LLM Agents as Static Level-k Players in Behavioural Games

Large Language Models (LLMs) are increasingly used as stand-ins in behavioural games. These stand-ins rely on the assumption that the LLM's distribution of choices meaningfully matches how humans play the same game. This study tests that assumption through two games. The first is a p-beauty contest, and the second one is a public goods game. The study first investigates five local-model settings within the same model family. These settings are varied together in a 360-cell factorial, which balances temperature, scale (0.5-32B), quantisation, instruct vs base, and framing. Each cell's distribution is then compared against whole choice distributions in published human data. Each deployment setting, except for quantisation, governs a different aspect of fidelity. Mechanically, while the dispersion of human players can be somewhat recovered through deployment settings, the strategic process behind it cannot. Through the lens of the level-k cognitive theory, we find that LLMs act as static, category-retrieved level-k players, where k is set by the model scale. The models also do not run within-game belief-updating or backward induction throughout multiple-round horizon settings. While human contributions decayed in the public goods game, LLMs stayed flat or rose at every scale. When the horizon test was administered, LLMs were more cooperative under an indefinite horizon compared to a finite one. However, LLMs ignore their relative round position, so no last-round defection was displayed. This implies that LLMs retrieved levels relative to the horizon category rather than working out iteratively from the specific game setting.