Equilibrium or Retrieval? Do LLM Agents Derive or Recall Game-Theoretic Equilibria, and What That Means for Mechanism Design

Large language models (LLMs) play textbook games well, but a model can reach the textbook answer in two ways: by deriving it from the rules, or by recognising a famous game and recalling its answer. We separate the two with isomorphic twins. Each question is asked in its standard wording and in disguised wordings with identical numbers, rules and information but no game vocabulary, and every answer is paired with a competence check in the same wording. In four pre-registered studies, four models from four developers answer 20,368 questions about a Vickrey auction, a Cournot duopoly, a finitely repeated prisoner's dilemma and a public goods game. The answer differs by model. Two models solve a Cournot game they no longer recognise and are not fooled by traps that make the textbook answer wrong. A third loses the equilibrium step by step as recognition falls, and gives the textbook answer about five times as often in its standard wording. A fourth loses it as soon as the game's familiar vocabulary goes. In the prisoner's dilemma, recognition retrieves a script of cooperation rather than the equilibrium, and logit quantal response equilibrium (QRE) fails pre-registered calibration for every model. We then derive mechanisms for QRE agents, including a closed-form cooperation subsidy, and prove a ceiling: below the subsidy's halfway point, no regular QRE produces cooperation above one half, at any precision. In a pre-registered test, the subsidy derived from fitted QRE reaches its target in every model but overpays by up to twenty times, and classical theory predicts the response better. The low fitted precision reflected cooperation that QRE cannot represent, not insensitivity to incentives. Pre-registrations (OSF): osf.io/2uqpj, osf.io/6hz8d, osf.io/cqnsy, osf.io/qg43y. Code, data and analysis plans: https://github.com/mmehta4/llm-mechanism-design