A Conflict between Finite Additivity and Avoiding Dutch Book

Abstract

For Savage (1954) as for de Finetti (1974), the existence of subjective (personal) probability is a consequence of the normative theory of preference. (De Finetti achieves the reduction of belief to desire with his generalized Dutch-Book argument for previsions.) Both Savage and de Finetti rebel against legislating countable additivity for subjective probability. They require merely that probability be finitely additive. Simultaneously, they insist that their theories of preference are weak, accommodating all but self-defeating desires. In this paper we dispute these claims by showing that the following three cannot simultaneously hold:(i) Coherent belief is reducible to rational preference, i.e. the generalized Dutch-Book argument fixes standards of coherence.(ii) Finitely additive probability is coherent.(iii) Admissible preference structures may be free of consequences, i.e. they may lack prizes whose values are robust against all contingencies.