Subjective probability and expected utility without additivity. (English) Zbl 0672.90011

Schmeidler, David

Econometrica 57, No. 3, 571-587 (1989).

Summary: An act maps states of nature to outcomes: deterministic outcomes as well as random outcomes are included. Two acts f and g are comonotonic, by definition, if it never happens that f(s)\(\succ f(t)\) and g(t)\(\succ g(s)\) for some states of nature s and t. An axiom of comonotonic independence is introduced here. It weakens the von Neumann-Morgenstern axiom of independence as follows: If \(f\succ g\) and if f, g, and h are comonotonic, then \(\alpha f+(1-\alpha)h\succ g+(1-\alpha)h.\)
If a nondegenerate, continuous, and monotonic (state independent) weak order over acts satisfies comonotonic independence, then it induces a unique non-(necessarily-)additive probability and a von Neumann- Morgenstern utility. Furthermore, one can compute the expected utility of an act with respect to the nonadditive probability, using the Choquet integral.
This extension of the expected utility theory covers situations, as the Ellsberg paradox, which are inconsistent with additive expected utility. The concept of uncertainty aversion and interpretation of comonotonic independence in the context of social welfare functions are included.

MSC:

91B16

Utility theory

Keywords:

subjective probability; deterministic outcomes; random outcomes; comonotonic independence; von Neumann-Morgenstern utility; expected utility; Ellsberg paradox; uncertainty aversion