Convex Markov Games: A Framework for Fairness, Imitation, and Creativity in Multi-Agent Learning

Expert imitation, behavioral diversity, and fairness preferences give rise to preferences in sequential decision making domains that do not decompose additively across time. We introduce the class of convex Markov games that allow general convex preferences over occupancy measures. Despite infinite time horizon and strictly higher generality than Markov games, pure strategy Nash equilibria exist under strict convexity. Furthermore, equilibria can be approximated efficiently by performing gradient descent on an upper bound of exploitability. Our experiments imitate human choices in ultimatum games, reveal novel solutions to the repeated prisoner's dilemma, and find fair solutions in a repeated asymmetric coordination game. In the prisoner's dilemma, our algorithm finds a policy profile that deviates from observed human play only slightly, yet achieves higher per-player utility while also being three orders of magnitude less exploitable.

Swim till You Sink: Computing the Limit of a Game

During 2023, two interesting results were proven about the limit behavior of game dynamics: First, it was shown that there is a game for which no dynamics converges to the Nash equilibria. Second, it was shown that the sink equilibria of a game adequately capture the limit behavior of natural game dynamics. These two results have created a need and opportunity to articulate a principled computational theory of the meaning of the game that is based on game dynamics. Given any game in normal form, and any prior distribution of play, we study the problem of computing the asymptotic behavior of a class of natural dynamics called the noisy replicator dynamics as a limit distribution over the sink equilibria of the game. When the prior distribution has pure strategy support, we prove this distribution can be computed efficiently, in near-linear time to the size of the best-response graph. When the distribution can be sampled -- for example, if it is the uniform distribution over all mixed strategy profiles -- we show through experiments that the limit distribution of reasonably large games can be estimated quite accurately through sampling and simulation.

States as Strings as Strategies: Steering Language Models with Game-Theoretic Solvers

Game theory is the study of mathematical models of strategic interactions among rational agents. Language is a key medium of interaction for humans, though it has historically proven difficult to model dialogue and its strategic motivations mathematically. A suitable model of the players, strategies, and payoffs associated with linguistic interactions (i.e., a binding to the conventional symbolic logic of game theory) would enable existing game-theoretic algorithms to provide strategic solutions in the space of language. In other words, a binding could provide a route to computing stable, rational conversational strategies in dialogue. Large language models (LLMs) have arguably reached a point where their generative capabilities can enable realistic, human-like simulations of natural dialogue. By prompting them in various ways, we can steer their responses towards different output utterances. Leveraging the expressivity of natural language, LLMs can also help us quickly generate new dialogue scenarios, which are grounded in real world applications. In this work, we present one possible binding from dialogue to game theory as well as generalizations of existing equilibrium finding algorithms to this setting. In addition, by exploiting LLMs generation capabilities along with our proposed binding, we can synthesize a large repository of formally-defined games in which one can study and test game-theoretic solution concepts. We also demonstrate how one can combine LLM-driven game generation, game-theoretic solvers, and imitation learning to construct a process for improving the strategic capabilities of LLMs.

Neural Population Learning beyond Symmetric Zero-sum Games

We study computationally efficient methods for finding equilibria in n-player general-sum games, specifically ones that afford complex visuomotor skills. We show how existing methods would struggle in this setting, either computationally or in theory. We then introduce NeuPL-JPSRO, a neural population learning algorithm that benefits from transfer learning of skills and converges to a Coarse Correlated Equilibrium (CCE) of the game. We show empirical convergence in a suite of OpenSpiel games, validated rigorously by exact game solvers. We then deploy NeuPL-JPSRO to complex domains, where our approach enables adaptive coordination in a MuJoCo control domain and skill transfer in capture-the-flag. Our work shows that equilibrium convergent population learning can be implemented at scale and in generality, paving the way towards solving real-world games between heterogeneous players with mixed motives.

Exploiting hidden structures in non-convex games for convergence to Nash equilibrium

A wide array of modern machine learning applications - from adversarial models to multi-agent reinforcement learning - can be formulated as non-cooperative games whose Nash equilibria represent the system's desired operational states. Despite having a highly non-convex loss landscape, many cases of interest possess a latent convex structure that could potentially be leveraged to yield convergence to equilibrium. Driven by this observation, our paper proposes a flexible first-order method that successfully exploits such "hidden structures" and achieves convergence under minimal assumptions for the transformation connecting the players' control variables to the game's latent, convex-structured layer. The proposed method - which we call preconditioned hidden gradient descent (PHGD) - hinges on a judiciously chosen gradient preconditioning scheme related to natural gradient methods. Importantly, we make no separability assumptions for the game's hidden structure, and we provide explicit convergence rate guarantees for both deterministic and stochastic environments.

Scalable AI Safety via Doubly-Efficient Debate

The emergence of pre-trained AI systems with powerful capabilities across a diverse and ever-increasing set of complex domains has raised a critical challenge for AI safety as tasks can become too complicated for humans to judge directly. Irving et al. [2018] proposed a debate method in this direction with the goal of pitting the power of such AI models against each other until the problem of identifying (mis)-alignment is broken down into a manageable subtask. While the promise of this approach is clear, the original framework was based on the assumption that the honest strategy is able to simulate deterministic AI systems for an exponential number of steps, limiting its applicability. In this paper, we show how to address these challenges by designing a new set of debate protocols where the honest strategy can always succeed using a simulation of a polynomial number of steps, whilst being able to verify the alignment of stochastic AI systems, even when the dishonest strategy is allowed to use exponentially many simulation steps.

A Quadratic Speedup in Finding Nash Equilibria of Quantum Zero-Sum Games

Recent developments in domains such as non-local games, quantum interactive proofs, and quantum generative adversarial networks have renewed interest in quantum game theory and, specifically, quantum zero-sum games. Central to classical game theory is the efficient algorithmic computation of Nash equilibria, which represent optimal strategies for both players. In 2008, Jain and Watrous proposed the first classical algorithm for computing equilibria in quantum zero-sum games using the Matrix Multiplicative Weight Updates (MMWU) method to achieve a convergence rate of $\mathcal{O}(d/\epsilon^2)$ iterations to $\epsilon$-Nash equilibria in the $4^d$-dimensional spectraplex. In this work, we propose a hierarchy of quantum optimization algorithms that generalize MMWU via an extra-gradient mechanism. Notably, within this proposed hierarchy, we introduce the Optimistic Matrix Multiplicative Weights Update (OMMWU) algorithm and establish its average-iterate convergence complexity as $\mathcal{O}(d/\epsilon)$ iterations to $\epsilon$-Nash equilibria. This quadratic speed-up relative to Jain and Watrous' original algorithm sets a new benchmark for computing $\epsilon$-Nash equilibria in quantum zero-sum games.

Stability of Multi-Agent Learning: Convergence in Network Games with Many Players

The behaviour of multi-agent learning in many player games has been shown to display complex dynamics outside of restrictive examples such as network zero-sum games. In addition, it has been shown that convergent behaviour is less likely to occur as the number of players increase. To make progress in resolving this problem, we study Q-Learning dynamics and determine a sufficient condition for the dynamics to converge to a unique equilibrium in any network game. We find that this condition depends on the nature of pairwise interactions and on the network structure, but is explicitly independent of the total number of agents in the game. We evaluate this result on a number of representative network games and show that, under suitable network conditions, stable learning dynamics can be achieved with an arbitrary number of agents.

Discovering How Agents Learn Using Few Data

Decentralized learning algorithms are an essential tool for designing multi-agent systems, as they enable agents to autonomously learn from their experience and past interactions. In this work, we propose a theoretical and algorithmic framework for real-time identification of the learning dynamics that govern agent behavior using a short burst of a single system trajectory. Our method identifies agent dynamics through polynomial regression, where we compensate for limited data by incorporating side-information constraints that capture fundamental assumptions or expectations about agent behavior. These constraints are enforced computationally using sum-of-squares optimization, leading to a hierarchy of increasingly better approximations of the true agent dynamics. Extensive experiments demonstrated that our approach, using only 5 samples from a short run of a single trajectory, accurately recovers the true dynamics across various benchmarks, including equilibrium selection and prediction of chaotic systems up to 10 Lyapunov times. These findings suggest that our approach has significant potential to support effective policy and decision-making in strategic multi-agent systems.

Multiplicative Updates for Online Convex Optimization over Symmetric Cones

We study online convex optimization where the possible actions are trace-one elements in a symmetric cone, generalizing the extensively-studied experts setup and its quantum counterpart. Symmetric cones provide a unifying framework for some of the most important optimization models, including linear, second-order cone, and semidefinite optimization. Using tools from the field of Euclidean Jordan Algebras, we introduce the Symmetric-Cone Multiplicative Weights Update (SCMWU), a projection-free algorithm for online optimization over the trace-one slice of an arbitrary symmetric cone. We show that SCMWU is equivalent to Follow-the-Regularized-Leader and Online Mirror Descent with symmetric-cone negative entropy as regularizer. Using this structural result we show that SCMWU is a no-regret algorithm, and verify our theoretical results with extensive experiments. Our results unify and generalize the analysis for the Multiplicative Weights Update method over the probability simplex and the Matrix Multiplicative Weights Update method over the set of density matrices.