Extending an order over a set to its power set

Abstract

This thesis investigates the problem of extending a (complete) order over a set to its power set. We interpret the set under consideration as a set of alternatives and we conceive orders as individual preferences. The elements of the power sets are the non-resolute outcomes. To determine how an individual with a given preference over alternatives is required to rank certain sets, we need a concept of extension axioms. In the this part, the final outcome is determined by an ì(external) chooserî which is a resolute choice function. The individual whose preference is under consideration confronts a set of resolute choice functions which reflects the possible behaviors of the chooser. Every such set naturally induces an extension axiom (i.e., a rule that determines how an individual with a given preference over alternatives is required to rank certain sets). Our model allows to revisit various extension axioms of the literature. Interestingly, the Gardenfors (1976) and Kelly (1977) principles are singled-out as the only two extension axioms compatible with the non-resolute outcome interpretation. In the second part, the extension axioms we consider generate orderings over sets according to their expected utilities induced by some assignment of utilities over alternatives and probability distributions over sets. The model we propose gives a general and unified exposition of expected utility consistent extensions while it allows to emphasize various subtleties, the effects of which seem to be underestimated - particularly in the literature on strategy-proof social choice correspondences.