Abstract
Classic probability theory treats rare events as ‘outliers’ that are often disregarded and underestimated. Yet in a moment of change rare events can become frequent and frequent events rare. We therefore postulate new axioms for probability theory that require a balanced treatment for rare and frequent events, based on what we call “the topology of change”. The axioms extend the foundation of probability to integrate rare but potentially catastrophic events or black swans: natural hazards, market crashes, catastrophic climate change and major episodes of species extinction. The new results presented in this article include a characterization of a family of purely finitely additive measures that are—somewhat surprisingly—absolutely continuous with respect to the Lebesgue measure. This is a new development from an earlier characterization of all the probabilities measures implied by the new axioms as a combination of purely finitely additive and countably additive measures that was first established in Chichilnisky (2000, 2002, 2009b) and the results are contrasted here to the work of Kolmogorov (1950), De Groot (1970), Arrow (1971), Dubins and Savage (1965), Savage (1972), Von Neumann and Morgernstern (1944), and Herstein and Milnor (1953).
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Notes
- 1.
The theory presented here explains also Jump-Diffusion processes Chichilnisky (2012), the existence of ‘heavy tails’ in power law distributions, and the lumpiness of most of the physical systems that we observe and measure.
- 2.
\(\phi _{U}(x)=1 \) when \(x\in U\) and \(\phi _{U}(x)=0\) when \(x\notin U\).
- 3.
In this article we make no difference between probabilities and relative likelihoods.
- 4.
This is Savage (1972) definition of probability.
- 5.
Savage’s probabilities can be either purely finitely additive or countably additive. In that sense they include all the probabilities in this article. However this article will exclude probabilities that are either purely finitely additive, orthose that are countably additive, and therefore our characterization of a probability is strictly finer than that Savage (1972), and different from the view of a measure as a countably additive set function in De Groot (1970).
- 6.
An equivalent definition of Monotone Continuity is that for every two events \( E_{1}\) and \(E_{2}\) in \(\{E_{\alpha }\}_{=1,2 \ldots }\) with \(W(E_{1})>W(E_{2})\), there exists N such that altering arbitrarily the events \(E_{1}\) and \(E_{2 \text { }}\)on a subset \(E^{i},\) where \(i>N,\) does not alter the subjective probability ranking of the events, namely \(W(E_{1}^{\prime })>W(E_{2}^{\prime })\) where \(E_{1}^{\prime }\) and \(E_{2}^{\prime }\) are the altered events.
- 7.
See De Groot (1970, Chap. 6, p. 71).
- 8.
Here \(E^{c}\) denotes the complement of the set E.
References
Arrow, K. (1971). Essays in the theory of risk bearing. Amsterdam: North Holland.
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. Social Choice and Welfare, 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. In A. H. El-Shaarawi & W. W. Piegorsch (Eds.), Encyclopedia of environmetrics (Vol. 1, pp. 274–279). Chichester: Wiley.
Chichilnisky, G. (2009a). The limits of econometrics: Non parametric estimation in Hilbert spaces. Economic Theory 2009.
Chichilnisky, G. (2009b). The topology of fear. Journal of Mathematical Economics 2009.
Chichilnisky, G. (2010). The foundations of statistics with black swans. Mathematical Social Sciences, 59(2), 184–192.
Chichilnisky, G. (2011). Sustainable markets with short sales. Economic Theory, 49(2), 293–307.
Chichilnisky, G. (2012). Economic theory and the global environment. Economic Theory, 59(2), 217–225.
Chichilnisky, G., & Heal, G. M. (1997). Social choice with infinite populations. Social Choice and Welfare, 14, 303–319.
Chichilnisky, G., & Wu, H.-M. (2006). General equilibrium with endogenous uncertainty and default. Journal of Mathematical Economics, 42, 499–524.
De Groot, M. H. (1970 and 2004). Optimal statistical decisions. Hoboken, New Jersey: Wiley.
Debreu, G. (1953). Valuation equilibrium and pareto optimum. Proceedings of the National Academy of Sciences, 40, 588–592.
Dubins, L. (1975). Finitely additive conditional probabilities, conglomerability and disintegration. The Annals of Probability, 3, 89–99.
Dubins, L., & Savage, L. (1965). How to gamble if you must. New York: McGraw Hill.
Dunford, N., & Schwartz, J. T. (1958). Linear operators, Part I. New York: Interscience.
Godel, K. (1940). The consistency of the continuum hypothesis. Annals of mathematical studies 3. Princeton NJ, USA: Princeton University Press.
Herstein, I. N., & Milnor, J. (1953). An axiomatic approach to measurable utility. Econometrica, 21, 291–297.
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.
Kolmogorov, A.N. (1933/1950) Foundations of the theory of probability. New York: Chelsea Publishing Co.
Le Doux, J. (1996). The Emotional Brain. New York: Simon and Schuster.
Purves, R. W., & Sudderth., (1976). Some finitely additive probability. The Annals of Probability, 4, 259–276.
Savage, L. J. (1972). The foundations of statistics New York: Wiley. (1054, revised edition).
Villegas, C. (1964). On quantitative probability \(\sigma \)–algebras. Annals of Mathematical Statistics, 35, 1789–1800.
Von Neumann, J. (1932/1955). Mathematical foundations of quantuum theory. New Jersey: Princeton University Press.
Von Neumann, J., & Morgernstern, O. (1944). Theory of games and economic behavior. New Jersey: Princeton University Press.
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.
Acknowledgements
This article is an expression of gratitude to the memory of Jerry Marsden, a great mathematician and a wonderful man. As his first PhD student in pure Mathematics when he was a professor at the Mathematics Department of UC Berkeley, the author is indebted to Jerry Marsden for counseling and support in obtaining the first of her two PhDs at UC Berkeley, in pure Mathematics. The second PhD in Economics at UC Berkeley was obtained by the author with the counseling of the Nobel Laureate economist, Gerard Debreu. Jerry Marsden was critical to encourage the growth of the research in this article on new and more realistic axiomatic foundations of probability theory; he invited the author to organize a Workshop on Catastrophic Risks at the Fields Institute in 1996 where this research was introduced, and strongly encouraged since 1996 the continuation and growth of this research.
The author is Director, Columbia Consortium for Risk Management (CCRM) Columbia University, and Professor of Economics and of Mathematical Statistics, Columbia University, New York 10027, 335 Riverside Drive, NY NY 10025, tel. 212 678 1148, email: chichilnisky@columbia.edu; website: www.chichilnisky.com. We acknowledge support from Grant No 5222-72 of the US Air Force Office of Research directed by Professor Jun Zheng, Washington DC from 2009 to 2012. Initial results on Sustainable Development were presented at Stanford University’s 1993 Seminar on Reconsideration of Values, organized by Kenneth Arrow, at the National Bureau of Economic Research 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 former Professor Christopher Heyde at the Department of Statistics of Columbia University, Fall 2007, 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. In December 8 2012, the work presented here and its applications were presented at an invited Plenary Key Note Presentation by the author to the Annual Meetings of the Canadian Mathematical Society Montreal Canada, December 8 2012. The work presented in this article is also the subject of a forthoming Plenary Key Note Presentation to the Annual Meeting of the Australian Mathematical Society in Sidney, Australia, December 18, 2013. We are grateful to the above institutions and individuals for supporting the research, and for helpful comments and suggestions.
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Chichilnisky, G. (2016). The Topology of Change Foundations of Probability with Black Swans Dedicated to the Memory of Jerrold Marsden . 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_7
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