How to co-exist with nonexistent expectations

Dozens of articles have addressed the challenge that gambles having undefined expectation pose for decision theory. This paper makes two contributions. The first is incremental: we evolve Colyvan's "Relative Expected Utility Theory" into a more viable "conservative extension of expected utility theory" by formulating and defending emendations to a version of this theory that was proposed by Colyvan and Hajek. The second is comparatively more surprising. We show that, so long as one assigns positive probability to the theory that there is a uniform bound on the expected utility of possible gambles (and assuming a uniform bound on the amount of utility that can accrue in a fixed amount of time), standard principles of anthropic reasoning (as formulated by Brandon Carter) place lower and upper bounds on the expected values of gambles advertised as having no expectation-even assuming that with positive probability, all gambles advertised as having infinite expected utility are administered faithfully. Should one accept the uniform bound premises, this reasoning thus dissolves (or nearly dissolves, in some cases) several puzzles in infinite decision theory.