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K-Level Reasoning: Establishing Higher Order Beliefs in Large Language Models for Strategic Reasoning

Yadong Zhang, Shaoguang Mao, Tao Ge, Xun Wang, Yan Xia, Man Lan, Furu Wei

TL;DR

The paper introduces K-Level Reasoning with Large Language Models (K-R), a recursive framework that enables LLMs to form higher-order beliefs about other agents in dynamic multi-agent settings. By recursively simulating opponents at varying thinking levels and integrating environmental and historical information, K-R achieves deeper strategic reasoning than static prompting baselines. Empirical results across game theory tasks (G0.8A, SAG) and social intelligence benchmarks (NEG, SOTOPIA) show K-R improves rationality, prediction accuracy, and adaptability, with notable gains even for weaker base models and in open-source LLMs. The work provides theoretical justification via implicit Bayesian interpretation of in-context learning and establishes a foundation for future theory-of-m mind research in LLMs, while also noting computational costs and ethical considerations in real-world use.

Abstract

Strategic reasoning is a complex yet essential capability for intelligent agents. It requires Large Language Model (LLM) agents to adapt their strategies dynamically in multi-agent environments. Unlike static reasoning tasks, success in these contexts depends on anticipating other agents' beliefs and actions while continuously adjusting strategies to achieve individual goals. LLMs and LLM agents often struggle with strategic reasoning due to the absence of a reasoning framework that enables them to dynamically infer others' perspectives and adapt to changing environments. Inspired by the Level-K framework from game theory and behavioral economics, which extends reasoning from simple reactions to structured strategic depth, we propose a novel framework: "K-Level Reasoning with Large Language Models (K-R)." This framework employs recursive mechanisms to enable LLMs to achieve varying levels of strategic depth, allowing agents to form higher order beliefs - beliefs about others' beliefs. We validate this framework through rigorous testing on four testbeds: two classical game theory problems and two social intelligence tasks. The results demonstrate the advantages of K-R in strategic reasoning. Our work presents the first recursive implementation of strategic depth in large language models (LLMs). It establishes a foundation for future research into theory of mind and strategic reasoning in LLMs.

K-Level Reasoning: Establishing Higher Order Beliefs in Large Language Models for Strategic Reasoning

TL;DR

The paper introduces K-Level Reasoning with Large Language Models (K-R), a recursive framework that enables LLMs to form higher-order beliefs about other agents in dynamic multi-agent settings. By recursively simulating opponents at varying thinking levels and integrating environmental and historical information, K-R achieves deeper strategic reasoning than static prompting baselines. Empirical results across game theory tasks (G0.8A, SAG) and social intelligence benchmarks (NEG, SOTOPIA) show K-R improves rationality, prediction accuracy, and adaptability, with notable gains even for weaker base models and in open-source LLMs. The work provides theoretical justification via implicit Bayesian interpretation of in-context learning and establishes a foundation for future theory-of-m mind research in LLMs, while also noting computational costs and ethical considerations in real-world use.

Abstract

Strategic reasoning is a complex yet essential capability for intelligent agents. It requires Large Language Model (LLM) agents to adapt their strategies dynamically in multi-agent environments. Unlike static reasoning tasks, success in these contexts depends on anticipating other agents' beliefs and actions while continuously adjusting strategies to achieve individual goals. LLMs and LLM agents often struggle with strategic reasoning due to the absence of a reasoning framework that enables them to dynamically infer others' perspectives and adapt to changing environments. Inspired by the Level-K framework from game theory and behavioral economics, which extends reasoning from simple reactions to structured strategic depth, we propose a novel framework: "K-Level Reasoning with Large Language Models (K-R)." This framework employs recursive mechanisms to enable LLMs to achieve varying levels of strategic depth, allowing agents to form higher order beliefs - beliefs about others' beliefs. We validate this framework through rigorous testing on four testbeds: two classical game theory problems and two social intelligence tasks. The results demonstrate the advantages of K-R in strategic reasoning. Our work presents the first recursive implementation of strategic depth in large language models (LLMs). It establishes a foundation for future research into theory of mind and strategic reasoning in LLMs.
Paper Structure (40 sections, 13 equations, 21 figures, 14 tables, 1 algorithm)

This paper contains 40 sections, 13 equations, 21 figures, 14 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. K-Level Reasoning with Large Language Models
  3. Methodology
  4. Theoretical Analysis
  5. Experiments: Game Theory
  6. Task Definition and Metrics
  7. Guessing 0.8 of the Average (G0.8A)
  8. Survival Auction Game (SAG)
  9. Base Techniques
  10. Experimental Settings
  11. Results
  12. Experiments: Social Intelligence
  13. Task Definition and Metrics
  14. Negotiation (NEG)
  15. SOTOPIA Benchmark
  16. ...and 25 more sections

Figures (21)

  • Figure 1: Level-K Framework: In first-level thinking, agents respond directly to the environment. In second-level thinking, agents consider the first-level thinking of others. This process continues iteratively, with agents forming higher order beliefs based on assumptions about others' thoughts.
  • Figure 2: The illustration of three reasoning problems in dynamic, interactive environments in this paper. Left: Guessing 0.8 of the Average; Middle: Survival Auction Game; Right: Negotiation.
  • Figure 3: The deviation in prediction during the G0.8A between PCoT and K-Level Reasoning.
  • Figure 4: Prompts used in Guessing 0.8 of the Average game.
  • Figure 5: Prompts used in Survival Auction Game.
  • ...and 16 more figures