We study learnability of mixed-strategy Nash Equilibrium (NE) in general finite games using higher-order replicator dynamics as well as classes of higher-order uncoupled heterogeneous dynamics. In higher-order uncoupled learning dynamics, players have no access to utilities of opponents (uncoupled) but are allowed to use auxiliary states to further process information (higher-order). We establish a link between uncoupled learning and feedback stabilization with decentralized control. Using this association, we show that for any finite game with an isolated completely mixed-strategy NE, there exist higher-order uncoupled learning dynamics that lead (locally) to that NE. We further establish the lack of universality of learning dynamics by linking learning to the control theoretic concept of simultaneous stabilization. We construct two games such that any higher-order dynamics that learn the completely mixed-strategy NE of one of these games can never learn the completely mixed-strategy NE of the other. Next, motivated by imposing natural restrictions on allowable learning dynamics, we introduce the Asymptotic Best Response (ABR) property. Dynamics with the ABR property asymptotically learn a best response in environments that are asymptotically stationary. We show that the ABR property relates to an internal stability condition on higher-order learning dynamics. We provide conditions under which NE are compatible with the ABR property. Finally, we address learnability of mixed-strategy NE in the bandit setting using a bandit version of higher-order replicator dynamics.

To understand the complexity of the dynamic of learning in differential games, we decompose the game into components where the dynamic is well understood. One of the possible tools is Helmholtz's theorem, which can decompose a vector field into a potential and a harmonic component. This has been shown to be effective in finite and normal-form games. However, applying Helmholtz's theorem by connecting it with the Hodge theorem on $\mathbb{R}^n$ (which is the strategy space of differential game) is non-trivial due to the non-compactness of $\mathbb{R}^n$. Bridging the dynamic-strategic disconnect through Hodge/Helmoltz's theorem in differential games is then left as an open problem \cite{letcher2019differentiable}. In this work, we provide two decompositions of differential games to answer this question: the first as an exact scalar potential part, a near vector potential part, and a non-strategic part; the second as a near scalar potential part, an exact vector potential part, and a non-strategic part. We show that scalar potential games coincide with potential games proposed by \cite{monderer1996potential}, where the gradient descent dynamic can successfully find the Nash equilibrium. For the vector potential game, we show that the individual gradient field is divergence-free, in which case the gradient descent dynamic may either be divergent or recurrent.

In this paper, we propose a second-order extension of the continuous-time game-theoretic mirror descent (MD) dynamics, referred to as MD2, which provably converges to mere (but not necessarily strict) variationally stable states (VSS) without using common auxiliary techniques such as time-averaging or discounting. We show that MD2 enjoys no-regret as well as an exponential rate of convergence towards strong VSS upon a slight modification. MD2 can also be used to derive many novel continuous-time primal-space dynamics. We then use stochastic approximation techniques to provide a convergence guarantee of discrete-time MD2 with noisy observations towards interior mere VSS. Selected simulations are provided to illustrate our results.

We present a simulation-based approach for solution of mean field games (MFGs), using the framework of empirical game-theoretical analysis (EGTA). Our primary method employs a version of the double oracle, iteratively adding strategies based on best response to the equilibrium of the empirical MFG among strategies considered so far. We present Fictitious Play (FP) and Replicator Dynamics as two subroutines for computing the empirical game equilibrium. Each subroutine is implemented with a query-based method rather than maintaining an explicit payoff matrix as in typical EGTA methods due to a representation issue we highlight for MFGs. By introducing game model learning and regularization, we significantly improve the sample efficiency of the primary method without sacrificing the overall learning performance. Theoretically, we prove that a Nash equilibrium (NE) exists in the empirical MFG and show the convergence of iterative EGTA to NE of the full MFG with either subroutine. We test the performance of iterative EGTA in various games and show that it outperforms directly applying FP to MFGs in terms of iterations of strategy introduction.

Consider a weather forecaster who announces each day a probability p that there will be rain tomorrow. The forecaster is said to be calibrated if, for each forecast p that is used, the relative frequency of rainy days out of those days in which the forecast was p is equal to p in the long run. The surprising result of Foster and Vohra (1998) is that calibration can be guaranteed, no matter what the weather will be. There are various proofs of this result, and there is a large literature on calibration and its uses; see the survey of Olszewski (2015) and the more recent paper of Foster and Hart (2021). A simple proof of the existence of calibrated forecasts, based on the minimax theorem, was provided by the author in 1995.

In this paper, we introduce a class of learning dynamics for general quantum games, that we call "follow the quantum regularized leader" (FTQL), in reference to the classical "follow the regularized leader" (FTRL) template for learning in finite games. We show that the induced quantum state dynamics decompose into (i) a classical, commutative component which governs the dynamics of the system's eigenvalues in a way analogous to the evolution of mixed strategies under FTRL; and (ii) a non-commutative component for the system's eigenvectors which has no classical counterpart. Despite the complications that this non-classical component entails, we find that the FTQL dynamics incur no more than constant regret in all quantum games. Moreover, adjusting classical notions of stability to account for the nonlinear geometry of the state space of quantum games, we show that only pure quantum equilibria can be stable and attracting under FTQL while, as a partial converse, pure equilibria that satisfy a certain "variational stability" condition are always attracting. Finally, we show that the FTQL dynamics are Poincar\'e recurrent in quantum min-max games, extending in this way a very recent result for the quantum replicator dynamics.

Recent years have witnessed significant advances in technologies and services in modern network applications, including smart grid management, wireless communication, cybersecurity as well as multi-agent autonomous systems. Considering the heterogeneous nature of networked entities, emerging network applications call for game-theoretic models and learning-based approaches in order to create distributed network intelligence that responds to uncertainties and disruptions in a dynamic or an adversarial environment. This paper articulates the confluence of networks, games and learning, which establishes a theoretical underpinning for understanding multi-agent decision-making over networks. We provide an selective overview of game-theoretic learning algorithms within the framework of stochastic approximation theory, and associated applications in some representative contexts of modern network systems, such as the next generation wireless communication networks, the smart grid and distributed machine learning. In addition to existing research works on game-theoretic learning over networks, we highlight several new angles and research endeavors on learning in games that are related to recent developments in artificial intelligence. Some of the new angles extrapolate from our own research interests. The overall objective of the paper is to provide the reader a clear picture of the strengths and challenges of adopting game-theoretic learning methods within the context of network systems, and further to identify fruitful future research directions on both theoretical and applied studies.

Occasionally, I manage to be clever when I am not even trying to be clever, which isn't often. In a recent conversation about the new class of doomsday scenarios inspired by AlphaGo beating the Korean trash-talker Lee Sedol, I came up with the phrase human complete (HC) to characterize certain kinds of problems: the hardest problems of being human. An example of (what I hypothesize is) an HC problem is earning a living. I think human complete is a very clever phrase that people should use widely, and credit me for, since I can't find other references to it. I suspect there may be money in it. Here is a picture of the phrase that I will explain in a moment. In this post, I want to explore a particular bunny trail: the relationship between being human and the ability to solve infinite game problems in the sense of James Carse. I think this leads to an interesting perspective on the meaning and purpose of AI. The phrase human complete is constructed via analogy to the term AI complete, an ambiguously defined class of problems, including machine vision and natural language processing, that is supposed to contain the hardest problems in AI. That term itself is a reference to a much more precise one used in computer science: NP complete, which is a class of the hardest problems in computer science in a certain technical sense. NP complete is a subset of a larger class known as NP, which is the set of all problems for a certain class of non-God-level computers.