Ranking Alternative Non-Combinable Prospects: A Stochastic Dominance Based Route to the Second Best Solution

The problem considered here is that of dealing with the "incompleteness" property of Stochastic Dominance Orderings by quantifying the extent to which distributions differ when there is no dominant distribution at a given order. For example consider a policymaker's choice problem when facing a set of distinct, non-combinable policy options. When policies are not combinable, the classic comparative static or first best solution to the choice problem is not available. The approach proposed here is an elaboration of a technique employed in the optimal statistical testing literature. It is supposed that policies could be combined so that the ideal first best "stochastically dominant" optimal envelope policy outcome is constructed under the policymaker's given imperative. Then the second best policy whose outcome most closely approximates this ideal is selected by employing a statistic that measures proximity of alternative policies to that ideal. The statistic is shown to obey an Independence of Irrelevant Alternatives proposition. The paper concludes with 3 illustrative examples of its use.