Abstract
We prove a coincidence of the class of multi-preference hyper-relations and the class of decent hyper-relations (DHR), that is the class of binary relations on opportunity sets satisfying monotonicity, no-dummy, stability with respect to contraction and extension, and the union property. We study subclasses of DHR. In order to pursue our analysis, we establish a canonical bijection between DHR and the class of no-dummy heritage choice functions. From this we obtain that the no-dummy heritage choice functions have multi-criteria rationalizations with reflexive binary relations. We also prove that the restriction of this bijection to two subclasses of DHR, namely the transitive decent hyper-relations, and the ample hyper-relations, is a bijection between these subclasses and the classes of closed no-dummy choice functions and no-dummy path-independent choice functions (Plott functions), respectively.
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Notes
It is worth noticing here that the idea to connect hyper-relations and choice functions goes back to Puppe (1996).
Here an arrow \(x\rightarrow z\) means \(x>z\) and so on.
Here, the term ‘specification’ means that the list of comparisons \(a\preceq A\) determines a decent hyper-relation.
The reader can assume for simplicity that \(X\) is a finite set, although the finiteness of \(X\) is almost nowhere used.
Let \(\preceq \) and \(\preceq ^{\prime }\) be two binary relations, we set \(\preceq \subseteq \preceq ^{\prime }\) if, for any \(A\), \(B\subseteq X\), such that \(A\preceq B\), it holds that \( A\preceq ^{\prime }B\).
There are some other reformulation of this axiom. For instance, in the literature on stable matchings (Roth and Sotomayor (1990), Definiton 6.2 p. 173), the Heritage axiom is the Substitutability property: if a worker \(a\) is hired by a firm from a list \(B\) she will be also hired from any shorter list \(A\subseteq B\).
We provide a proof of the latter claim. Suppose that \(A\not \preceq _{f}^{\prime \prime }B\), that is there exists \(a\in f(A\cup B)\) such that \(a\not \in B\). Then, since \(f\) is a heritage choice function and \(a\in A\backslash B\), we obtain that \(a\in f(a\cup B)\) \((a\cup B\subseteq A\cup B\) and \(a\in f(A\cup B)\) ). Hence, \(A\not \preceq _{f}B\), since \(A\preceq _{f}B\) implies \(a\notin f(a\cup B)\)).
The peeling order is a variant of the peeling rank in Statistics. Namely, for a finite set of points \(X\) in an Euclidean space, let us consider the following chain of sets \(X_{0}:=X\), \(X_{1}:=X_{0}\setminus ex(X_{0})\), where \(ex(Y)\) denotes the set of points of \(Y\) which belong to the boundary of its convex hull, \(X_{2}:=X_{1}\setminus ex(X_{1})\), \(\ldots \) . Then, for a subset \(A\subseteq X\) its rank \(r\) is defined as maximal number \( r\) such that \(A\subseteq X_{r}\). The peeling rank is used in non-parametric rank tests [Eddy (1984)].
A mapping \(\mu :2^{X}\rightarrow 2^{X}\) is extensive if \(A\subseteq \mu \left( A\right) \) for all \(A\in 2^{X}\). An extensive mapping is no-dummy if \(\mu (\emptyset )=\emptyset \).
Notice that the transitive decent hyper-relations have been studied in the literature on implication systems with the name of complete implication systems, CIS, see, for example Falmagne and Doignon (2011) and Domenach and Leclerc (2004). Specifically, CIS are the dual of transitive decent relations. In 1974, Armstrong (1974) has shown that CIS are in a one-to-one correspondence with the closure operators. Kreps proves the same result [Lemma 2, Kreps (1979)].
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Acknowledgments
We thank referees for useful remarks and suggestions. We also thank Clemens Puppe and the participants of his seminar for useful discussions and remarks.
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Danilov, V., Koshevoy, G. & Savaglio, E. Hyper-relations, choice functions, and orderings of opportunity sets. Soc Choice Welf 45, 51–69 (2015). https://doi.org/10.1007/s00355-014-0844-5
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DOI: https://doi.org/10.1007/s00355-014-0844-5