Work supported by MATS and SPAR. Code at https://github.com/ArjunPanickssery/math_problems_debate/.

Three measures for evaluating debate are

1. whether the debate judge outperforms a naive-judge baseline where the naive judge answers questions without hearing any debate arguments.
2. whether the debate judge outperforms a consultancy baseline where the judge hears argument(s) from a single "consultant" assigned to argue for a random answer.
3. whether the judge can continue to supervise the debaters as the debaters are optimized for persuasiveness. We can measure whether judge accuracy increases as the debaters vary in persuasiveness (measured with Elo ratings). This variation in persuasiveness can come from choosing different models, choosing the best of N sampled arguments for different values of N, or training debaters for persuasiveness (i.e. for winning debates) using RL.

Radhakrishan (Nov 2023),  Khan et al. (Feb 2024), and Kenton et al. (July 2024) study an information-gap setting where judges answer multiple-choice questions about science-fiction stories whose text they can't see, both with and without a debate/consultancy transcript that includes verified quotes from the debaters/consultant.

Past results from the QuALITY information-gap setting are seen above. Radhakrishnan (top row) finds no improvement to judge accuracy as debater Elo increases, while Khan et al. (middle row) and Kenton et al. (bottom row) do find a positive trend. Radhakrishnan varied models using RL while Khan et al. used best-of-N and critique-and-refinement optimizations. Kenton et al. vary the persuasiveness of debaters by using models with different capability levels. Both Khan et al. and Kenton et al. find that in terms of judge accuracy, debate > consultancy > naive judge for this setting.

In addition to the information-gap setting, consider a reasoning-gap setting where the debaters are distinguished from the judge not by their extra information but by their stronger ability to answer the questions and explain their reasoning. Kenton et al. run debates on questions from MMLU, TruthfulQA, PrOntoQA (logical reasoning), GQPA, and GSM8K (grade-school math). For the Elo-calculation experiments they use Gemini Pro 1.0 and Pro 1.5 judges with five debaters: Gemma7B, GPT-3.5, Gemini Pro 1.0, Gemini Pro 1.5 (all with best-of-N=1), and Gemini Pro 1.5 with best-of-N=4.

They find (top row) that debate slightly outperforms consultancy but outperforms the naive-judge baseline for only one of the four judges; they don't find that more persuasive debaters lead to higher judge accuracy. We get similar results (bottom row), specifically by

1. Generating 100 wrong answers and proofs to GSM8K questions to create binary-choice questions.
2. Computing the judge accuracy in naive, consultancy, and single-turn debate settings using four judges (Llama2-7B, Llama3-8B, GPT-3.5 Turbo, and GPT-4o) and seven debaters (Claude-3.5 Sonnet, Claude-3 Sonnet, GPT-3.5 Turbo, GPT-4o, Llama2-13B, Llama2-7B, and Llama3-8B).
3. Generating Elo scores from round-robin matchups between the seven models, using the same method as Kenton et al. 

We basically replicate the results. We find that

1. Debate doesn't consistently outperform the naive-judge baseline, and only slightly outperforms the consultancy baseline.
2. The positive relationship between debater persuasiveness and judge accuracy seen in the information-gap setting doesn't transfer to the reasoning-gap setting. (Results are shown below colored by debater rather than by judge). 

We also find some evidence of a self-preference bias (Panickssery et al., Apr 2024) where debaters have a higher Elo rating when judged by similar models. The GPT-3.5 Turbo judge sides most often with the GPT-3.5 Turbo debater while the Llama2-7B judge sides most often with the Llama2-13B debater, and the two GPT-based judges rank Llama2-7B last while the two Llama-based judges don't. Below is the rank order of the debaters' Elo ratings for each of the four judges.

Sorted ratings for llama2_7b (descending):
['llama2_13b', 'claude35_sonnet', 'llama2_7b', 'gpt35_turbo', 'claude3_sonnet', 'llama3_8b', 'gpt4o']

Sorted ratings for llama3_8b (descending):
['claude3_sonnet', 'gpt35_turbo', 'llama2_13b', 'claude35_sonnet', 'llama3_8b', 'llama2_7b', 'gpt4o']

Sorted ratings for gpt35_turbo (descending):
['gpt35_turbo', 'claude3_sonnet', 'llama3_8b', 'claude35_sonnet', 'llama2_13b', 'gpt4o', 'llama2_7b']

Sorted ratings for gpt4o (descending):
['gpt35_turbo', 'claude3_sonnet', 'llama3_8b', 'claude35_sonnet', 'gpt4o', 'llama2_13b', 'llama2_7b']

Appendix: Cross table of wins

Cells show the win rate of the row-debater over the column-debater.