Symmetries of value

Abstract

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.