Four notions of mean-preserving increase in risk, risk attitudes and applications to the rank-dependent expected utility model

This article presents various notions of risk generated by the intuitively appealing single-crossing operations between distribution functions. These stochastic orders, Bickel & Lehmann dispersion or (its equal-mean version) Quiggin's monotone mean-preserving increase in risk and Jewitt's location-independent risk, have proved to be useful in the study of Pareto allocations, ordering of insurance premia and other applications in the Expected Utility setup. These notions of risk are also relevant tothe Quiggin-Yaari Rank-dependent Expected Utility (RDEU) model of choice among lotteries. Risk aversion is modeled in the vNM Expected Utility model by Rothschild & Stiglitz's Mean Preserving Increase in Risk (MPIR). Realizing that in the broader rank-dependent set-up this order is too weak to classify choice, Quiggin developed the stronger monotone MPIR for this purpose. This paper reviews four notions of mean-preserving increase in risk - MPIR, monotoneMPIR and two versions of location-independent risk (renamed here left and right monotone MPIR) - and shows which choice questions are consistently modeled by each of these four orders.

[1]  Robert B. Wilson.  THE THEORY OF SYNDICATES , 1968 .

[2]  Isaac Meilijson,et al.  More pessimism than greediness: a characterization of monotone risk aversion in the rank-dependent expected utility model , 2005 .

[3]  J. Quiggin.  A theory of anticipated utility , 1982 .

[4]  J. Stiglitz,et al.  Increases in risk and in risk aversion , 1974 .

[5]  John Quiggin,et al.  Increasing Risk: Another Definition , 1991 .

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

[7]  H. Levy,et al.  Efficiency analysis of choices involving risk , 1969 .

[8]  M. Rothschild,et al.  Increasing risk: I. A definition , 1970 .

[9]  I. Jewitt.  Choosing between risky prospects: the characterization of comparative statics results, and location independent risk , 1989 .

[10]  M. Yaari.  The Dual Theory of Choice under Risk , 1987 .

[11]  P. Bickel,et al.  Descriptive Statistics for Nonparametric Models. III. Dispersion , 1976 .

[12]  I. Meilijson,et al.  Lotteries, insurance, and star-shaped utility functions , 1990 .

[13]  P. Bickel,et al.  DESCRIPTIVE STATISTICS FOR NONPARAMETRIC MODELS IV. SPREAD , 1979 .

[14]  C. S. Hong,et al.  Risk aversion in the theory of expected utility with rank dependent probabilities , 1987 .

[15]  I. Meilijson,et al.  The generating process and an extension of Jewitt's location independent risk concept , 1994 .

[16]  J. Pratt,et al.  Increasing Risk: Some Direct Constructions , 1997 .

[17]  Michèle Cohen,et al.  Risk-aversion concepts in expected- and non-expected-utility models. , 1996 .

[18]  Alain Chateauneuf,et al.  On the use of capacities in modeling uncertainty aversion and risk aversion , 1991 .

[19]  Stephen E. Fienberg,et al.  Testing Statistical Hypotheses , 2005 .

[20]  M. Rothschild,et al.  Increasing risk II: Its economic consequences , 1971 .

[21]  Isaac Meilijson,et al.  Co-monotone allocations, Bickel-Lehmann dispersion and the Arrow-Pratt measure of risk aversion , 1994, Ann. Oper. Res..

[22]  Michèle Cohen,et al.  Risk-Aversion Concepts in Expected- and Non-Expected-Utility Models , 1995 .

[23]  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 .

[24]  Edi Karni,et al.  Two theorems on optimal stopping with backward solicitation , 1977, Journal of Applied Probability.

[25]  E. Rowland.  Theory of Games and Economic Behavior , 1946, Nature.

[26]  Harris Schlesinger,et al.  Arrow's theorem on the optimality of deductibles: A stochastic dominance approach , 1996 .

[27]  Ian Jewitt,et al.  Risk Aversion and the Choice Between Risky Prospects: The Preservation of Comparative Statics Results , 1987 .

[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]  William G. Cochran,et al.  Contributions to statistics , 1983 .

[30]  Isaac Meilijson,et al.  Demand for risky financial assets: A portfolio analysis , 1990 .

[31]  D. A. Edwards.  On the existence of probability measures with given marginals , 1978 .

[32]  A. Chikán,et al.  Progress in decision, utility and risk theory , 1991 .

[33]  M. Yaari,et al.  A controversial proposal concerning inequality measurement , 1988 .

[34]  Risk aversion without diminishing marginal utility , 1984 .

[35]  M. O. Lorenz,et al.  Methods of Measuring the Concentration of Wealth , 1905, Publications of the American Statistical Association.

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

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

[38]  Patrick Moyes.  Inequality Reducing and Inequality Preserving Transformations of Incomes: Symmetric and Individualistic Transformations , 1994 .

[39]  Edi Karni,et al.  Search theory: The case of search with uncertain recall , 1977 .

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