Enforcing Natural Properties of Choice Functions, with Application for Combination

One important and natural representation of preferences is a choice function, which returns the preferred options amongst any given subset of the alternatives. There are some very intuitive coherence conditions that might be assumed for an agent's choice function, in particular path independence, and a consistency condition stating that there is always at least one preferred alternative among any non-empty set. However, an elicited choice function may not satisfy path independence, because of the elicitation being incomplete, or because of there being some incoherence in the agent's reported choice function (despite the agent assenting to the general coherence conditions). Furthermore, if we wish to combine the choice functions of more than one agent, simple natural combination operations can lose path independence. This paper develops methods for enforcing path independence and restoring consistency, thus, making the user preferences coherent; this method also leads to approaches for combining two choice functions, in order to suggest the most promising alternatives for a pair of agents.