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Ranking Sets of Possibly Interacting Objects Using Shapley Extensions

Last modified: 2012-05-17

#### Abstract

We deal with the problem of how to extend a preference relation over a set

*X*of "objects" to the set of all subsets of X. This problem has been carried out in the tradition of the literature on extending an order on a set to its power set with the objective to analyze the axiomatic structure of families of rankings over subsets. In particular, most of these approaches make use of axioms aimed to prevent any kind of interaction among the objects in*X*. In this paper, we apply coalitional games to study the problem of extending preferences over a finite set X to its power set 2^{X.}A coalitional game can be seen as a numerical representation of a preference extension on 2^{X.}. We focus on a particular class of extensions on 2^{X.}such that the ranking induced by the Shapley value of each coalitional game representing an extension in this class, coincides with the original preference on*X*. Some properties of Shapley extensions are discussed, with the objective to justify and contextualize the application of Shapley extensions to the problem of ranking sets of possibly interacting objects.We deal with the problem of how to extend a preference relation over a set*X*of "objects" to the set of all subsets of*X*. This problem has been carried out in the tradition of the literature on extending an order on a set to its power set with the objective to analyze the axiomatic structure of families of rankings over subsets. In particular, most of these approaches make use of axioms aimed to prevent any kind of interaction among the objects in*X*. In this paper, we apply coalitional games to study the problem of extending preferences over a finite set*X*to its power set 2^{X.}. A coalitional game can be seen as a numerical representation of a preference extension on 2^{X.}. We focus on a particular class of extensions on 2^{X.}such that the ranking induced by the Shapley value of each coalitional game representing an extension in this class, coincides with the original preference on*X*. Some properties of Shapley extensions are discussed, with the objective to justify and contextualize the application of Shapley extensions to the problem of ranking sets of possibly interacting objects.
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