The Foundations of Probability with Black Swans

We extend the foundation of probability in samples with rareevents that are potentially catastrophic, called black swans, such as naturalhazards, 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 additivebut a combination of both. They exclude countably additive probabilitiesas in De Groot (1970) and Arrow (1971) and are a strict subset ofSavage (1954) probabilities that are finitely additive measures. Our subjectiveprobabilities are standard distributions when the sample has noblack swans. The finitely additive part assigns however more weight torare events than do standard distributions and in that sense explains thepersistent 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 treatmentsof 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)