Abstract
We extend the foundation of subjective probability to integrate rare events that are potentially catastrophic, called black swans, such as natural hazards, market crashes, catastrophic climate change and major episodes of species extinction. Such events are generally treated as ‘outliers’ and often disregarded. We propose a new axiomatization of subjective probability requiring equal treatment for rare and frequent events—the Swan Axiom—and characterize the subjective probabilities that the axioms imply: these are neither finitely additive nor countably additive but a combination of both. They exclude countably additive probabilities as in De Groot (1970) and Arrow (1971), and are a strict subset of Savage (1972) probabilities that are finitely additive measures. Our subjective probabilities are standard distributions when the sample has no black swans. The finitely additive part assigns however more weight to rare events than do standard distributions and in that sense explains the persistent observation of ‘power laws’ and ‘heavy tails’ that eludes classic theory. The axioms extend earlier work in Chichilnisky (2000; 2002) to encompass the foundation of subjective probability, and axiomatic treatments of subjective probability by Savage (1972), Villegas (1964) and the foundations of choice under uncertainty in Arrow (1971).
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Allais, M. (1953). Le Comportement de l’Homme Rationnel devant le Risque: Critique des Postulats et Axiomes de l’Ecole Americaine. Econometrica, 21(4), 503–546.
Arrow, K. (1971). Essays in the theory of risk bearing. Amsterdam: North Holland.
Bernstein, P. L. (1996). Against the gods: A remarkable story of risk. New York: Wiley.
Chanel, O., & Chichilnisky, G. (2009a). The influence of fear in decisions: Experimental evidence. Journal of Risk and Uncertainty, 39(3),
Chanel, O., & Chichilnisky, G. (2009b). The value of life: Theory and experiments working paper, GREQE. New York: Universite de Marseille and Columbia University
Chichilnisky, G. (1996). An axiomatic approach to sustainable development. Soc Choice Welf, 13, 257–321.
Chichilnisky, G. (2000). An axiomatic approach to choice under uncertainty with catastrophic risks. Resource Energy Economics, 22, 221–231.
Chichilnisky, G. (2002). Catastrophic risk, encyclopedia of environmetrics. In A.H. El-Shaarawi & W.W. Piegorsch (Eds.) (Vol. 1, pp. 274-279). Chichester: Wiley.
Chichilnisky, G. (2009a). The limits of econometrics: Non parametric estimation in Hilbert spaces. Journal of Economic Theory.
Chichilnisky, G. (2009b). Choice under uncertainty: The work and legacy of kenneth arrow encyclopedia of quantitative finance, ed Cont R. Chichester: Wiley (in press).
Chichilnisky, G. (2009c). The topology of fear. Journal of Mathematical Economics.
Chichilnisky, G., & Heal, G. M. (1997). Social choice with infinite populations. Soc Choice Welf, 14, 303–319.
Chichilnisky, G., & Wu, H.-M. (2006). General equilibrium with endogenous uncertainty and default. Journal of Mathematical Economics, 42, 499–524.
Debreu, G. (1953). Valuation equilibrium and pareto optimum. Proceedings of the National Academy of Sciences, 40, 588–592.
De Groot, M. H. (1970, 2004). Optimal statistical decisions. Hoboken, New Jersey: Wiley.
Dreze, J. H. (1987). Essays on economic decisions under uncertainty. Cambridge: Cambridge University Press.
Dunford, N., & Schwartz, J. T. (1958). Linear operators, Part I. New York: Interscience.
Ellsberg, D. (1961). Risk, ambiguity, and the savage axioms. Quartarly Journal of Economics, 75, 643–669.
Godel, K. (1940). The consistency of the continuum hypothesis. Annals of Mathematical Studies 3. Princeton: Princeton University Press.
Kadane, J. B., & O’Hagan, A. (1995). Using finitely additive probability: Uniform distributions of the natural numbers. Journal of the American Statistical Association, 90, 525–631.
Kahneman, D., & Tversky, A. (1979). Prospect theory: An analysis of decisions under risk. Econometrica, 47, 263–291.
Le Doux, J. (1996). The emotional brain. New York: Simon and Schuster.
Savage, L. (1972). The foundations of statistics. Wiley: New York (1054, revised edition).
Schmeidler, D. (1989). Subjective probability and expected utility without additivity. Econometrica, 47, 263–291.
Tversky, A., & Kahneman, D. (1992). Advances in prospect theory: Cumulative representation of uncertainty. Journal of Risk and Uncertainty, 5, 297–323.
Yosida, K. (1974). Functional analysis (4th ed.). New York, Heidelberg: Springer.
Yosida, K., & Hewitt, E. (1952). Finitely level independent measures. Transactions of the American Mathematical Society, 72, 46–66.
Villegas, C. (1964). On quantitative probability \(\sigma \)–algebras. Annals of Mathematical Statistics, 35, 1789–1800.
Acknowledgements
This research was conducted at Columbia University’s Program on Information and Resources and its Columbia Consortium for Risk Management (CCRM). We acknowledge support from Grant No 5222-72 of the US Air Force Office of Research directed by Professor Jun Zheng, Washington DC. The initial results (Chichilnisky (1996)) were presented at Stanford University’s 1993 Seminar on Reconsideration of Values, organized by Professor Kenneth Arrow, at a 1996 Workshop on Catastrophic Risks organized at the Fields Institute for Mathematical Sciences of the University of Toronto, Canada, at the NBER Conference Mathematical Economics: The Legacy of Gerard Debreu at UC Berkeley, October 21, 2005, the Department of Economics of the University of Kansas National Bureau of Economic Research General Equilibrium Conference, September 2006, at the Departments of Statistics of the University of Oslo, Norway, Fall 2007, at a seminar organized by the late Professor Chris Heyde at the Department of Statistics of Columbia University, Fall 2007, and at seminars organized by Drs. Charles Figuieres and Mabel Tidball at LAMETA Universite de Montpellier, France December 19 and 20, 2008, and by Professor Alan Kirman at GREQAM Universite de Marseille, December 18 2008. We are grateful to the above institutions and individuals for supporting the research, and for helpful comments and suggestions. An anonymous referee provided insightful comments that improved this article.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Chichilnisky, G. (2016). The Foundations of Uncertainty with Black Swans. In: Chichilnisky, G., Rezai, A. (eds) The Economics of the Global Environment. Studies in Economic Theory, vol 29. Springer, Cham. https://doi.org/10.1007/978-3-319-31943-8_6
Download citation
DOI: https://doi.org/10.1007/978-3-319-31943-8_6
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-31941-4
Online ISBN: 978-3-319-31943-8
eBook Packages: Economics and FinanceEconomics and Finance (R0)