Classical Lottery in Action: Quantifying Risk and Evaluating Uncertainty

The rarity of objectively known probabilities undermines the risk-ambiguity dichotomy, challenging the practical relevance of related theories. We return to classical lotteries—coins, dice, and similar devices—which inspired early probability theories through the idea of equiprobable outcomes and are widely considered strong candidates for objective probability. We adapt axioms of expected utility for risk to advocate average utility for classical lotteries, highlighting their conceptual affinity. Any general unknown event is conceived as a collection of possible classical-lottery frequencies, consistent with the ambiguity literature, and we suggest normative principles for their aggregation into an indifferent matching frequency. These principles identify a model in the spirit of the smooth model of ambiguity; the agent assigns subjective probabilities over the frequencies and aggregates them via a nonlinear mixture. The nonlinearity reflects the agent’s distinct attitudes toward classical-lottery frequencies versus subjective probabilities; a concave mixture, for example, captures ambiguity aversion. Our approach provides a concrete justification for the distinct attitudes and presents several advantages for model elicitation. We illustrate the theory’s applicability through examples and, especially, social policy evaluation where the veil of ignorance can be seen as a classical lottery.