Ghossoub, Mario (2011): Monotone equimeasurable rearrangements with non-additive probabilities.
Download (371Kb) | Preview
In the classical theory of monotone equimeasurable rearrangements of functions, “equimeasurability” (i.e. the fact the two functions have the same distribution) is defined relative to a given additive probability measure. These rearrangement tools have been successfully used in many problems in economic theory dealing with uncertainty where the monotonicity of a solution is desired. However, in all of these problems, uncertainty refers to the classical Bayesian understanding of the term, where the idea of ambiguity is absent. Arguably, Knighitan uncertainty, or ambiguity is one of the cornerstones of modern decision theory. It is hence natural to seek an extension of these classical tools of equimeasurable rearrangements to situations of ambiguity. This paper introduces the idea of a monotone equimeasurable rearrangement in the context of non-additive probabilities, or capacities that satisfy a property that I call strong nonatomicity. The latter is a strengthening of the notion of nonatomicity, and these two properties coincide for additive measures and for submodular (i.e. concave) capacities. To illustrate the usefulness of these tools in economic theory, I consider an application to a problem arising in the theory of production under uncertainty.
|Item Type:||MPRA Paper|
|Original Title:||Monotone equimeasurable rearrangements with non-additive probabilities|
|Keywords:||Ambiguity, Capacity, Non-Additive Probability, Choquet Integral, Monotone Equimeasurable Rearrangement, Production under Uncertainty|
|Subjects:||C - Mathematical and Quantitative Methods > C6 - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling > C65 - Miscellaneous Mathematical Tools
C - Mathematical and Quantitative Methods > C0 - General > C02 - Mathematical Methods
D - Microeconomics > D8 - Information, Knowledge, and Uncertainty > D89 - Other
D - Microeconomics > D2 - Production and Organizations > D24 - Production; Cost; Capital; Capital, Total Factor, and Multifactor Productivity; Capacity
|Depositing User:||Mario Ghossoub|
|Date Deposited:||27. Mar 2012 22:04|
|Last Modified:||14. Feb 2013 21:43|
 C.D. Aliprantis and K.C. Border. Infinite Dimensional Analysis - 3rd edition. Springer-Verlag, 2006.
 M. Amarante. Foundations of Neo-Bayesian Statistics. Journal of Economic Theory, 144(5):2146–2173, 2009. 4,
 M. Amarante, M. Ghossoub, and E. Phelps. The Entrepreneurial Economy I: Contracting under Knightian Uncertainty. Columbia University, Center on Capitalism and Society, Working Paper No. 68 (April 2011).
 C. Burgert and L. R¨uschendorf. On the Optimal Risk Allocation Problem. Statistics & Decisions, 24(1):153–171, 2006.
 S. Cambanis, G. Simons, and W. Stout. Inequalities for Ek(X,Y) when the Marginals are Fixed. Zeitschrift fur Wahrscheinlichkeitstheorie und verwandte Gebiete, 36(4):285–294, 1976.
 C.F. Camerer. Individual Decision Making. In J.H. Kagel and A.E. Roth (eds.), Handbook of Experimental Economics. Princeton,: Princeton University Press, 1995.
 C.F. Camerer and M. Weber. Recent Developments in Modeling Preferences: Uncertainty and Ambiguity. Journal of Risk and Uncertainty, 5(4):325–370, 1992.
 G. Carlier and R.A. Dana. Core of Convex Distortions of a Probability. Journal of Economic Theory, 113(2):199–222, 2003.
 G. Carlier and R.A. Dana. Pareto Efficient Insurance Contracts when the Insurer’s Cost Function is Discontinuous. Economic Theory, 21(4):871–893, 2003.
 G. Carlier and R.A. Dana. Existence and Monotonicity of Solutions to Moral Hazard Problems. Journal of Mathematical Economics, 41(7):826–843, 2005.
 G. Carlier and R.A. Dana. Rearrangement Inequalities in Non-convex Insurance Models. Journal of Mathematical Economics, 41(4-5):483–503, 2005.
 G. Carlier and R.A. Dana. Two-persons Efficient Risk-sharing and Equilibria for Concave Law-invariant Utilities. Economic Theory, 36(2):189–223, 2008.
 G. Carlier and R.A. Dana. Optimal Demand for Contingent Claims when Agents Have Law Invariant Utilities. Mathematical Finance, 21(2):169–201, 2011.
 G. Carlier, R.A. Dana, and A. Galichon. Pareto Efficiency for the Concave Order and Multivariate Comonotonicity. Journal of Economic Theory, 147(1):207–229, 2012.
 G. Carlier and A. Lachapelle. A Numerical Approach for a Class of Risk-Sharing Problems. Journal of Mathematical Economics, 47(1):1–13, 2011.
 G. Carlier and L. Renou. Existence and Monotonicity of Optimal Debt Contracts in Costly State Verification Models. Economics Bulletin, 7(5):1–9, 2003.
 N.L. Carothers. Real Analysis. Cambridge University Press, 2000.
 A. Castaldo, F. Maccheroni, and M. Marinacci. Random Correspondences as Bundles of Random Variables. Sankhya: The Indian Journal of Statistics, 66(3):409–427, 2004.
 A. Castaldo and M. Marinacci. A Lusin Theorem for a Class of Choquet Capacities. Statistical Papers, 43(1):137–142, 2002.
 R.G. Chambers and J. Quiggin. Uncertainty, Production, Choice, and Agency: The State-Contingent Approach. Cambridge University Press, 2000.
 A. Chateauneuf and J. Lefort. Some Fubini Theorems on Product algebras for Non-Additive Measures. International Journal of Approximate Reasoning, 48(3):686–696, 2008.
 A. Chateauneuf and Y. Rebille. A Yosida-Hewitt decomposition for totally monotone games. Mathematical Social Sciences, 48(1):1–9, 2004.
 V. Chernozhukov, I. Fernandez-Val, and A. Galichon. Improving Point and Interval Estimators of Monotone Functions by Rearrangement. Biometrika, 96(3):559–575, 2009.
 V. Chernozhukov, I. Fernandez-Val, and A. Galichon. Rearranging Edgeworth-Cornish-Fisher Expansions. Economic Theory, 42(2):419–435, 2009.
 V. Chernozhukov, I. Fernandez-Val, and A. Galichon. Quantile and Probability Curves without Crossing. Econometrica, 78(3):1093–1125, 2010.
 K.M. Chong and N.M. Rice. Equimeasurable Rearrangements of Functions. Queens Papers in Pure and Applied Mathematics, 28, 1971.
 D.L. Cohn. Measure Theory. Birkhauser, 1980.
 R.A. Dana. Market Behavior when Preferences are Generated by Second-Order Stochastic Dominance. Journal of Mathematical Economics, 40(6):619–639, 2004.
 R.A. Dana and M. Scarsini. Optimal Risk Sharing with Background Risk. Journal of Economic Theory, 133(1):152–176, 2007.
 P.W. Day. Rearrangements of Measurable Functions. PhD thesis, California Institute of Technology, 1970.
 P.W. Day. Rearrangement Inequalities. Canadian Journal of Mathematics, 24(5):930–943, 1972.
 B. De Finetti. La Prevision: Ses Lois Logiques, Ses Sources Subjectives. Annales de l’Institut Henri Poincare, 7(1):1–68, 1937.
 D. Denneberg. Non-Additive Measure and Integral. Kluwer Academic Publishers, 1994.
 J. Dieudonne. Foundations of Modern Analysis. Academic press, 1969.
 J.L. Doob. Measure Theory. Springer, 1994.
 D. Ellsberg. Risk, Ambiguity, and the Savage Axioms. Quarterly Journal of Economics, 75(4):643–669, 1961.
 J.B. Epperson. A Class of Monotone Decreasing Rearrangements. Journal of Mathematical Analysis and Applications, 150(1):224–236, 1990.
 P. Ghirardato. On Independence for Non-Additive Measures, with a Fubini Theorem. Journal of Economic Theory, 73(2):261–291, 1997.
 P. Ghirardato, F.Maccheroni, and M. Marinacci. Differentiating Ambiguity and Ambiguity Attitude. Journal of Economic Theory, 118(2):133–173, 2004.
 M. Ghossoub. Belief Heterogeneity in the Arrow-Borch-Raviv Insurance Model. mimeo (2011).
 M. Ghossoub. On a Class of Monotone Comparative Statics Problems under Heterogeneous Uncertainty. mimeo (2011). 2  M. Ghossoub. Contracting under Heterogeneous Beliefs. PhD thesis, University of Waterloo – Dept. of Statistics & Actuarial Science, 2011.
 I. Gilboa and M. Marinacci. Ambiguity and the Bayesian Paradigm. mimeo (2011).
 I. Gilboa and D. Schmeidler. Maxmin Expected Utility with a Non-Unique Prior. Journal of Mathematical Economics, 18(2):141–153, 1989.
 G.H. Hardy, J.E. Littlewood, and G. P´olya. Inequalities. Cambridge University Press, Cambridge. Reprint of the 1952 edition, 1988.
 X.D. He and X.Y. Zhou. Portfolio Choice via Quantiles. Mathematical Finance, 21(2):203–231, 2011.
 E. Hewitt and K. Stromberg. Real and Abstract Analysis. Springer-Verlag: New York, 1965.
 H. Jin and X.Y. Zhou. Behavioral Portfolio Selection in Continous Time. Mathematical Finance, 18(3):385–426, 2008.
 F.H. Knight. Risk, Uncertainty, and Profit. Boston/New York: Houghton Mifflin, 1921.
 M. Landsberger and I. Meilijson. Co-monotone Allocations, Bickel-Lehmann Dispersion and the Arrow-Pratt Measure of Risk Aversion. Annals of Operations Research, 52(2):97–106, 1994.
 G.G. Lorentz. An Inequality for Rearrangements. The American Mathematical Monthly, 60(3):176–179, 1953.
 W.A.J. Luxemburg. Rearrangement Invariant Banach Function Spaces. In Queens Papers in Pure and Applied Mathematics, vol. 10: Proceeding of the Symposium in Analysis, pages 83–144. Queen’s University, Kingston, Ontario, 1967.
 F. Maccheroni and M. Marinacci. A Strong Law of Large Numbers for Capacities. Annals of probability, p. 1171–1178, 2005.
 M. Marinacci. Limit Laws for Non-Additive Probabilities and their Frequentist Interpretation. Journal of Economic Theory, 84(2):145–195, 1999.
 M. Marinacci and L. Montrucchio. Introduction to the Mathematics of Ambiguity. In I. Gilboa (ed.), Uncertainty in Economic Theory. Routledge, London, 2004.
 E. Pap. Null-Additive Set Functions. Kluwer Academic Publishers, 1995.
 Y. Rebille. Law of Large Numbers for Non-AdditiveMeasures. Journal of Mathematical Analysis and Applications, 352(2):872–879, 2009.
 S.I. Resnick. A Probability Path. Birkhauser, 1999.
 L.J. Savage. The Foundations of Statistics (2nd revised edition) – 1st ed. 1954. New York: Dover Publications, 1972.
 D. Schmeidler. Subjective Probability and Expected Utility without Additivity. Econometrica, 57(3):571–587, 1989.
 J. von Neumann and O. Morgenstern. Theory of Games and Economic Behavior. Princeton University Press, 1947.