Savage Without Monotonicity

Shuo Li Liu, Jingni Yang

Keywords: decision theory, Savage representation theorem, AI alignment, reward modeling, finite additivity, monotonicity

TL;DR: Savage's full-domain representation separates belief from taste, but dominance requires an extra principle when finite additivity permits a positive event built from null parts.

Abstract: Savage separates belief from taste: coherent preferences over acts can be represented by subjective probabilities over states and utilities over consequences. We identify a conditional boundary of that separation. Savage's full-domain ideal ranges over both state-contingent acts and events, and its representation delivers a finitely additive probability. Those rich infinite domains are not added here as a special counterexample; they are the standard Savage-style machinery that lets preference data separate probabilistic belief from utility. Under either a countable convex-range route or a constructibility-based null-ideal route, that package can supply a positive event $A$ partitioned into disjoint null cells $E_n$. On a real-valued outcome scale, dominance supplies strictly better consequences whose utility gains tend to zero. Expected utility can then tie an act with its statewise improvement: dominance says the improved act is strictly preferred because the improvement occurs throughout a behaviorally non-null event, while representation says the acts are indifferent. The contribution is a philosophical premise separation, not a claim that finitely additive pathologies themselves are new: representation, dominance, and the repairs that connect them must be distinguished. Countable additivity, an explicit dominance axiom, act-domain restrictions, consequence restrictions, or state-space restrictions block the construction. Disentangling belief and taste does not itself supply monotonicity.

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Submission Number: 73