Catastrophic Risks with Finite or Infinite States

Catastrophic risks are rare events with major consequences, such as market crashes, catastrophic climate change, asteroids or the extinction of a species. We show that classic expected utility theory based on Von Neumann axioms is insensitive to rare events no matter how catastrophic. Its insensitivity emerges from a requirement of continuity (e.g. Arrow's Monotone Continuity Axiom, and its relatives as defined by De Groot, Hernstein and Milnor) that anticipate average responses to extreme events. This leads to countably additive measures and `expected utility' that are insensitive to extreme risks. In a new axiomatic extension, the author (Chichilnisky 1996, 2000, 2002) requires equal treatment of rare and frequent events, deriving the new decision criterion the axioms imply. These are expected utility combined with purely finitely additive measures that focus on catastrophes, and explain the presistent observations of distributions with "fat tails" in earth sciences and financial markets. Continuity is based on the `topology of fear' introduced in Chichilnisky (2009), and is linked to Debreu's 1953 work on Adam Smith's Invisible Hand. The balance between the classic and the new axioms tests the limits of non- parametric estimation in Hilbert spaces, Chichilnisky (2008).. extending the foundations of probability & statistics (Chichilnisky 2009 and 2010) to include "black swans" or rare events, and finite as well as infinite state spaces.