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26-12-2022 | Original Paper

# Families of abstract decision problems whose admissible sets intersect in a singleton

Author: Michele Gori

Published in: Social Choice and Welfare

## Abstract

An abstract decision problem is an ordered pair where the first component is a nonempty and finite set of alternatives and the second component is an irreflexive relation on that set, called dominance relation. The admissible set of an abstract decision problem is the set of the maximal elements of the reflexive and transitive closure of the dominance relation. Given a finite sequence of abstract decision problems on the same set of alternatives, we give conditions on the dominance relations that guarantee that the intersection of all the admissible sets of the considered problems is a singleton as well as conditions that guarantee that the intersection is nonempty. We show then that such results allow to deduce some interesting facts about the resoluteness of the Schulze network solution and the Schulze social choice correspondence as well as some information about the existence of a (unique) common recurrent state for finite families of discrete-time homogeneous Markov chains.

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Footnotes
1

2
Note that when (XR) is a tournament, that is, R is asymmetric and, for every $$x,y\in X$$ with $$x\ne y$$, we have that $$(x,y)\in R$$ or $$(y,x)\in R$$, then A(XR) coincides with the top cycle associated with (XR).

3
See, for instance, Theorem 4.1 in Shenoy (1980).

4
That fact is, for instance, a consequence of Propositions 2 and 54 in Bubboloni and Gori (2018). Note that Proposition 54 is actually about networks having capacity whose values are nonnegative integers but it can be easily generalized to the case of networks with real-valued capacity.

5
See, for instance, Theorem 5 in Kalai and Schmeidler (1977).

6
See Proposition 1.4.2 in Bang-Jensen and Gutin (2008).

7
See Lemma 1b in Sen (1970).

8
See Proposition 1.4.2 in Bang-Jensen and Gutin (2008).

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Title
Families of abstract decision problems whose admissible sets intersect in a singleton
Author
Michele Gori
Publication date
26-12-2022
Publisher
Springer Berlin Heidelberg
Published in
Social Choice and Welfare
Print ISSN: 0176-1714
Electronic ISSN: 1432-217X
DOI
https://doi.org/10.1007/s00355-022-01443-1