Hyperreal Expected Utilities and Pascal's Wager

This paper re-examines two major concerns about the validity of Pascal's Wager: (1) The classical von Neumann-Morgenstern Theorem seems to contradict the rationality of maximising expected utility when the utility function's range contains in nite numbers (McClennen 1994). (2) Apparently, the utility of salvation cannot be re exive under addition by real numbers (which Pascal's Pensée 233 demands) and strictly irre exe under multiplication by scalars < 1 at the same time (Hájek 2003). Robinsonian nonstandard analysis is used to establish a hyperreal version of the von Neumann-Morgenstern Theorem: an a ne utility representation theorem for internal, complete, transitive, independent and in nitesimally continuous preference orderings on lotteries with hyperreal probabilities. (Herein, a preference relation1 on lotteries is called in nitesimally continuous if and only if for all x ≺ y ≺ z, there exist hyperreal, possibly in nitesimal, numbers p, q such that the perturbed preference ordering px+(1−p)z ≺ y ≺ qx+(1−q)z holds. In nitesimal Continuity is hence a much weaker condition than continuity.) This Hyperreal von Neumann-Morgenstern Theorem yields a hyperreal version of the Expected Utility Theorem a rming a conjecture by Sobel (1996). This responds to objection (1). To address objection (2), a convex linearly ordered superset S of the reals whose maximum is both re exive under addition by nite numbers and strictly irre exive under multiplication by scalars < 1 is constructed. If the Wagerer is indi erent among the pure outcomes except salvation (a common soteriological position) and some technical conditions hold, then the Hyperreal Expected Utility Theorem allows to represent the Wagerer's preference ordering through an S-valued (not just hyperreal-valued) utility function, answering objections (1) and (2) simultaneously. Behold, I set before you the way of life and the way of death. But your happiness? Let us weigh the gain and the loss in wagering that God is. Let us estimate these two chances. If you gain, you gain all; if you lose, you lose nothing. Wager, then, without hesitation that He is. Jeremiah 21,8 Blaise Pascal, Pensées, Ÿ233 (King James Version) (Trotter translation)