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Social Choice and Welfare

Erschienen in:

01.08.2014 | Original Paper

Choice in ordered-tree-based decision problems

verfasst von: Saptarshi Mukherjee

Erschienen in: Social Choice and Welfare | Ausgabe 2/2014

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Abstract

Decision makers often have to choose alternatives which appear in an exogenous structure. For example, consider choosing a dish in a restaurant after going through different menu cards (a menu card for buffet, another for combo offers etc.). The decision maker observes the items displayed (as a list) on a menu card and she switches to another menu card to see the items displayed on it (again in the form of another list). Thus the set of all items appears as an “ordered-tree” to the decision maker. There are other examples of decision making where alternatives appear in the form of an ordered-tree. In this paper we consider the cases of choice from ordered-trees (in particular, “lists of lists”) and characterize the choice functions. We impose the axioms of Backward Consistency and Replacement Indifference on choice functions and obtain characterization results for \(k\)-ary ordered-trees and more general ordered-trees. We show that the results for \(k=2\) are similar to those of (Rubinstein and Salant 2006), in which the cases of choice from lists are considered. However the results for \(k \ge 3\) and general ordered-trees are different and allow for instance, a richer class of tie-breaking rules in special cases.

Vorheriger Artikel A new approach for bounding awards in bankruptcy problems

Nächster Artikel On the representation of preference orders on sequence spaces

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Formally, the set of alternatives appears in the structure of an ordered-tree. Henceforth by tree we mean ordered tree only.

A choice function is a mapping from a set of trees to the set of alternatives such that for every tree the chosen alternative is one of those appearing in the lists that are listed in that tree.

A binary relation is an ordering if it is complete and transitive, i.e. for any two alternatives \(x,y \in X\), either \(x \succeq y\), or \( y \succeq x \). Also if \( x \succeq y, y \succeq z\), then \( x \succeq z\), i.e. \(\succeq \) is transitive. Also if for \(x \ne y,\; x \succeq y, y \succeq x\), then we say that \(x\) and \(y\) are indifferent. An ordering \(\succ \) is called strict ordering if it is antisymmetric.

\(k\)th list in a tree is the next list to the \((k-1)\)th list from the left.

Backward induction is widely used in economics, for instance in sub-game perfect equilibrium in sequential games and in various dynamic optimization problems.

For instance, Sen’s Property Sen (1993) \(\alpha \) (or Contraction Consistency): for each pair of sets \(S\) and \(T\), and for each \(x \in S\), if \(x \in C(T)\) and \( S \subset T\), then \(x \in C(S)\).

We have already demonstrated that alternatives in a tree \(t\) appear in an ordered sequence of tuples. Thus the first alternative in \( M^o( X(t), \succeq )\) is the alternative in \( M( X(t), \succeq )\) which lies to the left of the other alternatives in \( M( X(t), \succeq )\) in \(t\).

We can also directly show that a CON is an MC. Let \(C\) be a CON from binary trees. One can form a weak ordering \(\succeq \) that induces two indifference sets of good and bad alternatives. Also there exists an admissible binary indicator function \(\delta \), such that for all alternatives in “good” class \(\delta \) is \(left\), and for all bad alternatives also \(\delta \) is \(left\).

As we mentioned earlier, the admissible preferences also appear in Ehlers (2002) in the context of a housing allocation model and preferences over houses. Ehlers (2002) justifies these preferences on the grounds that the DM does not possess (or does not have the incentives to acquire) information about the bottom-ranked alternatives and hence ranks them as indifferent to each other.

For the sake of simplicity, we denote any elementary tree \(t\equiv (X^1)\), where \(X^1\) is a list, by \(t \equiv X_1\). For example, we represent a tree \(t\equiv ((a,b))\) as \(t\equiv (a,b)\).

Proofs of some lemmas are omitted because they follow the same logic as the proofs of the other lemmas. However, all proofs are available upon request from the author.

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Titel: Choice in ordered-tree-based decision problems
verfasst von: Saptarshi Mukherjee
Publikationsdatum: 01.08.2014
Verlag: Springer Berlin Heidelberg
Erschienen in: Social Choice and Welfare / Ausgabe 2/2014
Print ISSN: 0176-1714
Elektronische ISSN: 1432-217X
DOI: https://doi.org/10.1007/s00355-013-0778-3

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