A set of axioms for the utility theory with rational probabilities
Classical utility theory assigns preference-order-preserving linear expected utility \(u(c,p;d,1-p)\) (\(c\) happens with probability \(p,\) while \(d\) with probability \(1-p\)) to games (gambles, lotteries). Thrall (1954; not in the references) is quoted as saying that in lotteries irrational \(p\) 's cannot occur (although it seems that they occasionally can, since no lottery wheel is exactly symmetric [reviewer]). The authors state two systems of axioms for the existence of such (expected) utility functions \(u\) just for rational or even just for dyadic probabilities, one by \textit{J. C. Stepherdson} [J. Math. Econ. 7, 91-113 (1980; Zbl 0472.90009)] and their own in which Stepherdson's continuity axiom is weekened, while a new ``convexity'' axiom is added. They offer a proof that the two systems are equivalent. [For another theory of gambles, see \textit{R. D. Luce}, ``Utility of gains and losses: measurement-theoretical and experimental approaches'' (Erlbaum, London) (2000; Zbl 0997.91500)].
scientific article; zbMATH DE number 1643042