A Von Neumann-Morgenstern Representation Result Without Weak Continuity Assumption

14 Pages Posted: 12 Mar 2010 Last revised: 11 Mar 2021

See all articles by Michael Kupper

Michael Kupper

Vienna Institute of Finance

Freddy Delbaen

Swiss Federal Institute of Technology at Zurich

Samuel Drapeau

China Academy of Financial Research (SAIF) and School of Mathematical Sciences

Date Written: March 11, 2010

Abstract

In the paradigm of von Neumann-Morgenstern, a representation of affine preferences in terms of an expected utility can be obtained under the assumption of weak continuity. Since the weak topology is coarse, this requirement is a priori far from being negligible. In this work, we replace the assumption of weak continuity by monotonicity. More precisely, on the space of lotteries on a real open interval, it is shown that any affine numerical representation of a preference order monotone with respect to the first stochastic order, admits a representation in terms of an expected utility for some nondecreasing utility function. As a consequence, any affine numerical representation on the subset of lotteries with compact support monotone with respect to the second stochastic order can be represented in terms of an expected utility for some nondecreasing concave utility function. We also provide such representations for affine preference order on the subset of those lotteries which fulfills some integrability conditions. The subtleties of the weak topology are also illustrated by some examples.

Keywords: von Neumann and Morgenstern representation, affine preference orders, automatic continuity, first stochastic order

JEL Classification: D08, D8, D81

Suggested Citation

Kupper, Michael and Delbaen, Freddy and Drapeau, Samuel, A Von Neumann-Morgenstern Representation Result Without Weak Continuity Assumption (March 11, 2010). Available at SSRN: https://ssrn.com/abstract=1568765 or http://dx.doi.org/10.2139/ssrn.1568765

Michael Kupper

Vienna Institute of Finance ( email )

Nordbergstrasse 15
Vienna, 1090
Austria

Freddy Delbaen

Swiss Federal Institute of Technology at Zurich ( email )

ETH-Zentrum
CH-8092 Zurich
Switzerland

Samuel Drapeau (Contact Author)

China Academy of Financial Research (SAIF) and School of Mathematical Sciences ( email )

Shanghai Jiao Tong University
211 West Huaihai Road
Shanghai, 200030
China

HOME PAGE: http://www.samuel-drapeau.info

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