We extend the foundation of probability in samples with rare
events that are potentially catastrophic, called

The article refines and extends the formulation of probability in an uncertain world. It provides an argument, and formalization, that probabilities must be additive functionals on

Savage [

The results are based on an axiomatic approach to choice under uncertainty and sustainable development introduced by Chichilnisky [

There are other approaches to subjective probability such as Choquet Expected Utility Model (CEU, Schmeidler, [

Uncertainty is described by a set of distinctive and exhaustive possible

For every two events

This axiom is equivalent to requiring that the probability of the sets along a vanishing sequence goes to zero. Observe that the decreasing sequence could consist of infinite intervals of the form

If

The following proposition establishes that the two axioms presented above are one and the same; both imply countable additivity.

A relative likelihood (subjective probability) satisfies the Monotone Continuity Axiom if and only if it satisfies Axiom

Assume that De Groot's axiom

The next section shows that the two axioms, Monotone Continuity and

The best way to explain the role of

To overcome the bias we introduce an axiom that is the logical negation of MC: this means that sometimes MC holds and others it does not. We call this the

Our Axiom provides a reasonable resolution to this dilemma that is realistic and consistent with the experimental evidence. It implies that there exist catastrophic outcomes such as the risk of death, so terrible that one is unwilling to face a small probability of death to obtain one cent versus nothing, no matter how small the probability may be. According to our Axiom, no probability of death may be acceptable when one cent is involved. Our Axiom also implies that in other cases there may be a small enough probability that the lottery involving death may be acceptable, for example if the payoff is large enough to justify the small risk. This is a possibility discussed by Arrow [

This axiom is the logical negation of Monotone Continuity: There exist events

A probability

A subjective probability satisfies Monotone Continuity if and only if it is biased against rare events.

This is immediate from the definitions of both [

Countably additive probabilities are biased against rare events.

It follows from Propositions

Purely finitely additive probabilities are biased against frequent events.

See example in the appendix.

A subjective probability that satisfies the Swan Axiom is neither biased against rare events, nor biased against frequent events.

This is immediate from the definition.

This section proposes an axiomatic foundation for subjective probability that is unbiased against rare and frequent events. The axioms are as follows:

Subjective probabilities are continuous and additive.

Subjective probabilities are unbiased against rare events.

Subjective probabilities are unbiased against frequent events.

Additivity is a natural condition and

A subjective probability that satisfies the Swan Axiom is neither finitely additive nor countably additive; it is a strict convex combination of both.

This follows from Propositions

Theorem

Theorem

From here on events are the Borel sets of the real line

A subjective probabiliy satisfying the classic axioms by De Groot [

The next step is to introduce the new axioms, show existence and characterize all the distributions that satisfy the axioms. We need more definitions. A subjective probability

The following three axioms are identical to the axioms in last section, specialized to the case at hand.

The first and the second axiom agree with classic theory and standard likelihoods satisfy them. The third axiom is new.

A standard probability satisfies Axioms

Consider

There exists a subjective probability

This result follows from the representation theorem by Chichilnisky [

The following illustrates the additional weight that the new axioms assign to rare events; in this example in a form suggesting “heavy tails.” The finitely additive measure

In samples without rare events, a subjective probability that satisfies Axioms

Axiom

There is a connection between the new axioms presented here and the Axiom of Choice that is at the foundation of mathematics (Godel, [

Consider a possible measure

One can illustrate a function on

One can use Hahn—Banach's theorem to extend the function

This illustrates why any attempts to construct

Consider the function

Consider the family

The space of continuous linear functions on

Illustration of a Finitely Additive Measure that is not Countably Additive

See Example

This research was conducted at Columbia University's Program on Information and Resources and its Columbia Consortium for Risk Management (CCRM). The author acknowledges support from Grant no 5222-72 of the US Air Force Office of Research directed by Professor Jun Zheng, Arlington VA. The initial results (Chichilnisky [