Aspect-wise Pairwise Tournaments for Test-Time Selection

Sampling-based test-time scaling enhances LLM reasoning by aggregating multiple reasoning paths for the same query. Existing methods use answer frequencies or path confidence to compute candidate scores, which do not explicitly characterize comparisons between competing answers. Inspired by the Condorcet principle in social choice theory, we introduce pairwise confidence comparisons between paths supporting different answers to capture answer-level comparative advantage. We combine this comparative evidence with voting support through a KL-regularized objective. Theoretically, we show that this comparative-advantage is equivalent to the adjustment to the traditional voting logits. In this way, our proposed Pairwise-Guided Logit Adjustment (PGLA) framework allows comparative evidence to compensate for weaker voting support and correct voting-based ranking errors. Experiments on 10 challenging benchmarks including mathematical reasoning, knowledge reasoning, and code generation with 7 models ranging from 1.7B to 32B parameters speak to the effectiveness of this plug-and-play term. Additional evaluations with external reward scores and online sampling further demonstrate its applicability.

Many applications seek to optimize LLM outputs at test time by iteratively proposing, scoring, and refining candidates over a discrete output space. Existing methods use a calibrated scalar evaluator for the target objective to guide search, but for many tasks such scores are unavailable, too sparse, or unreliable. Pairwise comparisons, by contrast, are often easier to elicit, still provide useful signal on improvement directions, and can be obtained from the LLM itself without external supervision. Building on this observation, we introduce Duel-Evolve, an evolutionary optimization algorithm that replaces external scalar rewards with pairwise preferences elicited from the same LLM used to generate candidates. Duel-Evolve aggregates these noisy candidate comparisons via a Bayesian Bradley-Terry model, yielding uncertainty-aware estimates of candidate quality. These quality estimates guide allocation of the comparison budget toward plausible optima using Double Thompson Sampling, as well as selection of high-quality parents to generate improved candidates. We evaluate Duel-Evolve on MathBench, where it achieves 20 percentage points higher accuracy over existing methods and baselines, and on LiveCodeBench, where it improves over comparable iterative methods by over 12 percentage points. Notably, the method requires no reward model, no ground-truth labels during search, and no hand-crafted scoring function. Results show that pairwise self-preferences provide strong optimization signal for test-time improvement over large, discrete output spaces.

Test-time scaling evaluates reasoning LLMs by sampling multiple outputs per prompt, but ranking models in this regime remains underexplored. We formalize dense benchmark ranking under test-time scaling and introduce Scorio, a library that implements statistical ranking methods such as paired-comparison models, item response theory (IRT) models, voting rules, and graph- and spectral-based methods. Across $20$ reasoning models on four Olympiad-style math benchmarks (AIME'24, AIME'25, HMMT'25, and BrUMO'25; up to $N=80$ trials), most full-trial rankings agree closely with the Bayesian gold standard $\mathrm{Bayes}_{\mathcal{U}}@80$ (mean Kendall's $τ_b = 0.93$--$0.95$), and $19$--$34$ methods recover exactly the same ordering. In the single-trial regime, the best methods reach $τ_b \approx 0.86$. Using greedy decoding as an empirical prior ($\mathrm{Bayes}_{\mathbf{R}_0}@N$) reduces variance at $N=1$ by $16$--$52\%$, but can bias rankings when greedy and stochastic sampling disagree. These results identify reliable ranking methods for both high- and low-budget test-time scaling. We release Scorio as an open-source library at https://github.com/mohsenhariri/scorio.