On the hierarchical nature of partial preferences over lotteries

In this work we consider preference relations that might not be total. Partial preferences may be helpful to represent those situations where, due to lack of information or vacillating desires, the decision maker would like to maintain different options “alive” and defer the final decision. In particular, we show that, when totality is relaxed, different axiomatizations of classical Decision Theory are no longer equivalent but form a hierarchy where some of them are more restrictive than others. We compare such axiomatizations with respect to theoretical aspects—such as their ability to propagate comparability/incomparability over lotteries and the induced topology—and to different preference elicitation methodologies that are applicable in concrete domains. We also provide a polynomial-time procedure based on the bipartite matching problem to determine whether one lottery is preferred to another.