2.4 - A Conflict Between Finite Additivity and Avoiding Dutch Book

Summary

ABSTRACT

For Savage (1954) as for deFinetti (1974), the existence of subjective (personal) probability is a consequence of the normative theory of preference. (DeFinetti achieves the reduction of belief to desire with his generalized Dutch-Book argument for previsions.) Both Savage and deFinetti 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 chapter we dispute these claims by showing that the following three cannot simultaneously hold:

1. i. Coherent belief is reducible to rational preference, i.e. the generalized Dutch- Book argument fixes standards of coherence.

2. ii. Finitely additive probability is coherent.

3. iii. Admissible preference structures may be free of consequences, i.e. they may lack prizes whose values are robust against all contingencies.

I. INTRODUCTION

One of the most important results of the subjectivist theories of Savage and deFinetti is the thesis that, normatively, preference circumscribes belief. Specifically, these authors argue that the theory of subjective probability is reducible to the theory of reasonable preference, i.e. coherent belief is a consequence of rational desire. In Savage's (1954) axiomatic treatment of preference, the existence of a quantitative subjective probability is assured once the postulates governing preference are granted. In deFinetti's (1974) discussion of prevision, avoidance of a (uniform) loss for certain is thought to guarantee agreement with the requirements of subjective probability (sometimes called the avoidance of “Dutch Book”).