Differential Voting: Loss Functions For Axiomatically Diverse Aggregation of Heterogeneous Preferences

TL;DR

This paper introduces Differential Voting, a framework for designing differentiable loss functions that align with various classical voting rules, enabling explicit control over preference aggregation in reinforcement learning from human feedback.

Contribution

It develops differentiable surrogates for multiple voting rules, analyzes their properties, and links loss design choices to normative aggregation behavior in preference modeling.

Findings

Differentiable surrogates for BTL, Copeland, and Kemeny rules.

Calibration and consistency with social choice axioms.

Trade-offs between axiomatic guarantees and optimization stability.

Abstract

Reinforcement learning from human feedback (RLHF) implicitly aggregates heterogeneous human preferences into a single utility function, even though the underlying utilities of the participants are in practice diverse. Hence, RLHF can be viewed as a form of voting, where the aggregation mechanism is defined by the loss function. Although Arrow's Impossibility Theorem suggests that different mechanisms satisfy different sets of desirable axioms, most existing methods rely on a single aggregation principle, typically the Bradley-Terry-Luce (BTL) model, which corresponds to Borda count voting. This restricts the axiomatic properties of the learned reward and obscures the normative assumptions embedded in optimization. In this work, we introduce Differential Voting, a unifying framework that constructs instance-wise, differentiable loss functions whose population-level optima provably correspond to distinct classical voting rules. We develop differentiable surrogates for majority-based aggregation (BTL), Copeland, and Kemeny rules, and formally analyze their calibration properties, gradient fields, and limiting behavior as smoothing parameters vanish. For each loss, we establish consistency with the corresponding social choice rule and characterize the axioms it satisfies or violates. Our analysis shows how design choices in loss geometry-such as margin sensitivity and boundary concentration-directly translate into normative aggregation behavior. Differential Voting makes preference aggregation an explicit and controllable design choice in RLHF, enabling principled trade-offs between axiomatic guarantees and optimization stability. Code to reproduce our experiments is open-sourced.