A Decision-Theoretical Measure of Uncertainty

Abstract

Uncertain or ambiguous events cannot be objectively measured by probabilities, i.e. different decision makers may disagree about their likelihood of occurrence. This paper proposes a new decision-theoretical approach how to measure ambiguity (Knightian uncertainty) that is independent of subjective preference orders and beliefs of decision makers and that is analogous to axiomatic risk measurement in finance. A decision-theoretical measure of ambiguity is a function from choice alternatives (acts) to non-negative real numbers. Our proposed measure of ambiguity is derived from a novel assumption that ambiguity of any choice alternative can be decomposed into a left-tail ambiguity (uncertainty in the realization of relatively undesirable outcomes) and a right-tail ambiguity (uncertainty in the realization of relatively desirable outcomes). This decomposability assumption is combined with two standard assumptions: ambiguity sources (events) are independent (separable) from outcomes (consequences) and any elementary increase in uncertainty (increasing a more desirable outcome in a binary act) necessarily increases ambiguity.