  Large language models (LLMs) exhibit positional bias in how they use context,
which especially complicates listwise ranking. To address this, we propose
permutation self-consistency, a form of self-consistency over ranking list
outputs of black-box LLMs. Our key idea is to marginalize out different list
orders in the prompt to produce an order-independent ranking with less
positional bias. First, given some input prompt, we repeatedly shuffle the list
in the prompt and pass it through the LLM while holding the instructions the
same. Next, we aggregate the resulting sample of rankings by computing the
central ranking closest in distance to all of them, marginalizing out prompt
order biases in the process. Theoretically, we prove the robustness of our
method, showing convergence to the true ranking in the presence of random
perturbations. Empirically, on five list-ranking datasets in sorting and
passage reranking, our approach improves scores from conventional inference by
up to 7-18% for GPT-3.5 and 8-16% for LLaMA v2 (70B), surpassing the previous
state of the art in passage reranking. Our code is at
https://github.com/castorini/perm-sc.

Computation and Language;Machine Learning