SCOPE: Selective Conformal Optimized Pairwise LLM Judging

Large language models (LLMs) are increasingly used as judges to replace costly human preference labels in pairwise evaluation. Despite their practicality, LLM judges remain prone to miscalibration and systematic biases. This paper proposes SCOPE (Selective Conformal Optimized Pairwise Evaluation), a framework for selective pairwise judging with finite-sample statistical guarantees. Under exchangeability, SCOPE calibrates an acceptance threshold such that the error rate among non-abstained judgments is at most a user-specified level $\alpha$. To provide SCOPE with a bias-neutral uncertainty signal, we introduce Bidirectional Preference Entropy (BPE), which queries the judge under both response positions, aggregates the implied preference probabilities to enforce invariance to response order, and converts the aggregated probability into an entropy-based uncertainty score. Across MT-Bench, RewardBench, and Chatbot Arena, BPE improves uncertainty quality over standard confidence proxies, providing a stronger selection signal that enables SCOPE to consistently meet the target risk level while retaining good coverage across judge scales. In particular, at $\alpha = 0.10$, SCOPE consistently satisfies the risk bound across all benchmarks and judge scales (empirical risk $\approx 0.097$ to $0.099$), while retaining substantial coverage, reaching $0.89$ on RewardBench with Qwen-14B and $0.98$ on RewardBench with Qwen-32B. Compared to na\"ive baselines, SCOPE accepts up to $2.4\times$ more judgments on MT-Bench with Qwen-7B under the same target risk constraint, demonstrating that BPE enables reliable and high-coverage LLM-based evaluation