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A von Neumann–Morgenstern representation result without weak continuity assumption

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  • Delbaen, Freddy
  • Drapeau, Samuel
  • Kupper, Michael

Abstract

In the paradigm of von Neumann and Morgenstern (1947), 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 an interval of the real line, it is shown that any affine preference order which is 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 preference order on the subset of lotteries with compact support, which is 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 orders on the subset of those lotteries which fulfill some integrability conditions. The subtleties of the weak topology are illustrated by some examples.

Suggested Citation

  • Delbaen, Freddy & Drapeau, Samuel & Kupper, Michael, 2011. "A von Neumann–Morgenstern representation result without weak continuity assumption," Journal of Mathematical Economics, Elsevier, vol. 47(4-5), pages 401-408.
  • Handle: RePEc:eee:mateco:v:47:y:2011:i:4:p:401-408
    DOI: 10.1016/j.jmateco.2011.04.002
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    References listed on IDEAS

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    1. Hadar, Josef & Russell, William R, 1969. "Rules for Ordering Uncertain Prospects," American Economic Review, American Economic Association, vol. 59(1), pages 25-34, March.
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    Cited by:

    1. Adriana Piazza & Bernardo Pagnoncelli, 2015. "The stochastic Mitra–Wan forestry model: risk neutral and risk averse cases," Journal of Economics, Springer, vol. 115(2), pages 175-194, June.
    2. Harin, Alexander, 2014. "Problems of utility and prospect theories. Certainty effect near certainty," MPRA Paper 61026, University Library of Munich, Germany.
    3. Harin, Alexander, 2014. "Problems of utility and prospect theories. A discontinuity of Prelec’s function," MPRA Paper 61027, University Library of Munich, Germany.
    4. Harin, Alexander, 2015. "Problems of utility and prospect theories. A “certain–uncertain” inconsistency within their experimental methods," MPRA Paper 67911, University Library of Munich, Germany.
    5. Spinu, Vitalie & Wakker, Peter P., 2013. "Expected utility without continuity: A comment on Delbaen et al. (2011)," Journal of Mathematical Economics, Elsevier, vol. 49(1), pages 28-30.

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