Having argued for the legitimacy of assuming transitivity and normality, I now turn to the independence principle (defined in Section 1.3). Independence does not have the kind of popular endorsement that transitivity does. Nevertheless, I show that independence does follow from premises that are widely endorsed. The argument that I give differs at least subtly from those in the literature, and I explain why I regard my argument as superior. I also explain why arguments against independence strike me as uncompelling.

VIOLATIONS

In Chapter 2 we saw that most people endorse transitivity and normality. With independence, on the other hand, there is fairly strong prima facie evidence that many people reject it. Part of this evidence consists of decision problems in which a substantial proportion of people make choices that appear inconsistent with independence; there is also evidence that to many people the axiom itself does not seem compelling.

The Allais problems

Maurice Allais was one of the earliest critics of independence, and backed up his position by formulating decision problems in which many people have preferences that appear to violate independence. I will present these problems in the way Savage (1954, p. 103) formulated them, since this brings out the conflict with independence most clearly.

A ball is to be randomly drawn from an urn containing 100 balls, numbered from 1 to 100. There are two separate decision problems to consider, with options and outcomes as in Figure 3.1.