A game theory for foundation models shows new paths to rational cooperation through similarity inference
Authors: Alexander Meulemans, Maciej Wołczyk, Marissa A. Weis, Rajai Nasser, Roberta Rocca, Seijin Kobayashi, Guillaume Lajoie, Angelika Steger, +6 more
Organizations: Google, Paradigms of Intelligence Team · Mila - Quebec AI Institute · Université de Montréal · CIFAR · ETH Zürich · McGill University · Google DeepMind · Google · Santa Fe Institute
As autonomous agents powered by foundation models are increasingly integrated into social and economic systems, understanding the principles governing their collective behavior is essential for ensuring safety and cooperation. Classical game theory, the dominant framework for modeling rational interaction, is built upon the assumption of decoupled agency,' where agents treat their own decision-making as independent of the environment and other actors. Modern AI agents, however, jointly predict their own future actions alongside external observations. Here, we report a striking finding: when interacting in stylized social dilemmas, foundation model agents engaging in optimal planning consistently converge to stable cooperation, directly contradicting classical game-theoretic predictions of mutual defection. To understand this phenomenon, we introduce the embedded Bayesian agent,' a theoretical model for foundation model agents. By shifting from decoupled to embedded agency, these agents model themselves as part of the universe they inhabit, maintaining epistemic uncertainty about their own decision-making algorithms. We show that by inferring whether others are behaviorally similar, an embedded agent treats its own deliberation during planning as evidence: a decision to cooperate predicts a similar decision by a similar partner. We formalize this mechanism of similarity inference through the `embedded equilibrium,' a novel solution concept replacing the Nash equilibrium to provide a foundational game theory for the social behavior of modern AI agents.
A key challenge for the safety of advanced AI systems is the possibility that multiple simpler agents might inadvertently form a collective agent with capabilities and goals distinct from those of any individual. More generally, determining when a group of agents can be viewed as a unified collective agent is a foundational question in the study of interactions and incentives in both biological and artificial systems. We adopt a behavioral perspective in answering this question, ascribing collective agency to a group when viewing the group's joint actions as rational and goal-directed successfully predicts its behavior. We formalize this perspective on collective agency using causal games -- which are causal models of strategic, multi-agent interactions -- and causal abstraction -- which formalizes when a simple, high-level model faithfully captures a more complex, low-level model. We use this framework to solve a puzzle regarding multi-agent incentives in actor-critic models and to make quantitative assessments of the degree of collective agency exhibited by different voting mechanisms. Our framework aims to provide a foundation for theoretical and empirical work to understand, predict, and control emergent collective agents in multi-agent AI systems.
Frederik Hytting Jørgensen, Sebastian Weichwald, Lewis Hammond
It is increasingly important that LLM agents interact effectively and safely with other goal-pursuing agents, yet, recent works report the opposite trend: LLMs with stronger reasoning capabilities behave less cooperatively in mixed-motive games such as the prisoner's dilemma and public goods settings. Indeed, our experiments show that recent models -- with or without reasoning enabled -- consistently defect in single-shot social dilemmas. To tackle this safety concern, we present the first comparative study of game-theoretic mechanisms designed to enable cooperative outcomes between rational agents in equilibrium. Across four social dilemmas testing distinct components of robust cooperation, we evaluate four families of mechanisms: (1) repeating the game for many rounds, (2) reputation systems, (3) third-party mediators to delegate decision making to, and (4) contract agreements for outcome-conditional payments between players. Among our findings, we establish that contracting and mediation are most effective in achieving cooperative outcomes between capable LLM models, and that repetition-induced cooperation deteriorates drastically when co-players vary. Moreover, we demonstrate that the mechanisms become more effective under evolutionary pressures to maximize individual payoffs.
Emanuel Tewolde, Xiao Zhang, David Guzman Piedrahita +2
Collective intelligence emerges across biological, physical, and artificial systems without central coordination, yet a unifying principle governing such behaviour remains elusive. The Free Energy Principle explains how individual agents adapt through variational inference, while game theory formalises strategic interactions. Here we introduce the Game-Theoretic Free Energy Principle, a unified framework showing that multi-agent systems performing local free-energy minimisation implicitly implement a stochastic game. We prove that, under bounded rationality and local information constraints, stationary points of collective free energy correspond to approximate Nash equilibria of an induced game. Conversely, a broad class of cooperative games admits a variational representation in which equilibria arise as Gibbs distributions over coalitions, establishing a bridge between Bayesian inference and strategic interaction. To characterise higher-order effects, we introduce a free-energy formulation of the Harsanyi dividend, isolating irreducible multi-agent synergy. This yields a predictive theory of cooperation, including a falsifiable non-monotonic relationship between sensory precision and agent influence. We validate this prediction across neural, biological, and artificial multi-agent systems. These results identify a common variational principle underlying inference, thermodynamics, and game-theoretic equilibrium.