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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.

We introduce the use of generative adversarial learning to compute equilibria in general game-theoretic settings, specifically the generalized Nash equilibrium (GNE) in pseudo-games, and its specific instantiation as the competitive equilibrium (CE) in Arrow-Debreu competitive economies. Pseudo-games are a generalization of games in which players' actions affect not only the payoffs of other players but also their feasible action spaces. Although the computation of GNE and CE is intractable in the worst-case, i.e., PPAD-hard, in practice, many applications only require solutions with high accuracy in expectation over a distribution of problem instances. We introduce Generative Adversarial Equilibrium Solvers (GAES): a family of generative adversarial neural networks that can learn GNE and CE from only a sample of problem instances. We provide computational and sample complexity bounds, and apply the framework to finding Nash equilibria in normal-form games, CE in Arrow-Debreu competitive economies, and GNE in an environmental economic model of the Kyoto mechanism., Comment: 41 pages, 13 figures

Achieving convergence of multiple learning agents in general $N$-player games is imperative for the development of safe and reliable machine learning (ML) algorithms and their application to autonomous systems. Yet it is known that, outside the bounds of simple two-player games, convergence cannot be taken for granted. To make progress in resolving this problem, we study the dynamics of smooth Q-Learning, a popular reinforcement learning algorithm which quantifies the tendency for learning agents to explore their state space or exploit their payoffs. We show a sufficient condition on the rate of exploration such that the Q-Learning dynamics is guaranteed to converge to a unique equilibrium in any game. We connect this result to games for which Q-Learning is known to converge with arbitrary exploration rates, including weighted Potential games and weighted zero sum polymatrix games. Finally, we examine the performance of the Q-Learning dynamic as measured by the Time Averaged Social Welfare, and comparing this with the Social Welfare achieved by the equilibrium. We provide a sufficient condition whereby the Q-Learning dynamic will outperform the equilibrium even if the dynamics do not converge., Comment: Accepted in AAMAS 2023

Gamification is an emerging trend in the field of machine learning that presents a novel approach to solving optimization problems by transforming them into game-like scenarios. This paradigm shift allows for the development of robust, easily implementable, and parallelizable algorithms for hard optimization problems. In our work, we use gamification to tackle the Best Separable State (BSS) problem, a fundamental problem in quantum information theory that involves linear optimization over the set of separable quantum states. To achieve this we introduce and study quantum analogues of common-interest games (CIGs) and potential games where players have density matrices as strategies and their interests are perfectly aligned. We bridge the gap between optimization and game theory by establishing the equivalence between KKT (first-order stationary) points of a BSS instance and the Nash equilibria of its corresponding quantum CIG. Taking the perspective of learning in games, we introduce non-commutative extensions of the continuous-time replicator dynamics and the discrete-time Baum-Eagon/linear multiplicative weights update for learning in quantum CIGs, which also serve as decentralized algorithms for the BSS problem. We show that the common utility/objective value of a BSS instance is strictly increasing along trajectories of our algorithms, and finally corroborate our theoretical findings through extensive experiments.

Equilibrium solution concepts of normal-form games, such as Nash equilibria, correlated equilibria, and coarse correlated equilibria, describe the joint strategy profiles from which no player has incentive to unilaterally deviate. They are widely studied in game theory, economics, and multiagent systems. Equilibrium concepts are invariant under certain transforms of the payoffs. We define an equilibrium-inspired distance metric for the space of all normal-form games and uncover a distance-preserving equilibrium-invariant embedding. Furthermore, we propose an additional transform which defines a better-response-invariant distance metric and embedding. To demonstrate these metric spaces we study $2\times2$ games. The equilibrium-invariant embedding of $2\times2$ games has an efficient two variable parameterization (a reduction from eight), where each variable geometrically describes an angle on a unit circle. Interesting properties can be spatially inferred from the embedding, including: equilibrium support, cycles, competition, coordination, distances, best-responses, and symmetries. The best-response-invariant embedding of $2\times2$ games, after considering symmetries, rediscovers a set of 15 games, and their respective equivalence classes. We propose that this set of game classes is fundamental and captures all possible interesting strategic interactions in $2\times2$ games. We introduce a directed graph representation and name for each class. Finally, we leverage the tools developed for $2\times2$ games to develop game theoretic visualizations of large normal-form and extensive-form games that aim to fingerprint the strategic interactions that occur within., Comment: 42 pages

The designs of many large-scale systems today, from traffic routing environments to smart grids, rely on game-theoretic equilibrium concepts. However, as the size of an $N$-player game typically grows exponentially with $N$, standard game theoretic analysis becomes effectively infeasible beyond a low number of players. Recent approaches have gone around this limitation by instead considering Mean-Field games, an approximation of anonymous $N$-player games, where the number of players is infinite and the population's state distribution, instead of every individual player's state, is the object of interest. The practical computability of Mean-Field Nash equilibria, the most studied Mean-Field equilibrium to date, however, typically depends on beneficial non-generic structural properties such as monotonicity or contraction properties, which are required for known algorithms to converge. In this work, we provide an alternative route for studying Mean-Field games, by developing the concepts of Mean-Field correlated and coarse-correlated equilibria. We show that they can be efficiently learnt in \emph{all games}, without requiring any additional assumption on the structure of the game, using three classical algorithms. Furthermore, we establish correspondences between our notions and those already present in the literature, derive optimality bounds for the Mean-Field - $N$-player transition, and empirically demonstrate the convergence of these algorithms on simple games.

The study of learning in games has thus far focused primarily on normal form games. In contrast, our understanding of learning in extensive form games (EFGs) and particularly in EFGs with many agents lags far behind, despite them being closer in nature to many real world applications. We consider the natural class of Network Zero-Sum Extensive Form Games, which combines the global zero-sum property of agent payoffs, the efficient representation of graphical games as well the expressive power of EFGs. We examine the convergence properties of Optimistic Gradient Ascent (OGA) in these games. We prove that the time-average behavior of such online learning dynamics exhibits $O(1/T)$ rate convergence to the set of Nash Equilibria. Moreover, we show that the day-to-day behavior also converges to Nash with rate $O(c^{-t})$ for some game-dependent constant $c>0$., To appear in SAGT 2022

In this paper we study two-player bilinear zero-sum games with constrained strategy spaces. An instance of natural occurrences of such constraints is when mixed strategies are used, which correspond to a probability simplex constraint. We propose and analyze the alternating mirror descent algorithm, in which each player takes turns to take action following the mirror descent algorithm for constrained optimization. We interpret alternating mirror descent as an alternating discretization of a skew-gradient flow in the dual space, and use tools from convex optimization and modified energy function to establish an $O(K^{-2/3})$ bound on its average regret after $K$ iterations. This quantitatively verifies the algorithm's better behavior than the simultaneous version of mirror descent algorithm, which is known to diverge and yields an $O(K^{-1/2})$ average regret bound. In the special case of an unconstrained setting, our results recover the behavior of alternating gradient descent algorithm for zero-sum games which was studied in (Bailey et al., COLT 2020).

The recent framework of performative prediction is aimed at capturing settings where predictions influence the target/outcome they want to predict. In this paper, we introduce a natural multi-agent version of this framework, where multiple decision makers try to predict the same outcome. We showcase that such competition can result in interesting phenomena by proving the possibility of phase transitions from stability to instability and eventually chaos. Specifically, we present settings of multi-agent performative prediction where under sufficient conditions their dynamics lead to global stability and optimality. In the opposite direction, when the agents are not sufficiently cautious in their learning/updates rates, we show that instability and in fact formal chaos is possible. We complement our theoretical predictions with simulations showcasing the predictive power of our results.

We study the dynamics of simple congestion games with two resources where a continuum of agents behaves according to a version of Experience-Weighted Attraction (EWA) algorithm. The dynamics is characterized by two parameters: the (population) intensity of choice $a>0$ capturing the economic rationality of the total population of agents and a discount factor $\sigma\in [0,1]$ capturing a type of memory loss where past outcomes matter exponentially less than the recent ones. Finally, our system adds a third parameter $b \in (0,1)$, which captures the asymmetry of the cost functions of the two resources. It is the proportion of the agents using the first resource at Nash equilibrium, with $b=1/2$ capturing a symmetric network. Within this simple framework, we show a plethora of bifurcation phenomena where behavioral dynamics destabilize from global convergence to equilibrium, to limit cycles or even (formally proven) chaos as a function of the parameters $a$, $b$ and $\sigma$. Specifically, we show that for any discount factor $\sigma$ the system will be destabilized for a sufficiently large intensity of choice $a$. Although for discount factor $\sigma=0$ almost always (i.e., $b \neq 1/2$) the system will become chaotic, as $\sigma$ increases the chaotic regime will give place to the attracting periodic orbit of period 2. Therefore, memory loss can simplify game dynamics and make the system predictable. We complement our theoretical analysis with simulations and several bifurcation diagrams that showcase the unyielding complexity of the population dynamics (e.g., attracting periodic orbits of different lengths) even in the simplest possible potential games., Comment: 30 pages, 4 figures

Transaction fee markets are essential components of blockchain economies, as they resolve the inherent scarcity in the number of transactions that can be added to each block. In early blockchain protocols, this scarcity was resolved through a first-price auction in which users were forced to guess appropriate bids from recent blockchain data. Ethereum's EIP-1559 fee market reform streamlines this process through the use of a base fee that is increased (or decreased) whenever a block exceeds (or fails to meet) a specified target block size. Previous work has found that the EIP-1559 mechanism may lead to a base fee process that is inherently chaotic, in which case the base fee does not converge to a fixed point even under ideal conditions. However, the impact of this chaotic behavior on the fee market's main design goal -- blocks whose long-term average size equals the target -- has not previously been explored. As our main contribution, we derive near-optimal upper and lower bounds for the time-average block size in the EIP-1559 mechanism despite its possibly chaotic evolution. Our lower bound is equal to the target utilization level whereas our upper bound is approximately 6% higher than optimal. Empirical evidence is shown in great agreement with these theoretical predictions. Specifically, the historical average was approximately 2.9% larger than the target rage under Proof-of-Work and decreased to approximately 2.0% after Ethereum's transition to Proof-of-Stake. We also find that an approximate version of EIP-1559 achieves optimality even in the absence of convergence.

Recent advances in quantum computing and in particular, the introduction of quantum GANs, have led to increased interest in quantum zero-sum game theory, extending the scope of learning algorithms for classical games into the quantum realm. In this paper, we focus on learning in quantum zero-sum games under Matrix Multiplicative Weights Update (a generalization of the multiplicative weights update method) and its continuous analogue, Quantum Replicator Dynamics. When each player selects their state according to quantum replicator dynamics, we show that the system exhibits conservation laws in a quantum-information theoretic sense. Moreover, we show that the system exhibits Poincare recurrence, meaning that almost all orbits return arbitrarily close to their initial conditions infinitely often. Our analysis generalizes previous results in the case of classical games., NeurIPS 2022

Games are natural models for multi-agent machine learning settings, such as generative adversarial networks (GANs). The desirable outcomes from algorithmic interactions in these games are encoded as game theoretic equilibrium concepts, e.g. Nash and coarse correlated equilibria. As directly computing an equilibrium is typically impractical, one often aims to design learning algorithms that iteratively converge to equilibria. A growing body of negative results casts doubt on this goal, from non-convergence to chaotic and even arbitrary behaviour. In this paper we add a strong negative result to this list: learning in games is Turing complete. Specifically, we prove Turing completeness of the replicator dynamic on matrix games, one of the simplest possible settings. Our results imply the undecicability of reachability problems for learning algorithms in games, a special case of which is determining equilibrium convergence., Comment: 18 pages, 1 figure

Under what conditions do the behaviors of players, who play a game repeatedly, converge to a Nash equilibrium? If one assumes that the players' behavior is a discrete-time or continuous-time rule whereby the current mixed strategy profile is mapped to the next, this becomes a problem in the theory of dynamical systems. We apply this theory, and in particular the concepts of chain recurrence, attractors, and Conley index, to prove a general impossibility result: there exist games for which any dynamics is bound to have starting points that do not end up at a Nash equilibrium. We also prove a stronger result for $\epsilon$-approximate Nash equilibria: there are games such that no game dynamics can converge (in an appropriate sense) to $\epsilon$-Nash equilibria, and in fact the set of such games has positive measure. Further numerical results demonstrate that this holds for any $\epsilon$ between zero and $0.09$. Our results establish that, although the notions of Nash equilibria (and its computation-inspired approximations) are universally applicable in all games, they are also fundamentally incomplete as predictors of long term behavior, regardless of the choice of dynamics., Comment: 25 pages

Formalization of mathematics is the process of digitizing mathematical knowledge, which allows for formal proof verification as well as efficient semantic searches. Given the large and ever-increasing gap between the set of formalized and unformalized mathematical knowledge, there is a clear need to encourage more computer scientists and mathematicians to solve and formalize mathematical problems together. With blockchain technology, we are able to decentralize this process, provide time-stamped verification of authorship and encourage collaboration through implementation of incentive mechanisms via smart contracts. Currently, the formalization of mathematics is done through the use of proof assistants, which can be used to verify programs and protocols as well. Furthermore, with the advancement in artificial intelligence (AI), particularly machine learning, we can apply automated AI reasoning tools in these proof assistants and (at least partially) automate the process of synthesizing proofs. In our paper, we demonstrate a blockchain-based system for collaborative formalization of mathematics and programs incorporating both human labour as well as automated AI tools. We explain how Token-Curated Registries (TCR) and smart contracts are used to ensure appropriate documents are recorded and encourage collaboration through implementation of incentive mechanisms respectively. Using an illustrative example, we show how formalized proofs of different sorting algorithms can be produced collaboratively in our proposed blockchain system., Comment: This is an extended version of our accepted paper at The 4th IEEE International Conference on Blockchain (IEEE Blockchain-2021)

In this paper, we provide a novel and simple algorithm, Clairvoyant Multiplicative Weights Updates (CMWU) for regret minimization in general games. CMWU effectively corresponds to the standard MWU algorithm but where all agents, when updating their mixed strategies, use the payoff profiles based on tomorrow's behavior, i.e. the agents are clairvoyant. CMWU achieves constant regret of $\ln(m)/\eta$ in all normal-form games with m actions and fixed step-sizes $\eta$. Although CMWU encodes in its definition a fixed point computation, which in principle could result in dynamics that are neither computationally efficient nor uncoupled, we show that both of these issues can be largely circumvented. Specifically, as long as the step-size $\eta$ is upper bounded by $\frac{1}{(n-1)V}$, where $n$ is the number of agents and $[0,V]$ is the payoff range, then the CMWU updates can be computed linearly fast via a contraction map. This implementation results in an uncoupled online learning dynamic that admits a $O (\log T)$-sparse sub-sequence where each agent experiences at most $O(nV\log m)$ regret. This implies that the CMWU dynamics converge with rate $O(nV \log m \log T / T)$ to a \textit{Coarse Correlated Equilibrium}. The latter improves on the current state-of-the-art convergence rate of \textit{uncoupled online learning dynamics} \cite{daskalakis2021near,anagnostides2021near}., Comment: Expanded on the uncoupled online nature of the dynamics

Recent advances in multiagent learning have seen the introduction ofa family of algorithms that revolve around the population-based trainingmethod PSRO, showing convergence to Nash, correlated and coarse corre-lated equilibria. Notably, when the number of agents increases, learningbest-responses becomes exponentially more difficult, and as such ham-pers PSRO training methods. The paradigm of mean-field games pro-vides an asymptotic solution to this problem when the considered gamesare anonymous-symmetric. Unfortunately, the mean-field approximationintroduces non-linearities which prevent a straightforward adaptation ofPSRO. Building upon optimization and adversarial regret minimization,this paper sidesteps this issue and introduces mean-field PSRO, an adap-tation of PSRO which learns Nash, coarse correlated and correlated equi-libria in mean-field games. The key is to replace the exact distributioncomputation step by newly-defined mean-field no-adversarial-regret learn-ers, or by black-box optimization. We compare the asymptotic complexityof the approach to standard PSRO, greatly improve empirical bandit con-vergence speed by compressing temporal mixture weights, and ensure itis theoretically robust to payoff noise. Finally, we illustrate the speed andaccuracy of mean-field PSRO on several mean-field games, demonstratingconvergence to strong and weak equilibria., AAMAS

A seminal result in game theory is von Neumann's minmax theorem, which states that zero-sum games admit an essentially unique equilibrium solution. Classical learning results build on this theorem to show that online no-regret dynamics converge to an equilibrium in a time-average sense in zero-sum games. In the past several years, a key research direction has focused on characterizing the day-to-day behavior of such dynamics. General results in this direction show that broad classes of online learning dynamics are cyclic, and formally Poincar\'{e} recurrent, in zero-sum games. We analyze the robustness of these online learning behaviors in the case of periodic zero-sum games with a time-invariant equilibrium. This model generalizes the usual repeated game formulation while also being a realistic and natural model of a repeated competition between players that depends on exogenous environmental variations such as time-of-day effects, week-to-week trends, and seasonality. Interestingly, time-average convergence may fail even in the simplest such settings, in spite of the equilibrium being fixed. In contrast, using novel analysis methods, we show that Poincar\'{e} recurrence provably generalizes despite the complex, non-autonomous nature of these dynamical systems., Comment: To appear at NeurIPS 2021

We study agents competing against each other in a repeated network zero-sum game while applying the multiplicative weights update (MWU) algorithm with fixed learning rates. In our implementation, agents select their strategies probabilistically in each iteration and update their weights/strategies using the realized vector payoff of all strategies, i.e., stochastic MWU with full information. We show that the system results in an irreducible Markov chain where agent strategies diverge from the set of Nash equilibria. Further, we show that agents will play pure strategies with probability 1 in the limit.

We study the emergence of chaotic behavior of Follow-the-Regularized Leader (FoReL) dynamics in games. We focus on the effects of increasing the population size or the scale of costs in congestion games, and generalize recent results on unstable, chaotic behaviors in the Multiplicative Weights Update dynamics to a much larger class of FoReL dynamics. We establish that, even in simple linear non-atomic congestion games with two parallel links and any fixed learning rate, unless the game is fully symmetric, increasing the population size or the scale of costs causes learning dynamics to become unstable and eventually chaotic, in the sense of Li-Yorke and positive topological entropy. Furthermore, we show the existence of novel non-standard phenomena such as the coexistence of stable Nash equilibria and chaos in the same game. We also observe the simultaneous creation of a chaotic attractor as another chaotic attractor gets destroyed. Lastly, although FoReL dynamics can be strange and non-equilibrating, we prove that the time average still converges to an exact equilibrium for any choice of learning rate and any scale of costs., 30 pages, 8 figures

Regret has been established as a foundational concept in online learning, and likewise has important applications in the analysis of learning dynamics in games. Regret quantifies the difference between a learner's performance against a baseline in hindsight. It is well-known that regret-minimizing algorithms converge to certain classes of equilibria in games; however, traditional forms of regret used in game theory predominantly consider baselines that permit deviations to deterministic actions or strategies. In this paper, we revisit our understanding of regret from the perspective of deviations over partitions of the full \emph{mixed} strategy space (i.e., probability distributions over pure strategies), under the lens of the previously-established $��$-regret framework, which provides a continuum of stronger regret measures. Importantly, $��$-regret enables learning agents to consider deviations from and to mixed strategies, generalizing several existing notions of regret such as external, internal, and swap regret, and thus broadening the insights gained from regret-based analysis of learning algorithms. We prove here that the well-studied evolutionary learning algorithm of replicator dynamics (RD) seamlessly minimizes the strongest possible form of $��$-regret in generic $2 \times 2$ games, without any modification of the underlying algorithm itself. We subsequently conduct experiments validating our theoretical results in a suite of 144 $2 \times 2$ games wherein RD exhibits a diverse set of behaviors. We conclude by providing empirical evidence of $��$-regret minimization by RD in some larger games, hinting at further opportunity for $��$-regret based study of such algorithms from both a theoretical and empirical perspective.

A key challenge of evolutionary game theory and multi-agent learning is to characterize the limit behavior of game dynamics. Whereas convergence is often a property of learning algorithms in games satisfying a particular reward structure (e.g., zero-sum games), even basic learning models, such as the replicator dynamics, are not guaranteed to converge for general payoffs. Worse yet, chaotic behavior is possible even in rather simple games, such as variants of the Rock-Paper-Scissors game. Although chaotic behavior in learning dynamics can be precluded by the celebrated Poincar��-Bendixson theorem, it is only applicable to low-dimensional settings. Are there other characteristics of a game that can force regularity in the limit sets of learning? We show that behavior consistent with the Poincar��-Bendixson theorem (limit cycles, but no chaotic attractor) can follow purely from the topological structure of the interaction graph, even for high-dimensional settings with an arbitrary number of players and arbitrary payoff matrices. We prove our result for a wide class of follow-the-regularized leader (FoReL) dynamics, which generalize replicator dynamics, for binary games characterized interaction graphs where the payoffs of each player are only affected by one other player (i.e., interaction graphs of indegree one). Since chaos occurs already in games with only two players and three strategies, this class of non-chaotic games may be considered maximal. Moreover, we provide simple conditions under which such behavior translates into efficiency guarantees, implying that FoReL learning achieves time-averaged sum of payoffs at least as good as that of a Nash equilibrium, thereby connecting the topology of the dynamics to social-welfare analysis.

We present a novel control-theoretic understanding of online optimization and learning in games, via the notion of passivity. Passivity is a fundamental concept in control theory, which abstracts energy conservation and dissipation in physical systems. It has become a standard tool in analysis of general feedback systems, to which game dynamics belong. Our starting point is to show that all continuous-time Follow-the-Regularized-Leader (FTRL) dynamics, which include the well-known Replicator Dynamic, are lossless, i.e. it is passive with no energy dissipation. Interestingly, we prove that passivity implies bounded regret, connecting two fundamental primitives of control theory and online optimization. The observation of energy conservation in FTRL inspires us to present a family of lossless learning dynamics, each of which has an underlying energy function with a simple gradient structure. This family is closed under convex combination; as an immediate corollary, any convex combination of FTRL dynamics is lossless and thus has bounded regret. This allows us to extend the framework of Fox and Shamma [Games, 2013] to prove not just global asymptotic stability results for game dynamics, but Poincar�� recurrence results as well. Intuitively, when a lossless game (e.g. graphical constant-sum game) is coupled with lossless learning dynamics, their feedback interconnection is also lossless, which results in a pendulum-like energy-preserving recurrent behavior, generalizing the results of Piliouras and Shamma [SODA, 2014] and Mertikopoulos, Papadimitriou and Piliouras [SODA, 2018]., In ICML 2021

We study a game-theoretic model of blockchain mining economies and show that griefing, a practice according to which participants harm other participants at some lesser cost to themselves, is a prevalent threat at its Nash equilibria. The proof relies on a generalization of evolutionary stability to non-homogeneous populations via griefing factors (ratios that measure network losses relative to deviator's own losses) which leads to a formal theoretical argument for the dissipation of resources, consolidation of power and high entry barriers that are currently observed in practice. A critical assumption in this type of analysis is that miners' decisions have significant influence in aggregate network outcomes (such as network hashrate). However, as networks grow larger, the miner's interaction more closely resembles a distributed production economy or Fisher market and its stability properties change. In this case, we derive a proportional response (PR) update protocol which converges to market equilibria at which griefing is irrelevant. Convergence holds for a wide range of miners risk profiles and various degrees of resource mobility between blockchains with different mining technologies. Our empirical findings in a case study with four mineable cryptocurrencies suggest that risk diversification, restricted mobility of resources (as enforced by different mining technologies) and network growth, all are contributing factors to the stability of the inherently volatile blockchain ecosystem.

Many recent AI architectures are inspired by zero-sum games, however, the behavior of their dynamics is still not well understood. Inspired by this, we study standard gradient descent ascent (GDA) dynamics in a specific class of non-convex non-concave zero-sum games, that we call hidden zero-sum games. In this class, players control the inputs of smooth but possibly non-linear functions whose outputs are being applied as inputs to a convex-concave game. Unlike general zero-sum games, these games have a well-defined notion of solution; outcomes that implement the von-Neumann equilibrium of the "hidden" convex-concave game. We prove that if the hidden game is strictly convex-concave then vanilla GDA converges not merely to local Nash, but typically to the von-Neumann solution. If the game lacks strict convexity properties, GDA may fail to converge to any equilibrium, however, by applying standard regularization techniques we can prove convergence to a von-Neumann solution of a slightly perturbed zero-sum game. Our convergence guarantees are non-local, which as far as we know is a first-of-its-kind type of result in non-convex non-concave games. Finally, we discuss connections of our framework with generative adversarial networks.

Understanding the behavior of no-regret dynamics in general $N$-player games is a fundamental question in online learning and game theory. A folk result in the field states that, in finite games, the empirical frequency of play under no-regret learning converges to the game's set of coarse correlated equilibria. By contrast, our understanding of how the day-to-day behavior of the dynamics correlates to the game's Nash equilibria is much more limited, and only partial results are known for certain classes of games (such as zero-sum or congestion games). In this paper, we study the dynamics of "follow-the-regularized-leader" (FTRL), arguably the most well-studied class of no-regret dynamics, and we establish a sweeping negative result showing that the notion of mixed Nash equilibrium is antithetical to no-regret learning. Specifically, we show that any Nash equilibrium which is not strict (in that every player has a unique best response) cannot be stable and attracting under the dynamics of FTRL. This result has significant implications for predicting the outcome of a learning process as it shows unequivocally that only strict (and hence, pure) Nash equilibria can emerge as stable limit points thereof., 24 pages, 7 figures, 1 table

We present volume analyses of Multiplicative Weights Updates (MWU) and Optimistic Multiplicative Weights Updates (OMWU) in zero-sum as well as coordination games. Such analyses provide new insights into these game dynamical systems, which seem hard to achieve via the classical techniques within Computer Science and Machine Learning. The first step is to examine these dynamics not in their original space (simplex of actions) but in a dual space (aggregate payoff space of actions). The second step is to explore how the volume of a set of initial conditions evolves over time when it is pushed forward according to the algorithm. This is reminiscent of approaches in Evolutionary Game Theory where replicator dynamics, the continuous-time analogue of MWU, is known to always preserve volume in all games. Interestingly, when we examine discrete-time dynamics, both the choice of the game and the choice of the algorithm play a critical role. So whereas MWU expands volume in zero-sum games and is thus Lyapunov chaotic, we show that OMWU contracts volume, providing an alternative understanding for its known convergent behavior. However, we also prove a no-free-lunch type of theorem, in the sense that when examining coordination games the roles are reversed: OMWU expands volume exponentially fast, whereas MWU contracts. Using these tools, we prove two novel, rather negative properties of MWU in zero-sum games: (1) Extremism: even in games with unique fully mixed Nash equilibrium, the system recurrently gets stuck near pure-strategy profiles, despite them being clearly unstable from game theoretic perspective. (2) Unavoidability: given any set of good points (with your own interpretation of "good"), the system cannot avoid bad points indefinitely., 20 pages, 4 figures

With the success of modern machine learning, it is becoming increasingly important to understand and control how learning algorithms interact. Unfortunately, negative results from game theory show there is little hope of understanding or controlling general n-player games. We therefore introduce smooth markets (SM-games), a class of n-player games with pairwise zero sum interactions. SM-games codify a common design pattern in machine learning that includes (some) GANs, adversarial training, and other recent algorithms. We show that SM-games are amenable to analysis and optimization using first-order methods., 18 pages, 3 figures

In this paper we investigate the Follow the Regularized Leader dynamics in sequential imperfect information games (IIG). We generalize existing results of Poincar�� recurrence from normal-form games to zero-sum two-player imperfect information games and other sequential game settings. We then investigate how adapting the reward (by adding a regularization term) of the game can give strong convergence guarantees in monotone games. We continue by showing how this reward adaptation technique can be leveraged to build algorithms that converge exactly to the Nash equilibrium. Finally, we show how these insights can be directly used to build state-of-the-art model-free algorithms for zero-sum two-player Imperfect Information Games (IIG)., 43 pages

The predominant paradigm in evolutionary game theory and more generally online learning in games is based on a clear distinction between a population of dynamic agents that interact given a fixed, static game. In this paper, we move away from the artificial divide between dynamic agents and static games, to introduce and analyze a large class of competitive settings where both the agents and the games they play evolve strategically over time. We focus on arguably the most archetypal game-theoretic setting -- zero-sum games (as well as network generalizations) -- and the most studied evolutionary learning dynamic -- replicator, the continuous-time analogue of multiplicative weights. Populations of agents compete against each other in a zero-sum competition that itself evolves adversarially to the current population mixture. Remarkably, despite the chaotic coevolution of agents and games, we prove that the system exhibits a number of regularities. First, the system has conservation laws of an information-theoretic flavor that couple the behavior of all agents and games. Secondly, the system is Poincar\'{e} recurrent, with effectively all possible initializations of agents and games lying on recurrent orbits that come arbitrarily close to their initial conditions infinitely often. Thirdly, the time-average agent behavior and utility converge to the Nash equilibrium values of the time-average game. Finally, we provide a polynomial time algorithm to efficiently predict this time-average behavior for any such coevolving network game., Comment: To appear in AAAI 2021

Gradient descent is arguably one of the most popular online optimization methods with a wide array of applications. However, the standard implementation where agents simultaneously update their strategies yields several undesirable properties; strategies diverge away from equilibrium and regret grows over time. In this paper, we eliminate these negative properties by introducing a different implementation to obtain finite regret via arbitrary fixed step-size. We obtain this surprising property by having agents take turns when updating their strategies. In this setting, we show that an agent that uses gradient descent obtains bounded regret -- regardless of how their opponent updates their strategies. Furthermore, we show that in adversarial settings that agents' strategies are bounded and cycle when both are using the alternating gradient descent algorithm., 15 pages

We establish that algorithmic experiments in zero-sum games "fail miserably" to confirm the unique, sharp prediction of maxmin equilibration. Contradicting nearly a century of economic thought that treats zero-sum games nearly axiomatically as the exemplar symbol of economic stability, we prove that no meaningful prediction can be made about the day-to-day behavior of online learning dynamics in zero-sum games. Concretely, Multiplicative Weights Updates (MWU) with constant step-size is Lyapunov chaotic in the dual (payoff) space. Simply put, let's assume that an observer asks the agents playing Matching-Pennies whether they prefer Heads or Tails (and by how much in terms of aggregate payoff so far). The range of possible answers consistent with any arbitrary small set of initial conditions blows up exponentially with time everywhere in the payoff space. This result is robust both algorithmically as well as game theoretically: 1) Algorithmic robustness: Chaos is robust to agents using any of a general sub-family of Follow-the-Regularized-Leader (FTRL) algorithms, the well known regret-minimizing dynamics, even when agents mix-and-match dynamics, use different or slowly decreasing step-sizes. 2) Game theoretic robustness: Chaos is robust to all affine variants of zero-sum games (strictly competitive games), network variants with arbitrary large number of agents and even to competitive settings beyond these. Our result is in stark contrast with the time-average convergence of online learning to (approximate) Nash equilibrium, a result widely reported as "(weak) convergence to equilibrium"., 25 pages, 1 figure, accepted to COLT 2019

We show for the first time, to our knowledge, that it is possible to reconcile in online learning in zero-sum games two seemingly contradictory objectives: vanishing time-average regret and non-vanishing step sizes. This phenomenon, that we coin ``fast and furious" learning in games, sets a new benchmark about what is possible both in max-min optimization as well as in multi-agent systems. Our analysis does not depend on introducing a carefully tailored dynamic. Instead we focus on the most well studied online dynamic, gradient descent. Similarly, we focus on the simplest textbook class of games, two-agent two-strategy zero-sum games, such as Matching Pennies. Even for this simplest of benchmarks the best known bound for total regret, prior to our work, was the trivial one of $O(T)$, which is immediately applicable even to a non-learning agent. Based on a tight understanding of the geometry of the non-equilibrating trajectories in the dual space we prove a regret bound of $\Theta(\sqrt{T})$ matching the well known optimal bound for adaptive step sizes in the online setting. This guarantee holds for all fixed step-sizes without having to know the time horizon in advance and adapt the fixed step-size accordingly. As a corollary, we establish that even with fixed learning rates the time-average of mixed strategies, utilities converge to their exact Nash equilibrium values.

We introduce $��$-Rank, a principled evolutionary dynamics methodology for the evaluation and ranking of agents in large-scale multi-agent interactions, grounded in a novel dynamical game-theoretic solution concept called Markov-Conley chains (MCCs). The approach leverages continuous- and discrete-time evolutionary dynamical systems applied to empirical games, and scales tractably in the number of agents, the type of interactions, and the type of empirical games (symmetric and asymmetric). Current models are fundamentally limited in one or more of these dimensions and are not guaranteed to converge to the desired game-theoretic solution concept (typically the Nash equilibrium). $��$-Rank provides a ranking over the set of agents under evaluation and provides insights into their strengths, weaknesses, and long-term dynamics. This is a consequence of the links we establish to the MCC solution concept when the underlying evolutionary model's ranking-intensity parameter, $��$, is chosen to be large, which exactly forms the basis of $��$-Rank. In contrast to the Nash equilibrium, which is a static concept based on fixed points, MCCs are a dynamical solution concept based on the Markov chain formalism, Conley's Fundamental Theorem of Dynamical Systems, and the core ingredients of dynamical systems: fixed points, recurrent sets, periodic orbits, and limit cycles. $��$-Rank runs in polynomial time with respect to the total number of pure strategy profiles, whereas computing a Nash equilibrium for a general-sum game is known to be intractable. We introduce proofs that not only provide a unifying perspective of existing continuous- and discrete-time evolutionary evaluation models, but also reveal the formal underpinnings of the $��$-Rank methodology. We empirically validate the method in several domains including AlphaGo, AlphaZero, MuJoCo Soccer, and Poker.

Zero-sum games are natural, if informal, analogues of closed physical systems where no energy/utility can enter or exit. This analogy can be extended even further if we consider zero-sum network (polymatrix) games where multiple agents interact in a closed economy. Typically, (network) zero-sum games are studied from the perspective of Nash equilibria. Nevertheless, this comes in contrast with the way we typically think about closed physical systems, e.g., Earth-moon systems which move perpetually along recurrent trajectories of constant energy. We establish a formal and robust connection between multi-agent systems and Hamiltonian dynamics -- the same dynamics that describe conservative systems in physics. Specifically, we show that no matter the size, or network structure of such closed economies, even if agents use different online learning dynamics from the standard class of Follow-the-Regularized-Leader, they yield Hamiltonian dynamics. This approach generalizes the known connection to Hamiltonians for the special case of replicator dynamics in two agent zero-sum games developed by Hofbauer. Moreover, our results extend beyond zero-sum settings and provide a type of a Rosetta stone (see e.g. Table 1) that helps to translate results and techniques between online optimization, convex analysis, games theory, and physics.

Replicator dynamics, the continuous-time analogue of Multiplicative Weights Updates, is the main dynamic in evolutionary game theory. In simple evolutionary zero-sum games, such as Rock-Paper-Scissors, replicator dynamic is periodic \cite{zeeman1980population}, however, its behavior in higher dimensions is not well understood. We provide a complete characterization of its behavior in zero-sum evolutionary games. We prove that, if and only if, the system has an interior Nash equilibrium, the dynamics exhibit Poincar�� recurrence, i.e., almost all orbits come arbitrary close to their initial conditions infinitely often. If no interior equilibria exist, then all interior initial conditions converge to the boundary. Specifically, the strategies that are not in the support of any equilibrium vanish in the limit of all orbits. All recurrence results furthermore extend to a class of games that generalize both graphical polymatrix games as well as evolutionary games, establishing a unifying link between evolutionary and algorithmic game theory. We show that two degrees of freedom, as in Rock-Paper-Scissors, is sufficient to prove periodicity.

We study a wide class of non-convex non-concave min-max games that generalizes over standard bilinear zero-sum games. In this class, players control the inputs of a smooth function whose output is being applied to a bilinear zero-sum game. This class of games is motivated by the indirect nature of the competition in Generative Adversarial Networks, where players control the parameters of a neural network while the actual competition happens between the distributions that the generator and discriminator capture. We establish theoretically, that depending on the specific instance of the problem gradient-descent-ascent dynamics can exhibit a variety of behaviors antithetical to convergence to the game theoretically meaningful min-max solution. Specifically, different forms of recurrent behavior (including periodicity and Poincar�� recurrence) are possible as well as convergence to spurious (non-min-max) equilibria for a positive measure of initial conditions. At the technical level, our analysis combines tools from optimization theory, game theory and dynamical systems., To appear in NeurIPS 2019 (Spotlight talk)

Owing to their connection with generative adversarial networks (GANs), saddle-point problems have recently attracted considerable interest in machine learning and beyond. By necessity, most theoretical guarantees revolve around convex-concave (or even linear) problems; however, making theoretical inroads towards efficient GAN training depends crucially on moving beyond this classic framework. To make piecemeal progress along these lines, we analyze the behavior of mirror descent (MD) in a class of non-monotone problems whose solutions coincide with those of a naturally associated variational inequality - a property which we call coherence. We first show that ordinary, "vanilla" MD converges under a strict version of this condition, but not otherwise; in particular, it may fail to converge even in bilinear models with a unique solution. We then show that this deficiency is mitigated by optimism: by taking an "extra-gradient" step, optimistic mirror descent (OMD) converges in all coherent problems. Our analysis generalizes and extends the results of Daskalakis et al. (2018) for optimistic gradient descent (OGD) in bilinear problems, and makes concrete headway for establishing convergence beyond convex-concave games. We also provide stochastic analogues of these results, and we validate our analysis by numerical experiments in a wide array of GAN models (including Gaussian mixture models, as well as the CelebA and CIFAR-10 datasets)., 26 pages, 14 figures

Price of anarchy quantifies the degradation of social welfare in games due to the lack of a centralized authority that can enforce the optimal outcome. At its antipodes, mechanism design studies how to ameliorate these effects by incentivizing socially desirable behavior and implementing the optimal state as equilibrium. In practice, the responsiveness to such measures depends on the wealth of each individual. This leads to a natural, but largely unexplored, question. Does optimal mechanism design entrench, or maybe even exacerbate, social inequality? We study this question in nonatomic congestion games, arguably one of the most thoroughly studied settings from the perspectives of price of anarchy as well as mechanism design. We introduce a new model that incorporates the wealth distribution of the population and captures the income elasticity of travel time. This allows us to argue about the equality of wealth distribution both before and after employing a mechanism. We start our analysis by establishing a broad qualitative result, showing that tolls always increase inequality in symmetric congestion games under any reasonable metric of inequality, e.g., the Gini index. Next, we introduce the iniquity index, a novel measure for quantifying the magnitude of these forces towards a more unbalanced wealth distribution and show it has good normative properties (robustness to scaling of income, no-regret learning). We analyze iniquity both in theoretical settings (Pigou's network under various wealth distributions) as well as experimental ones (based on a large scale field experiment in Singapore). Finally, we provide an algorithm for computing optimal tolls for any point of the trade-off of relative importance of efficiency and equality. We conclude with a discussion of our findings in the context of theories of justice as developed in contemporary social sciences., Comment: 32 pages, 16 figures

Routing games are one of the most successful domains of application of game theory. It is well understood that simple dynamics converge to equilibria, whose performance is nearly optimal regardless of the size of the network or the number of agents. These strong theoretical assertions prompt a natural question: How well do these pen-and-paper calculations agree with the reality of everyday traffic routing? We focus on a semantically rich dataset from Singapore's National Science Experiment that captures detailed information about the daily behavior of thousands of Singaporean students. Using this dataset, we can identify the routes as well as the modes of transportation used by the students, e.g. car (driving or being driven to school) versus bus or metro, estimate source and sink destinations (home-school) and trip duration, as well as their mode-dependent available routes. We quantify both the system and individual optimality. Our estimate of the Empirical Price of Anarchy lies between 1.11 and 1.22. Individually, the typical behavior is consistent from day to day and nearly optimal, with low regret for not deviating to alternative paths., 19 pages, 7 figures

The Multiplicative Weights Update (MWU) method is a ubiquitous meta-algorithm that works as follows: A distribution is maintained on a certain set, and at each step the probability assigned to element $\gamma$ is multiplied by $(1 -\epsilon C(\gamma))>0$ where $C(\gamma)$ is the "cost" of element $\gamma$ and then rescaled to ensure that the new values form a distribution. We analyze MWU in congestion games where agents use \textit{arbitrary admissible constants} as learning rates $\epsilon$ and prove convergence to \textit{exact Nash equilibria}. Our proof leverages a novel connection between MWU and the Baum-Welch algorithm, the standard instantiation of the Expectation-Maximization (EM) algorithm for hidden Markov models (HMM). Interestingly, this convergence result does not carry over to the nearly homologous MWU variant where at each step the probability assigned to element $\gamma$ is multiplied by $(1 -\epsilon)^{C(\gamma)}$ even for the most innocuous case of two-agent, two-strategy load balancing games, where such dynamics can provably lead to limit cycles or even chaotic behavior., Comment: 17 pages, 9 figures

We analyze a stylized model of co-evolution between any two purely competing species (e.g., host and parasite), both sexually reproducing. Similarly to a recent model of Livnat \etal~\cite{evolfocs14} the fitness of an individual depends on whether the truth assignments on $n$ variables that reproduce through recombination satisfy a particular Boolean function. Whereas in the original model a satisfying assignment always confers a small evolutionary advantage, in our model the two species are in an evolutionary race with the parasite enjoying the advantage if the value of its Boolean function matches its host, and the host wishing to mismatch its parasite. Surprisingly, this model makes a simple and robust behavioral prediction. The typical system behavior is \textit{periodic}. These cycles stay bounded away from the boundary and thus, \textit{learning-dynamics competition between sexual species can provide an explanation for genetic diversity.} This explanation is due solely to the natural selection process. No mutations, environmental changes, etc., need be invoked. The game played at the gene level may have many Nash equilibria with widely diverse fitness levels. Nevertheless, sexual evolution leads to gene coordination that implements an optimal strategy, i.e., an optimal population mixture, at the species level. Namely, the play of the many "selfish genes" implements a time-averaged correlated equilibrium where the average fitness of each species is exactly equal to its value in the two species zero-sum competition. Our analysis combines tools from game theory, dynamical systems and Boolean functions to establish a novel class of conservative dynamical systems., Comment: Innovations in Theoretical Computer Science (ITCS), 2018

We develop a quasi-polynomial time Las Vegas algorithm for approximating Nash equilibria in polymatrix games over trees, under a mild renormalizing assumption. Our result, in particular, leads to an expected polynomial-time algorithm for computing approximate Nash equilibria of tree polymatrix games in which the number of actions per player is a fixed constant. Further, for trees with constant degree, the running time of the algorithm matches the best known upper bound for approximating Nash equilibria in bimatrix games (Lipton, Markakis, and Mehta 2003). Notably, this work closely complements the hardness result of Rubinstein (2015), which establishes the inapproximability of Nash equilibria in polymatrix games over constant-degree bipartite graphs with two actions per player., Comment: Appeared in the proceedings of the 8th International Symposium on Algorithmic Game Theory (SAGT), 2015. 11 pages

A key question in biological systems is whether genetic diversity persists in the long run under evolutionary competition or whether a single dominant genotype emerges. Classic work by Kalmus in 1945 has established that even in simple diploid species (species with two chromosomes) diversity can be guaranteed as long as the heterozygote individuals enjoy a selective advantage. Despite the classic nature of the problem, as we move towards increasingly polymorphic traits (e.g. human blood types) predicting diversity and understanding its implications is still not fully understood. Our key contribution is to establish complexity theoretic hardness results implying that even in the textbook case of single locus diploid models predicting whether diversity survives or not given its fitness landscape is algorithmically intractable. We complement our results by establishing that under randomly chosen fitness landscapes diversity survives with significant probability. Our results are structurally robust along several dimensions (e.g., choice of parameter distribution, different definitions of stability/persistence, restriction to typical subclasses of fitness landscapes). Technically, our results exploit connections between game theory, nonlinear dynamical systems, complexity theory and biology and establish hardness results for predicting the evolution of a deterministic variant of the well known multiplicative weights update algorithm in symmetric coordination games which could be of independent interest., Comment: 24 pages, 2 figues

Covering and packing problems can be modeled as games to encapsulate interesting social and engineering settings. These games have a high Price of Anarchy in their natural formulation. However, existing research applicable to specific instances of these games has only been able to prove fast convergence to arbitrary equilibria. This paper studies general classes of covering and packing games with learning dynamics models that incorporate a central authority who broadcasts weak, socially beneficial signals to agents that otherwise only use local information in their decision-making. Rather than illustrating convergence to an arbitrary equilibrium that may have very high social cost, we show that these systems quickly achieve near-optimal performance. In particular, we show that in the public service advertising model, reaching a small constant fraction of the agents is enough to bring the system to a state within a log n factor of optimal in a broad class of set cover and set packing games or a constant factor of optimal in the special cases of vertex cover and maximum independent set, circumventing social inefficiency of bad local equilibria that could arise without a central authority. We extend these results to the learn-then-decide model, in which agents use any of a broad class of learning algorithms to decide in a given round whether to behave according to locally optimal behavior or the behavior prescribed by the broadcast signal. The new techniques we use for analyzing these games could be of broader interest for analyzing more general classic optimization problems in a distributed fashion.

Routing games are amongst the most well studied domains of game theory. How relevant are these pen-and-paper calculations to understanding the reality of everyday traffic routing? We focus on a semantically rich dataset that captures detailed information about the daily behavior of thousands of Singaporean commuters and examine the following basic questions: (i) Does the traffic equilibrate? (ii) Is the system behavior consistent with latency minimizing agents? (iii) Is the resulting system efficient? In order to capture the efficiency of the traffic network in a way that agrees with our everyday intuition we introduce a new metric, the stress of catastrophe, which reflects the combined inefficiencies of both tragedy of the commons as well as price of anarchy effects., Comment: 22 pages, 9 figures, this article supersedes arXiv:1703.01599v2