Weng Kin San

Standard decision theory ranks risky prospects by their expected utility. This ranking does not change if the values of all possible outcomes are uniformly shifted or dilated. Similarly, if the values of the outcomes are negated, the ranking of prospects by their expected utility is reversed. In settings with unbounded levels of utility, the expected utility of prospects is not always defined, but it is still natural to accept the affine symmetry principles, which say that the true ranking of prospects is unchanged by shifts and dilations, and is reversed by negation—even in hard cases where expected utilities are undefined. This paper investigates the affine symmetry principles and their consequences. The principles are found to be surprisingly powerful. Combined with orthodox axioms, they assign precise utility values to previously problematic cases: for example, to the Pasadena prospect (Nover & Hájek, 2004) and to the alternating St Petersburg prospect. They also have important structural consequences, notably vindicating Colyvan's (2008) Relative Expectation Theory. The paper then establishes the consistency of the affine symmetry principles. In light of their fruitful consequences, this consistency result supports the adoption of the affine symmetry principles as fundamental axioms of decision theory.

This paper argues that the agent concerned to have accurate (outright) beliefs will have a consistent and multi-premise closed belief set, and not a (merely) single-premise closed and (merely) pairwise consistent belief set, as has often been thought. This argument rests on the fact that we need a notion of accuracy coherence for belief that is belief-sensitive; sensitive to one's perspective, in a way that the standard belief-insensitive notion of accuracy coherence is not. The choice of the belief-sensitive over belief-insensitive notion is defended, as is the choice of this particular belief-sensitive kind of accuracy coherence over other belief-sensitive rivals. Our conclusion is that accurate believers are deductively cogent.