Axiomatic Definitions of Risk. L. A. Cox, Jr., Cox Associates, 503 Franklin Street, Denver, Colorado 80218
The most useful definition of "#34 has long been debated without yielding a generally accepted resolution. Probabilities and consequences appear in most definitions, but formulas for combining them to obtain quantitative risk measures are elusive. This paper proposes axioms that any satisfactory definition of risk might satisfy. Existence and uniqueness theorems are given, showing which axioms imply what definitions of risk. Two main axiomatic frameworks are considered. First is the measurement theory framework, which introduces a binary comparative relation ("at least as risky as" for comparing mathematical objects or "#34 (e.g., probability distributions or stochastic processes). Properties of the comparative relation are posited that imply functional forms assigning numbers to prospects so that one prospect is at least as risky as another if and only if it has at least as high a number assigned to it. These numbers are measures of risk and the formulas for computing them define risk. The second axiomatic framework is decision-analytic. It asks what a risk-averse decision maker must know about a prospect, other than its expected value, to make rational decisions. The answer defines new measures of risk as minimal sufficient statistics for prospects, summarizing what about them risk-averse decision-makers avoid. Subsets of axioms are identified for which these two formal approaches agree on the definition of "#34.
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