Departments of Economics and Mathematical Statistics, Columbia University, 335 Riverside Drive, New York, NY 10027, USA
Copyright © 2010 Graciela Chichilnisky. This is an open access article distributed under the
Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We extend the foundation of probability in samples with rare
events that are potentially catastrophic, called black swans, such as natural
hazards, market crashes, catastrophic climate change, and species extinction. Such events are generally treated as ‘‘outliers’’
and disregarded. We propose a new axiomatization of probability requiring equal treatment in the measurement of 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 (1954) 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 by Chichilnisky (1996, 2000, 2002, 2009) to encompass the foundation of subjective probability and axiomatic treatments
of subjective probability by Villegas (1964), De Groot (1963), Dubins and Savage (1965), Dubins
(1975) Purves and Sudderth (1976) and of choice under uncertainty by Arrow (1971).