Learning from human preference data is the dominant approach to aligning language models with human values. In linear social choice, rewards are assumed to be linear functions of a fixed feature representation of prompt‑response pairs. Ge et al. (2024) showed that fitting such a reward by minimizing any non‑decreasing convex loss—including BTL—fails Preference Orderability (PO) and Pairwise Majority Consistency (PMC). Moreover, any rule that only reads the majority relation cannot satisfy PO once the output is required to be linearly induced.\ \ To address this dilemma, we relax the linear model by allowing a per‑candidate slack. We compute the relaxed linear reward with the smallest total slack that satisfies the axioms, imposing a margin $$\eta$$ as the minimum required difference between any two reward values. This solution satisfies all axioms without any assumptions on voters or how comparisons were collected. We prove that when $$\eta \le O\left(\frac{1}{m^2}\right)$$ (with $m$ candidates), the optimal total slack is bounded by $O(1)$, and we exhibit an instance that forces this bound, showing the rate is tight up to constant factors.\ \ In practice, the number of candidates far exceeds the feature dimension, and only a linear reward can be evaluated on unseen responses. Accordingly, we introduce a new method that charges the linear part for each comparison it gets wrong while simultaneously minimizing total slack and the number of violations. A parameter $\lambda$ trades off between the two objectives. We show that the total slack is monotone but saturating in $\lambda$: increasing $\lambda$ reduces violations of the deployed linear reward and raises the slack, yet the slack never exceeds $$ (\eta + \Delta \sqrt{d}) \left\lfloor \frac{m^2}{4} \right\rfloor $$, where $\Delta$ and $d$ denote the diameter and dimension of the feature space, respectively.\ \ Experiments on both synthetic and real‑world preference datasets corroborate the theory, and the linear reward produced by our method consistently outperforms the traditional linear BTL baseline.\ \ Review