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Positive Operators, Riesz Spaces, and Economics pp 69–88Cite as

Functional Analytic Tools for Expected Utility Theory

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Part of the book series: Studies in Economic Theory ((ECON.THEORY,volume 2))

Abstract

Depending on the school of thought, expected utility theory states that choices among lotteries either should be made or actually made by maximizing the expected value of a real valued function of the outcomes—a utility function. This article provides a look at some of the functional analytic results used in expected utility theory. I concentrate on applications to the theory of stochastic dominance relations and the revealed preference approach to expected utility. Few of these results are deep, given the underlying tools, but many of them are not widely known, and their combination is novel. In particular, the revealed preference results of Border [4] are extended to higher degree stochastic dominance relations.

I thank Mike Maxwell for pointing out errors in an early draft of this paper.

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References

  1. C. D. Aliprantis and O. Burkinshaw, Principles of Real Analysis, 2nd Ed., Academic Press, New York, 1991.

    Google Scholar 

  2. D. Amir and Z. Ziegler, Generalized convexity cones and their duals, Pacific J. Math. 27 (1968), 425–440.

    Google Scholar 

  3. A. B. Atkinson, On the measurement of inequality, J. Econom. Theory 2 (1970), 224–263.

    Article  Google Scholar 

  4. K. C. Border, Revealed preference, stochastic dominance, and the expected utility hypothesis, forthcoming in J. Econom. Theory.

    Google Scholar 

  5. S. L. Brumelle and R. G. Vickson, A unified approach to stochastic dominance, in: W. T. Ziemba and R. G. Vickson eds., Stochastic Optimization Methods in Finance (Academic Press, New York, 1975), pp. 101–113.

    Google Scholar 

  6. G. Choquet, Lectures on Analysis, 3 vols, Benjamin, Reading, Massachusettes, 1969.

    Google Scholar 

  7. J. Diestel and J. J. Uhl, Jr., Vector Measures, Mathematical Surveys, # 15, Amer. Math. Society, Rhode Island, 1977.

    Google Scholar 

  8. N. Dunford and J. Schwartz, Linear Operators, I, Interscience, New York, 1957.

    Google Scholar 

  9. P. C. Fishburn, Decision and Value Theory, Wiley, New York, 1964.

    Google Scholar 

  10. P. C. Fishburn, Convex stochastic dominance with continuous distribution functions, J. Econom. Theory 7 (1974), 143–158.

    Article  Google Scholar 

  11. P. C. Fishburn, Separation theorems and expected utility, J. Econom. Theory 11 (1975), 16–34.

    Article  Google Scholar 

  12. P. C. Fishburn, Continua of stochastic dominance relations for bounded probability distributions, J. Math. Econom. 3 (1976), 295–311.

    Article  Google Scholar 

  13. P. C. Fishburn, Continua of stochastic dominance relations for unbounded probability distributions, J. Math. Econom. 7 (1980), 271–285.

    Article  Google Scholar 

  14. P. C. Fishburn and R. Vickson, Theoretical foundations of stochastic dominance, in: G. Whitmore and M. Findlay, eds., Stochastic Dominance: An Approach to Decision Making Under Risk (Heath, Lexington, Massachusetts, 1978), pp. 39–113.

    Google Scholar 

  15. J. Hadar and W. Russell, Rules for ordering uncertain prospects, Amer. Econom. Rev. 59 (1969), 25–34.

    Google Scholar 

  16. J. Hadar and W. R. Russell, Stochastic dominance and diversification, J. Econom. Theory 3 (1971), 288–305.

    Article  Google Scholar 

  17. J. Hadar and W. R. Russell, Applications in economic theory and analysis, in: G. Whitmore and M. Findlay, eds., Stochastic Dominance: An Approach to Decision Making Under Risk (Heath, Lexington, Massachusetts, 1978), pp. 293–333.

    Google Scholar 

  18. G. Hanoch and H. Levy, Efficiency analysis of choices involving risk, Rev. Econom. Stud. 36 (1969), 335–346.

    Article  Google Scholar 

  19. G. H. Hardy, J. E. Littlewood, and G. Pólya, Some simple inequalities satisfied by convex functions, Messenger of Math. 58 (1929), 145–152.

    Google Scholar 

  20. I. Jewitt, A note on comparative statics and stochastic dominance, J. Math. Econom. 15 (1986), 249–254.

    Article  Google Scholar 

  21. T. Kamae, U. Krengel, and G. L. O’Brien, Stochastic inequalities on partially ordered spaces, Ann. Probab. 5 (1977), 899–912.

    Article  Google Scholar 

  22. S. Karlin, Total Positivity, I, Stanford University Press, Stanford, CA, 1968.

    Google Scholar 

  23. S. Karlin and A. Novikoff, Generalized convex inequalities, Pacific J. Math. 13 (1963), 1251–1279.

    Google Scholar 

  24. S. Karlin and W. Studden, Tchebycheff Systems with Applications in Analysis and Statistics, Interscience, New York, 1966.

    Google Scholar 

  25. S. Karlin and Z. Ziegler, Generalized absolutely monotone functions, Israel J.Math. 3 (1965), 173–180.

    Article  Google Scholar 

  26. Y. K. Kwon, J. C. Fellingham, and D. P. Newman, Stochastic dominance and information value, J. Econom. Theory 20 (1979), 213–230.

    Article  Google Scholar 

  27. E. L. Lehmann, Ordered families of distributions, Ann. Math. Statist. 26 (1955), 399–419.

    Article  Google Scholar 

  28. R. T. Masson, Utility functions with jump discontinuities: Some evidence and implications from peasant agriculture, Econom. Inquiry 12 (1974), 559–566.

    Article  Google Scholar 

  29. C. Menezes, C. Geiss, and J. Tressler, Increasing downside risk, Amer. Econom. Rev. 70 (1980), 921–932.

    Google Scholar 

  30. P. A. Meyer, Probability and Potentials, Blaisdell, Waltham, Massachusetts, 1966.

    Google Scholar 

  31. L. Nachbin, Linear continuous functionals positive on the increasing continuous functions, Summa Brasiliensis Math. 2 (1951), 135–150. Reprinted as Appendix 5 of Topology and Order, Van Nostrand, Princeton, 1965.

    Google Scholar 

  32. R. Phelps, Lectures on Choquet Theory, Van Nostrand, New York, 1966.

    Google Scholar 

  33. J. Quirk and R. Saposnik, Admissibility and measurable utility functions, Rev. Econom. Stud. 29 (1962), 140–146.

    Article  Google Scholar 

  34. M. Rothschild and J. Stiglitz, Increasing risk I: A definition, J. Econom. Theory 2 (1970), 225–243.

    Article  Google Scholar 

  35. M. Rothschild and J. Stiglitz, Increasing risk II: Its economic consequences, J. Econom. Theory 3 (1971), 66–84.

    Article  Google Scholar 

  36. U. Segal, Two-stage lotteries without the reduction axiom, Econometrica 58 (1990), 349–377.

    Article  Google Scholar 

  37. D. Schmeidler, A bibliographical note on a theorem by Hardy, Littlewood and Pólya, J. Econom. Theory 20 (1979), 125–128.

    Article  Google Scholar 

  38. V. Strassen, The existence of probability measures with given marginals, Ann. Math. Statist. 36 (1965), 423–439.

    Article  Google Scholar 

  39. A. E. Taylor, General Theory of Functions and Integration, Blaisdell, New York, 1965.

    Google Scholar 

  40. G. A. Whitmore, Third-degree stochastic dominance, Amer. Econom. Rev. 60 (1970), 457–459.

    Google Scholar 

  41. G. A. Whitmore, The theory of skewness preference, J. of Bus. Admin. 6 (1975), 13–20.

    Google Scholar 

  42. G. A. Whitmore and M. Findlay, Eds., Stochastic Dominance: An Approach to Decision Making Under Risk, Heath, Lexington, Massachusetts, 1978.

    Google Scholar 

  43. D. V. Widder, The Laplace Transform, Princeton University Press, Princeton, 1941.

    Google Scholar 

  44. Z. Ziegler, Generalized convexity cones, Pacific J. Math. 17 (1966), 561–580.

    Google Scholar 

  45. Z. Ziegler, On the characterization of measures of the cone dual to a generalized convexity cone, Pacific J. Math. 24 (1968), 603–626.

    Google Scholar 

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Authors and Affiliations

  1. Division of the Humanities and Social Sciences, California Institute of Technology, 91125, Pasadena, CA, USA

    Kim C. Border

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  1. Kim C. Border
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© 1991 Springer-Verlag Berlin Heidelberg

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Border, K.C. (1991). Functional Analytic Tools for Expected Utility Theory. In: Positive Operators, Riesz Spaces, and Economics. Studies in Economic Theory, vol 2. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-58199-1_4

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  • DOI: https://doi.org/10.1007/978-3-642-58199-1_4

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-63502-1

  • Online ISBN: 978-3-642-58199-1

  • eBook Packages: Springer Book Archive

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