Consequentialism, Non-Archimedean Probabilities, and Lexicographic Expected Utility

Earlier work (Hammond, 1988a, b) on dynamically consistent “consequentialist” behaviour in decision trees was unable to treat zero probability events satisfactorily. Here the rational probability functions considered in Hammond (1994), as well as other non-Archimedean probabilities, are incorporated into decision trees. As before, the consequentialist axioms imply the existence of a preference ordering satisfying independence. In the case of rational probability functions, those axioms, together with continuity and a new refinement assumption, imply the maximization of a somewhat novel lexicographic expected utility preference relation. This is equivalent to maximization of expected utility in the ordering of the relevant non-Archimedean field. Non-Archimedean Expected Utility ma la natura la dà sempre scema similemente operando all’artista c’ha l’abito dell’arte e man che trema Dante, La Divina Commedia, Vol. III: Paradiso (Canto XIII, 76–78)†