Nash CoT: Multi-Path Inference with Preference Equilibrium

TL;DR

Nash CoT introduces a game-theoretic mechanism to balance role-guided reasoning and general reasoning in multi-path Chain-of-Thought prompting. By formulating a two-player Preference Equilibrium on each path and enforcing a unique NE through KL regularization, it enables answer selection via NE-frequency voting while reducing the required number of reasoning paths. Empirical results across Arabic Reasoning, Symbolic Inference, and Commonsense QA show Nash CoT matching or surpassing self-consistency with the same or fewer paths and delivering substantial inference-time savings. The method emphasizes balancing role immersion and reasoning diversity through carefully designed templates and points to future work on automating template selection to enhance robustness and scalability.

Abstract

Chain of thought (CoT) is a reasoning framework that can enhance the performance of Large Language Models (LLMs) on complex inference tasks. In particular, among various studies related to CoT, multi-path inference stands out as a simple yet effective improvement. However, there is no optimal setting for the number of inference paths. Therefore, we have to increase the number of inference paths to obtain better results, which in turn increases the inference cost. To address this limitation, we can utilize question-related role templates to guide LLMs into relevant roles, thereby increasing the possibility of correct inferences for each path and further reducing dependence on the number of inference paths while improving reasoning accuracy. However, placing LLMs into specific roles may reduce their reasoning diversity and performance on a few tasks where role dependence is low. To alleviate the excessive immersion of the LLM into a specific role, we propose Nash CoT by constructing a game system on each path that balances the generation from role-specific LLMs' and the general LLMs' generation, thereby ensuring both effective role adoption and diversity in LLM generation further maintaining the performance of multi-path inference while reducing the requirement of the number of inference paths. We evaluate Nash CoT across various inference tasks, including Arabic Reasoning, Commonsense Question Answering, and Symbolic Inference, achieving results that are comparable to or better than those of multi-path CoT with the equal number of inference paths.

Figures (5)

• Figure 1: Demonstrations of Nash Chain-of-Thought (Nash CoT). As shown in this figure, Nash CoT can be divided into three main steps. Step 1 involves bringing the LLM into a template-related role. Step 2 utilizes the role-immersed LLM and LLM under normal conditions to collect model predictions separately. Step 3 filters the responses to ensure the existence of a unique Nash Equilibrium (NE).
• Figure 2: General Performance Comparison. We compare the average performance of, zero-shot, and zero-shot CoT self-consistency (20 Paths) with our Nash CoT (10 Paths) on Mistral-Instruct and GLM4.
• Figure 3: We used GLM4-chat (9B) on the same type of GPU (A100) to evaluate Nash CoT and self-consistency across selected tasks (60 questions per task). Nash CoT, employing a total of 10 paths, requires nearly half the time of self-consistency, which has 20 paths in total.
• Figure 4: We use Mistral-Instruct (7B) to examine the impact of loop numbers on the inference performance of the large language model. Specifically, we used solid lines of specific colors to represent the experimental performance under certain $N_{\rm outer}$ as the $N_{\rm mini}$ changed. We marked self-consistency with 20 paths using dashed lines, and some results of Nash CoT, with total paths close to 20, were marked with stars.
• Figure 5: Comparison of Nash CoT and self-consistency. (right) Nash Chain of thought (Nash CoT). (left) self-consistency.

Theorems & Definitions (2)

• Definition 3.1: Preference Equilibrium
• Theorem 3.1: Uniqueness of Preference Equilibrium