669–685CrossRef

Titel: Path independent choice and the ranking of opportunity sets
verfasst von: Matthew Ryan
Publikationsdatum: 01.01.2014
Verlag: Springer Berlin Heidelberg
Erschienen in: Social Choice and Welfare / Ausgabe 1/2014
Print ISSN: 0176-1714
Elektronische ISSN: 1432-217X
DOI: https://doi.org/10.1007/s00355-012-0719-6

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