Nau Risk Ambiguity and State-Preference Theory.Pdf
Economic Theory, VolumeNo.Numbers 2-3, October 2011, 437-467 Noname manuscript 48, (will be inserted by the editor) DOI: 10.1007/s00199-011-0632-8
Risk, Ambiguity, and State-Preference Theory
Robert Nau
Received: 30 August, 2010 / Accepted: 16 May, 2011 © Springer-Verlag 2011 Received: date / Accepted: date
Abstract The state-preference framework for modeling choice under uncertainty, in which objects of choice are allocations of wealth or commodities across states of the world, is a natural one for modeling “smooth” ambiguity-averse preferences. It does not require reference to objective probabilities, personalistic consequences, or counterfactual acts, and it allows for state-dependence of utility and unobservable background risk. The decision maker’s local beliefs are encoded in her risk neutral probabilities (her relative marginal rates of substitution between states) and her local risk preferences are encoded in the matrix of derivatives of the risk neutral probabilities. This matrix plays a central but generally unappreciated role in the modeling of risk attitudes in the state-preference framework. It can be computed by inverting a bordered Slutsky matrix and vice versa, it generalizes the Arrow-Pratt measure for approximating local risk premia, and its structure reveals whether the decision maker’s risk preferences are ambiguity-averse as well as risk averse. Two versions of the smooth-ambiguity model are analyzed—the source-dependent risk aversion model and the second-order uncertainty (KMM) model—and it is shown that in both cases the overall premium for local uncertainty can be decomposed as the sum of a risk premium and an ambiguity premium.
1 Introduction The urn experiments devised by Ellsberg (1961) expose a subtle restriction imposed by the axioms of subjective expected utility, namely that the additivity property they confer on utility functions requires the decision maker to exhibit the same degree of risk aversion toward every kind of
Risk, Ambiguity, and State-Preference Theory
Robert Nau
Received: 30 August, 2010 / Accepted: 16 May, 2011 © Springer-Verlag 2011 Received: date / Accepted: date
Abstract The state-preference framework for modeling choice under uncertainty, in which objects of choice are allocations of wealth or commodities across states of the world, is a natural one for modeling “smooth” ambiguity-averse preferences. It does not require reference to objective probabilities, personalistic consequences, or counterfactual acts, and it allows for state-dependence of utility and unobservable background risk. The decision maker’s local beliefs are encoded in her risk neutral probabilities (her relative marginal rates of substitution between states) and her local risk preferences are encoded in the matrix of derivatives of the risk neutral probabilities. This matrix plays a central but generally unappreciated role in the modeling of risk attitudes in the state-preference framework. It can be computed by inverting a bordered Slutsky matrix and vice versa, it generalizes the Arrow-Pratt measure for approximating local risk premia, and its structure reveals whether the decision maker’s risk preferences are ambiguity-averse as well as risk averse. Two versions of the smooth-ambiguity model are analyzed—the source-dependent risk aversion model and the second-order uncertainty (KMM) model—and it is shown that in both cases the overall premium for local uncertainty can be decomposed as the sum of a risk premium and an ambiguity premium.
1 Introduction The urn experiments devised by Ellsberg (1961) expose a subtle restriction imposed by the axioms of subjective expected utility, namely that the additivity property they confer on utility functions requires the decision maker to exhibit the same degree of risk aversion toward every kind of
References: Afriat S (1980) Demand Theory and the Slutsky Matrix. Princeton University Press Arrow K (1964) The role of securities in the optimal allocation of risk-bearing. Quarterly J. Econ 31:91–96 Arrow K (1965) Aspects of the Theory of Risk Bearing. Yrjo Jahnssonin, Helsinki Bewley T (1986) Knightian decision theory, part I. Cowles Foundation Discussion Paper No. 807, Published in Decisions in Economics and Finance (2002)25:79110 Chew S, Sagi J (2008) Small worlds: modeling attitudes toward sources of uncertainty. J. Econ. Theory 139:1–24 Condie S, Ganguli J (2010) Ambiguity and rational expectations equilibria. Rev. Econ. Stud. , forthcoming Condie S, Ganguli J (2011) Informational efficiency with ambiguous information. Econ. Theory : this issue Debreu G (1959) Theory of Value: An Axiomatic Analysis of Economic Equilibrium. Yale University Press, New Haven Dickhaut J, Lunawat R, Pronin K, Stecher J (2011) Decision making and trade without probabilities. Econ. Theory : this issue Dooley P (1983) Slutsky’s equation is Pareto’s solution. History of Political Economy pp 513–517 Duncan G (1977) A matrix measure of multivariate local risk aversion. Econometrica 45:895–903 Eichberger J, Kelsey D (2000) Non-additive beliefs and strategic equilibria. Games and Economic Behavior 30:183–215 Eichberger J, Kelsey D (2011) Are the treasures of game theory ambiguous? Econ. Theory : this issue Ellsberg D (1961) Risk, ambiguity, and the Savage axioms. Quarterly J. Econ. 75:643–669 Epstein L (2010) A paradox for the ‘smooth ambiguity’ model of preference. Econometrica 78:2085–2099 Epstein L, Schneider M (2003) Recursive multiple-priors. J. Econ. Theory 113:1–31 Ergin H, Gul F (2009) A subjective theory of compound lotteries. J. Econ. Theory 144:899–929 Etner MJ J., Tallon J (2011) Decision theory under ambiguity. J. Econ. Surveys de Finetti B (1937) La pr´vision: ses lois logiques, ses sources subjectives. Ann. e Inst. Henri Poincar´ 7:1–68, translation reprinted in H.E. Kyburg and H.E. e Smokler, Eds., Studies in Subjective Probability, 2nd ed. (1980), Robert Krieger, New York, 53-118 de Finetti B (1952) Sulla preferibilit´. Giornale degli Economistii e Annali di a Economia 11:658–709 de Finetti B (1974) Theory of Probability. Wiley, New York Galaabaatar T, Karni E (2010) Objective and subjective expected utility with incomplete preferences. Working paper Gilboa I, Schmeidler D (1989) Maxmin expected utility with non-unique prior. J. Math. Econ. 18:141–153 Hayashi T (2005) Intertemporal substitution, risk aversion, and ambiguity aversion. Econ. Theory 25:933–956 Risk, Ambiguity, and State-Preference Theory 29 Karni DS E., Vind K (1983) On state-dependent preferences and subjective probabilities. Econometrica 51:1021–1031 Karni E (1979) On multivariate risk aversion. Econometrica 47:1391–1401 Karni E (1985) Decision Making Under Uncertainty: The Case of State-Dependent Preferences. Harvard University Press Klibanoff P, Marinacci M, Mukerji S (2005) A smooth model of decision making under ambiguity. Econometrica 73:1849–1892 Klibanoff P, Marinacci M, Mukerji S (2010) On the smooth ambiguity model: a reply. Working paper Klibanoff P, Marinacci M, Mukerji S (2011) Definitions of ambiguous events and the smooth ambiguity model. Econ Theory : this issue Knight F (1921) Uncertainty and profit. Houghton Mifflin, Boston Koopman B (1940) Axioms and algebra of intuitive probability,. Ann. Math. Stat. 41:269–292 Levi I (1970) The Enterprise of Knowledge. MIT Press, Cambridge Machina M (2004) Almost-objective uncertainty. Econ. Theory 24:1–54 Machina M (2005) ‘Expected utility/subjective probability’ analysis without the sure-thing principle or probabilistic sophistication. Econ. Theory 26:1–62 Machina M (2011) Event separability in the Ellsberg urn. Econ. Theory : this issue Nau R (2001a) Uncertainty aversion with second-order probabilities and utilities. Electronic Proceedings of the Second International Symposium on Imprecise Probabilities and their Applications http://www.sipta.org/isipta01/ proceedings/063.html Nau R (2001b) De Finetti was right: Probability does not exist. Theory and Decision 51:89–124 Nau R (2003) A generalization of Pratt-Arrow measure to non-expected-utility preferences and inseparable probability and utility. Managment Sci. 49:1089– 1104 Nau R (2006a) The shape of incomplete preferences. Ann. Statist. 34:2430–2448 Nau R (2006b) Uncertainty aversion with second-order utilities and probabilities. Management Sci. 52:136–145 Nau R (2011) Imprecise probabilities in non-cooperative games. Working paper Ok E, Ortoleva P, Riella G (2008) Incomplete preferences under uncertainty: indecisiveness in beliefs vs. tastes. Working paper Ozsoylev H, Werner J (2011) Liquidity and asset prices in rational expectations equilibrium with ambiguous information. Econ. Theory : this issue Pratt J (1964) Risk aversion in the small and in the large. Econometrica 32:122– 136 Rigotti L, Shannon C (2005) Uncertainty and risk in financial markets. Econometrica 73:203–243 Schervish M, Seidenfeld T, Kadane J (1990) State-dependent utilities. J. Amer. Statist. Assoc. 85:840–847 Schmeidler D (1989) Subjective probability and expected utility without additivity. Econometrica 57:571–587 Seidenfeld T, Schervish M, Kadane J (1990) Decisions without ordering. In: Sieg W (ed) Acting and Reflecting, Reidel, Dordrecht Seidenfeld T, Schervish M, Kadane J (1995) A representation of partially ordered preferences. Ann. Statist. 23:2168–2217 30 Robert Nau Seo K (2009) Ambiguity and second-order belief. Econometrica 77:1575–1605 Smith C (1961) Consistency in statistical inference and decision. J. Roy. Stat. Soc. B 23:1–25 Stern N (1986) A note on commodity taxation: the choice of variable and the Slutsky, Hessian, and Antonelli matrices (SHAM). Rev. Econ. Stud. 53:293–299 Wakker P (1989) Additive Representations of Preferences: a New Foundation for Decision Analysis. Kluwer, Dordrecht Walley P (1989) Statistical Reasoning With Imprecise Probabilities. Chapman and Hall, London Yaari M (1969) Some remarks on measures of risk aversion and their uses. J. Econ. Theory 1:315–329