Abstract
Fishburn and Vickson (Stochastic dominance: an approach to decision-making under risk, Lexington Books, D.C. Heath and Company, Lexington, pp. 39–113, 1978) showed that, when applied to random alternatives with an equal mean, 3rd-degree and decreasing absolute risk aversion stochastic dominances represent equivalent rules. The present paper generalizes this result to higher degrees. Specifically, higher-degree stochastic dominance rules and common preference by all decision makers with decreasing higher-order absolute risk aversion are shown to coincide under appropriate constraints on the respective moments of the random variables to be compared.
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Notes
This intuition is based on an alternative proof of the Fishburn and Vickson theorem in Liu and Meyer (2012a).
For example, Menezes et al. (1980) defined a third-degree or downside risk increase as a change in a random variable that moves risk from right to left while keeping the mean and variance intact. And Ekern (1980) defined an \(n\)th-degree risk increase as an \(n\)th-degree stochastically dominated change in a random variable that keeps the first \(n-1\) moments the same.
Notice that we impose here a strict inequality to define risk apportionment, to avoid dividing by zero when defining \(A_{u,n,m}\) as ratios of derivatives of the utility function \(u\).
References
Arrow, K. J. (1971). Essays in the theory of risk-bearing. New York: North-Holland.
Brockett, P. L., & Golden, L. L. (1987). A class of utility functions containing all the common utility functions. Management Science, 33, 955–964.
Caballe, J., & Pomansky, A. (1996). Mixed risk aversion. Journal of Economic Theory, 71, 485–513.
Chiu, H. W. (2005). Skewness preference, risk aversion and the precedence relations on stochastic changes. Management Science, 51, 1816–1828.
Denuit, M., & Eeckhoudt, L. (2010a). A general index of absolute risk attitude. Management Science, 56, 712–715.
Denuit, M., & Eeckhoudt, L. (2010b). Stronger measures of higher-order risk attitudes. Journal of Economic Theory, 145, 2027–2036.
Denuit, M., & Eeckhoudt, L. (2012). Risk attitudes and the value of risk transformations. International Journal of Economic Theory, in press.
Denuit, M., & Eeckhoudt, L. (2013). Improving your chances: A new result. Economics Letters, 118, 475–477.
Denuit, M., Lefevre, C., & Scarsini, M. (2001). On s-convexity and risk aversion. Theory and Decision, 50, 239–248.
Denuit, M., & Rey, B. (2010). Prudence, temperance, edginess, and risk apportionment as decreasing sensitivity to detrimental changes. Mathematical Social Sciences, 60, 137–143.
Ebert, S. (2013). Moment characterization of higher-order risk preferences. Theory and Decision, 74, 267–284.
Eeckhoudt, L., & Schlesinger, H. (2006). Putting risk in its proper place. American Economic Review, 96, 280–289.
Eeckhoudt, L., & Schlesinger, H. (2009). On the utility premium of Friedman and Savage. Economics Letters, 105, 46–48.
Eeckhoudt, L., Schlesinger, H., & Tsetlin, I. (2009). Apportioning of risks via stochastic dominance. Journal of Economic Theory, 144, 994–1003.
Ekern, S. (1980). Increasing \(n\)th degree risk. Economics Letters, 6, 329–333.
Fishburn, P., & Vickson, R. (1978). Theoretical foundations of stochastic dominance. In G. Whitmore & M. Findlay (Eds.), Stochastic dominance: An approach to decision-making under risk (pp. 39–113). Lexington, MA: Lexington Books, D.C. Heath and Company.
Kimball, M. S. (1990). Precautionary savings in the small and in the large. Econometrica, 58, 53–73.
Kimball, M. S. (1992). Precautionary motives for holding assets. In P. Newman, M. Milgate, & J. Eatwell (Eds.), The new Palgrave dictionary of money and finance (Vol. 3, pp. 158–161). London: McMillan.
Liu, L., & Meyer, J. (2012a). Decreasing absolute risk aversion, prudence and increased downside risk aversion. Journal of Risk and Uncertainty, 44, 243–260.
Liu, L., & Meyer, J. (2012b). Risk aversion orders and intensity measures of a higher degree. Working paper available at http://perc.tamu.edu/perc/Publication/workingpapers/1203.pdf.
Menezes, C., Geiss, C., & Tressler, J. (1980). Increasing downside risk. American Economic Review, 70, 921–932.
Menezes, C. F., & Wang, H. (2005). Increasing outer risk. Journal of Mathematical Economics, 41, 875–886.
Pratt, J. W. (1964). Risk aversion in the small and in the large. Econometrica, 32, 122–136.
Roger, P. (2011). Mixed risk aversion and preference for risk disaggregation: A story of moments. Theory and Decision, 70, 27–44.
Rothschild, M., & Stiglitz, J. E. (1970). Increasing risk: I. A definition. Journal of Economic Theory, 2, 225–243.
Acknowledgments
The authors wish to warmly thank Jack Meyer for stimulating discussions. Michel Denuit acknowledges the financial support from the contract “Projet d’Actions de Recherche Concertées” No 12/17-045 of the “Communauté Française de Belgique”, granted by the “Académie Universitaire Louvain”.
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Denuit, M., Liu, L. Decreasing higher-order absolute risk aversion and higher-degree stochastic dominance. Theory Decis 76, 287–295 (2014). https://doi.org/10.1007/s11238-013-9374-3
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DOI: https://doi.org/10.1007/s11238-013-9374-3