Third-degree stochastic dominance and axioms for a convex marginal utility function

Abstract Given a set of random incomes on a finite probability space, necessary and sufficient conditions for third-degree stochastic dominance over this set are studied. These conditions provide a simple set of axioms for convexity of the marginal utility function of income. The axioms are compared with those of decreasing absolute risk aversion. Random incomes efficient with respect to third-degree dominance are characterized by efficiency prices. Furthermore, the notion of a partial ordering being induced by a semi-group of affine transformations is defined, and it is shown that third-degree dominance is induced by such a semi-group. This approach provides a unified analytical framework for stochastic dominance of the first, second, and third degree.

[1]  I. Olkin,et al.  Inequalities: Theory of Majorization and Its Applications , 1980 .

[2]  Peter C. Fishburn,et al.  Transfer principles in income redistribution , 1984 .

[3]  E. Maskin,et al.  Optimal Auctions with Risk Averse Buyers , 1984 .

[4]  Stephen A. Ross,et al.  Portfolio Efficient Sets , 1982 .

[5]  James E. Foster,et al.  ? 1987 the Society for Economic Analysis Limited Transfer Sensitive Inequality Measures , 2022 .

[6]  Steven A. Lippman,et al.  An Alternative Characterization of Decreasing Absolute Risk Aversion , 1983 .

[7]  Ronny Aboudi,et al.  Efficient Algorithms for Stochastic Dominance Tests Based on Financial Market Data , 1994 .

[8]  S. Ross.  SOME STRONGER MEASURES OF RISK AVERSION IN THE SMALL AND THE LARGE WITH APPLICATIONS , 1981 .

[9]  V. Bawa.  OPTIMAL, RULES FOR ORDERING UNCERTAIN PROSPECTS+ , 1975 .

[10]  L. Auslander,et al.  Introduction to Differentiable Manifolds , 1963 .

[11]  L. Thorlund-Petersen,et al.  Stochastic dominance and Friedman-Savage utility functions , 1988 .

[12]  Peter C. Fishburn,et al.  Moment-Preserving Shifts and Stochastic Dominance , 1982, Math. Oper. Res..

[13]  Roger J.-B. Wets,et al.  Order-preserving functions: applications to majorization and order statistics-Pacific Journal of Mat , 1967 .

[14]  David Schmeidler,et al.  A bibliographical note on a theorem of Hardy, Littlewood, and Polya , 1979 .

[15]  L. J. Savage,et al.  The Utility Analysis of Choices Involving Risk , 1948, Journal of Political Economy.

[16]  R. G. Vickson,et al.  Stochastic Dominance Tests for Decreasing Absolute Risk Aversion. I. Discrete Random Variables , 1975 .

[17]  M. Chapman Findlay,et al.  Stochastic dominance : an approach to decision-making under risk , 1978 .

[18]  J. Pratt.  RISK AVERSION IN THE SMALL AND IN THE LARGE11This research was supported by the National Science Foundation (grant NSF-G24035). Reproduction in whole or in part is permitted for any purpose of the United States Government. , 1964 .

[19]  K. Arrow.  Essays in the theory of risk-bearing , 1958 .

[20]  D. L. Hanson,et al.  ON A NEGLECTED ASPECT OF THE THEORY OF RISK AVERSION , 1971 .

[21]  Josef Hadar,et al.  Rules for Ordering Uncertain Prospects , 1969 .

[22]  L. Thorlund-Petersen,et al.  The value of perfect information in a simple investment problem , 1993 .

[23]  Lars Thorlund-Petersen,et al.  Comparison of probability measures: Dominance of the third degree , 2000, Oper. Res. Lett..

[24]  J. Stiglitz,et al.  Sharecropping and the Interlinking of Agrarian Markets , 1982 .

[25]  M. Yaari,et al.  Some remarks on measures of risk aversion and on their uses , 1969 .

[26]  Bezalel Peleg,et al.  A Price Characterization of Efficient Random Variables , 1975 .

[27]  P. Fishburn.  Third-degree stochastic dominance and random variables , 1985 .

[28]  J. Pratt.  RISK AVERSION IN THE SMALL AND IN THE LARGE11This research was supported by the National Science Foundation (grant NSF-G24035). Reproduction in whole or in part is permitted for any purpose of the United States Government. , 1964 .

[29]  L. J. Savage,et al.  The Foundations of Statistics , 1955 .

[30]  John D. Tressler,et al.  Increasing Downside Risk , 1980 .

[31]  G. Debreu,et al.  Stochastic Choice and Cardinal Utility , 1958 .