# Orderings of Opportunity Sets

###### Working Paper 2012-282

**Abstract**

We consider an extension of the class of multi-utility hyper-relations, the class of semi-decent hyper-relations. A semi-decent hyper-relation satis
es monotonicity, stability with respect to contraction, and the union property. We analyze the class of semi-decent hyper-relations both associating them to an appropriate class of choice functions and considering decomposition of a

decent relations via elementaryones. Doing so, we consider images in the set of choice functions

of three subclasses of semi-decent hyper-relations: the decent hyper-relations, the transitive decent hyper-relations , and transitive decent hyper-relations which satisfy the condition LE of lattice equivalence. We prove that the image of the set of decent hyper-relations coincides with of the set of heritage choice functions; the image of the set of transitive decent hyper-relations coincides with the set of closed choice functions; the image of the set of transitive decent hyper-relations which satisfy the LE coincides with the set of Plott functions. We consider, for each of the above subclasses of hyper-relations, the problem of the decomposition of a given hyper-relation into elementaryones, namely the representation of a given hyper-relation as the intersection of elementaryones.

**Authors**: Vladimir Danilov, Gleb Koshevoy, Ernesto Savaglio.

**Keywords**: Hyper-relations, opportuntiy sets, preference for flexibility.

**JEL**: D01, D71.