The Foundations of Uncertainty with Black Swans

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).