Decision Making for Symbolic Probability

This paper proposes a decision theory for a symbolic generalization of probability theory (SP). Darwiche and Ginsberg [2,3] proposed SP to relax the requirement of using numbers for uncertainty while preserving desirable patterns of Bayesian reasoning. SP represents uncertainty by symbolic supports that are ordered partially rather than completely as in the case of standard probability. We show that a preference relation on acts that satisfies a number of intuitive postulates is represented by a utility function whose domain is a set of pairs of supports. We argue that a subjective interpretation is as useful and appropriate for SP as it is for numerical probability. It is useful because the subjective interpretation provides a basis for uncertainty elicitation. It is appropriate because we can provide a decision theory that explains how preference on acts is based on support comparison.

💡 Research Summary

The paper develops a decision‑theoretic framework for Symbolic Probability (SP), a generalisation of classical probability that replaces numeric probabilities with symbolic “supports”. In SP each event is assigned a support drawn from a set S equipped with a partial order ≤, a bottom element ⊥, a top element ⊤, and two binary operations ⊕ (support addition) and ⊗ (support multiplication). These operations satisfy algebraic laws analogous to those of real‑valued probabilities (associativity, commutativity, distributivity, identity elements), but they do not require a total ordering or a normalisation constraint. A support function σ maps events in a σ‑algebra to supports, preserving the usual relationships: σ(∅)=⊥, σ(Ω)=⊤, σ(A∪B)=σ(A)⊕σ(B), σ(A∩B)=σ(A)⊗σ(B).

Decision problems are modelled in the standard act‑based style: a state space Ω, a consequence space X, and an act f:Ω→X. For each act the authors identify an “gain” event G_f and a “loss” event L_f, and define the corresponding supports α_f=σ(G_f) and β_f=σ(L_f). Preferences over acts are expressed by a binary relation ≽, which the authors constrain with six intuitive axioms: completeness & transitivity, certainty (preferences over deterministic outcomes follow the outcome ranking), independence (adding the same “sure” act to two acts does not change their order), continuity (small changes in supports lead to small changes in preference), monotonicity (larger gain support or smaller loss support improves the act), and composability (the preference over compound acts can be derived from the preferences over components).

Under these axioms the authors prove a representation theorem: there exists a utility function u mapping each act to a pair of supports u(f)=(α_f,β_f) such that (α₁,β₁)≽(α₂,β₂) iff α₁≥α₂ and β₁≤β₂. In other words, utility is a two‑dimensional vector ordered by the Pareto rule on the support lattice. The proof proceeds by first constructing an “expected support” for each act using σ, then separating it into gain and loss components, and finally showing that the Pareto order on these components reproduces the original preference relation.

A key contribution is the subjective interpretation of SP. Just as subjective probability allows an agent to express personal degrees of belief numerically, SP lets an agent express qualitative comparisons of supports (“A is more supported than B”, “C and D have equal support”). This makes elicitation feasible in domains where precise numbers are unavailable or unwarranted, such as expert judgement, sparse data environments, or when the underlying uncertainty is inherently vague. The authors illustrate this with examples from medical diagnosis and multi‑criteria decision making, showing how support‑pair utilities can be used to rank treatment options or alternatives without assigning exact probabilities.

The paper also discusses the relationship between SP‑based decision theory and classical expected‑utility theory. While the latter relies on a total order of probabilities and a real‑valued utility function, the SP approach works with a partial order and a vector‑valued utility, yet it retains the essential Bayesian rationality principles (e.g., updating via ⊕ and ⊗). This makes SP a natural extension for situations where the “complete” probabilistic information required by Bayesian updating is missing.

Limitations are acknowledged. Because the support order is partial, some pairs of acts may be incomparable, leaving the decision maker without a clear choice. The authors suggest augmenting the model with tie‑breaking rules or additional “refinement” supports. They also note that concrete implementations of ⊕ and ⊗, computational algorithms for large‑scale support calculations, and empirical validation remain open research avenues.

In conclusion, the paper establishes that a coherent, axiomatic decision theory can be built on Symbolic Probability. By representing preferences with support‑pair utilities, it preserves the spirit of Bayesian reasoning while relaxing the numeric requirement of classical probability, thereby offering a flexible tool for decision making under qualitative or incomplete uncertainty.