Combining two choice functions and enforcing natural properties

Abstract

This paper considers the problem of combining two choice functions (CFs), or setwise optimisation functions, based on use of intersection and composition. Each choice function represents preference information for an agent, saying, for any subset of a set of alternatives, which are the preferred, and which are the sub-optimal alternatives. The aim is to find a combination operation that maintains good properties of the choice function. We consider a family of natural properties of CFs, and analyse which hold for different classes of CF. We determine relationships between intersection and composition operations, and find out which properties are maintained by these combination rules. We go on to show how the most important of the CF properties can be enforced or restored, and use this kind of procedure to define combination operations that then maintain the desirable properties.