The Representation-Rationalizability Tradeoff in Reward Learning

In RLHF, each training example contains a prompt $x$ and two candidate responses $y,y'$, and annotators provide pairwise preferences between these responses. The learning problem is to convert these heterogeneous pairwise judgments into a single scalar reward $r(x,y)$ that measures response quality for each prompt. Classical social choice implies an impossibility because heterogeneous annotator samples can induce pooled preferences with Condorcet cycles, so no scalar reward can evaluate all compared response pairs consistently. A growing literature analyzes RLHF as a social-choice problem, but usually assumes a fixed finite set of alternatives, i.e., a pre-enumerated finite set of candidate responses for each prompt. Modern pipelines instead score responses through a learned representation $\phi(x,y)$ before a scalar head, so $\phi$ determines which responses are treated as distinguishable alternatives and which comparisons are visible to the reward model. Once this embedding is part of the problem, the impossibility results from social choice theory become a tradeoff. We show that the excess cross-entropy loss of any reward built on $\phi$ decomposes exactly into a representational term, which a richer $\phi$ shrinks, and an aggregation term, which a richer $\phi$ enlarges by exposing more comparisons that no scalar can rank consistently. The same results extend to direct preference optimization (DPO), and jointly training the embedding and the reward cannot guarantee to recover the sweet spot of this tradeoff. Experiments on synthetic data and real preference datasets