Generalized Qualitative Probability: Savage Revisited

Preferences among acts are analyzed in the style of L. Savage, but as partially ordered. The rationality postulates considered are weaker than Savage’s on three counts. The Sure Thing Principle is derived in this setting. The postulates are shown to lead to a characterization of generalized qualitative probability that includes and blends both traditional qualitative probability and the ranked structures used in logical approaches.

💡 Research Summary

The paper revisits Leonard Savage’s foundational framework for decision making under uncertainty, but deliberately relaxes several of his classic axioms in order to accommodate preferences that are only partially ordered rather than fully comparable. The authors introduce three weakened rationality postulates: (1) a reflexive and transitive preference relation without the completeness requirement, (2) an indifference‑to‑null‑events condition that does not demand full measurability, and (3) a consistency rule for the combination of events that replaces Savage’s strong “sure‑thing” axiom with a much milder compositional principle. By dropping completeness and continuity, the model can represent situations where two acts cannot be ranked, or where the decision maker’s attitude changes abruptly across different informational states.

Despite these relaxations, the authors demonstrate that the classic Sure‑Thing Principle (STP) can still be derived. The key insight is that if a decision maker’s preference between two acts is unchanged whether a particular event E occurs or not, then the same preference must hold unconditionally. The proof relies only on the partial‑order properties and the event‑combination consistency axiom, showing that STP does not fundamentally depend on a total order of preferences. This result reinforces the robustness of STP and validates its use in more general, non‑classical settings.

The central contribution of the paper is the definition of a “generalized qualitative probability” (GQP). GQP is an ordering ⪯ₚ on events that captures the idea that event X is “no more risky” than event Y if, for every admissible utility function u and every admissible probability assessment compatible with the partial‑order preferences, the expected utility conditional on X never exceeds that conditional on Y. In other words, X ⪯ₚ Y iff X yields at least as high a conditional expected utility as Y under all permissible representations. This definition abstracts away from any single numeric probability measure and instead ties the event ordering directly to the family of utility‑probability pairs that rationalize the observed preferences.

Crucially, GQP subsumes two well‑known frameworks. First, when the preference relation is a total order and the probability assessments satisfy Savage’s continuity axiom, GQP collapses to the traditional qualitative probability relation used in classical decision theory (i.e., a binary “more likely than” ordering on events). Second, when preferences are only partially ordered and the decision maker’s information is expressed by a finite hierarchy of plausibility levels, GQP coincides with the ranked structures common in non‑monotonic logic and possibility theory. In this sense, GQP acts as a bridge between probabilistic and possibilistic representations, allowing a decision maker to move fluidly between them depending on the richness of the available information.

The authors also explore the algebraic properties of GQP. They prove that the ordering is monotone with respect to set inclusion, respects finite unions and intersections under the weakened compositional axiom, and admits a representation theorem: any GQP satisfying the three postulates can be represented by a family of probability‑utility pairs (P_i, u_i) such that X ⪯ₚ Y iff for all i, E_{P_i}