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Mathematical reasoning has proven to be a critical yet challenging task for large language models (LLMs), as they often struggle with complex multi-step problems. To address these limitations, we introduce the Monte Carlo Nash Equilibrium Self-Refine Tree (MC-NEST) algorithm, an enhancement of the Monte Carlo Tree Self-Refine (MCTSr) approach. By integrating Nash Equilibrium strategies with LLM-based self-refinement and self-evaluation processes, MC-NEST aims to improve decision-making for complex mathematical reasoning tasks. This method ensures balanced exploration and exploitation of potential solutions, leveraging Upper Confidence Bound (UCT) scores and various selection policies. Through iterative critique and refinement, MC-NEST enhances the reasoning capabilities of LLMs, particularly for problems requiring strategic decision-making. Comparative analysis reveals that GPT-4o, equipped with MC-NEST using an Importance Sampling Policy, achieved superior accuracy in domains such as Number Theory and Geometry. These results suggest that both LLMs GPT-4o and Phi-3-mini can benefit from MC-NEST, with iterative self-refinement proving especially effective in expanding the reasoning capacity and problem-solving performance of LLMs. We evaluate the effectiveness of MC-NEST on challenging Olympiad-level benchmarks, demonstrating its potential to significantly boost complex mathematical reasoning performance in LLMs.

Multi-agent reinforcement learning (MARL) has achieved notable success in cooperative tasks, demonstrating impressive performance and scalability. However, deploying MARL agents in real-world applications presents critical safety challenges. Current safe MARL algorithms are largely based on the constrained Markov decision process (CMDP) framework, which enforces constraints only on discounted cumulative costs and lacks an all-time safety assurance. Moreover, these methods often overlook the feasibility issue (the system will inevitably violate state constraints within certain regions of the constraint set), resulting in either suboptimal performance or increased constraint violations. To address these challenges, we propose a novel theoretical framework for safe MARL with $\textit{state-wise}$ constraints, where safety requirements are enforced at every state the agents visit. To resolve the feasibility issue, we leverage a control-theoretic notion of the feasible region, the controlled invariant set (CIS), characterized by the safety value function. We develop a multi-agent method for identifying CISs, ensuring convergence to a Nash equilibrium on the safety value function. By incorporating CIS identification into the learning process, we introduce a multi-agent dual policy iteration algorithm that guarantees convergence to a generalized Nash equilibrium in state-wise constrained cooperative Markov games, achieving an optimal balance between feasibility and performance. Furthermore, for practical deployment in complex high-dimensional systems, we propose $\textit{Multi-Agent Dual Actor-Critic}$ (MADAC), a safe MARL algorithm that approximates the proposed iteration scheme within the deep RL paradigm. Empirical evaluations on safe MARL benchmarks demonstrate that MADAC consistently outperforms existing methods, delivering much higher rewards while reducing constraint violations.

We consider a class of Wasserstein distributionally robust Nash equilibrium problems, where agents construct heterogeneous data-driven Wasserstein ambiguity sets using private samples and radii, in line with their individual risk-averse behaviour. By leveraging relevant properties of this class of games, we show that equilibria of the original seemingly infinite-dimensional problem can be obtained as a solution to a finite-dimensional Nash equilibrium problem. We then reformulate the problem as a finite-dimensional variational inequality and establish the connection between the corresponding solution sets. Our reformulation has scalable behaviour with respect to the data size and maintains a fixed number of constraints, independently of the number of samples. To compute a solution, we leverage two algorithms, based on the golden ratio algorithm. The efficiency of both algorithmic schemes is corroborated through extensive simulation studies on an illustrative example and a stochastic portfolio allocation game, where behavioural coupling among investors is modeled., Comment: 14 pages, 5 figures

In this paper, we study the problem of the distributed Nash equilibrium seeking of N-player games over jointly strongly connected switching networks. The action of each player is governed by a class of uncertain nonlinear systems. Our approach integrates the consensus algorithm, the distributed estimator over jointly strongly connected switching networks, and some adaptive control technique. Furthermore, we also consider the disturbance rejection problem for bounded disturbances with unknown bounds. A special case of our results gives the solution of the distributed Nash equilibrium seeking for high-order integrator systems., Comment: 7 pages

We consider a repeatedly played generalized Nash equilibrium game. This induces a multi-agent online learning problem with joint constraints. An important challenge in this setting is that the feasible set for each agent depends on the simultaneous moves of the other agents and, therefore, varies over time. As a consequence, the agents face time-varying constraints, which are not adversarial but rather endogenous to the system. Prior work in this setting focused on convergence to a feasible solution in the limit via integrating the constraints in the objective as a penalty function. However, no existing work can guarantee that the constraints are satisfied for all iterations while simultaneously guaranteeing convergence to a generalized Nash equilibrium. This is a problem of fundamental theoretical interest and practical relevance. In this work, we introduce a new online feasible point method. Under the assumption that limited communication between the agents is allowed, this method guarantees feasibility. We identify the class of benign generalized Nash equilibrium problems, for which the convergence of our method to the equilibrium is guaranteed. We set this class of benign generalized Nash equilibrium games in context with existing definitions and illustrate our method with examples.

We prove input-to-state stability (ISS) of perturbed Newton methods for generalized equations arising from Nash equilibrium (NE) and generalized NE (GNE) problems. This ISS property allows the use of inexact computation in equilibrium-seeking to enable fast solution tracking in dynamic systems and plays a critical role in the analysis of interconnected plant-optimizer feedback systems such as model predictive control (MPC) with iterative optimization algorithms. For NE problems, we address the local convergence of perturbed Newton methods from the variational inequality (VI) stability analysis and point out that the ISS of the perturbed Josephy-Newton can be established under less restrictive stability/regularity conditions compared to the existing results established for nonlinear optimizations. Agent-distributed Josephy-Newton methods are then developed to enable agent-wise local computations. For GNE problems, since they cannot be reduced to VI problems in general, we use semismooth Newton methods to solve the semismooth equations arising from the Karush-Kuhn-Tucker (KKT) systems of the GNE problem and show that the ISS of the perturbed semismooth Newton methods can be established under a quasi-regularity condition. Agent-distributed updates are also developed for the GNE-seeking. To illustrate the use of the ISS in dynamic systems, applications to constrained game-theoretic MPC (CG-MPC) are studied. A real-time CG-MPC solver with both time- and agent- distributed computations is developed and is proven to have bounded solution tracking errors.

We introduce an Implicit Game-Theoretic MPC (IGT-MPC), a decentralized algorithm for two-agent motion planning that uses a learned value function that predicts the game-theoretic interaction outcomes as the terminal cost-to-go function in a model predictive control (MPC) framework, guiding agents to implicitly account for interactions with other agents and maximize their reward. This approach applies to competitive and cooperative multi-agent motion planning problems which we formulate as constrained dynamic games. Given a constrained dynamic game, we randomly sample initial conditions and solve for the generalized Nash equilibrium (GNE) to generate a dataset of GNE solutions, computing the reward outcome of each game-theoretic interaction from the GNE. The data is used to train a simple neural network to predict the reward outcome, which we use as the terminal cost-to-go function in an MPC scheme. We showcase emerging competitive and coordinated behaviors using IGT-MPC in scenarios such as two-vehicle head-to-head racing and un-signalized intersection navigation. IGT-MPC offers a novel method integrating machine learning and game-theoretic reasoning into model-based decentralized multi-agent motion planning., Comment: Submitted to 2025 Learning for Dynamics and Control Conference (L4DC)

The development of extremum seeking (ES) has progressed, over the past hundred years, from static maps, to finite-dimensional dynamic systems, to networks of static and dynamic agents. Extensions from ODE dynamics to maps and agents that incorporate delays or even partial differential equations (PDEs) is the next natural step in that progression through ascending research challenges. This paper reviews results on algorithm design and theory of ES for such infinite-dimensional systems. Both hyperbolic and parabolic dynamics are presented: delays or transport equations, heat-dominated equation, wave equations, and reaction-advection-diffusion equations. Nash equilibrium seeking (NES) methods are introduced for noncooperative game scenarios of the model-free kind and then specialized to single-agent optimization. Even heterogeneous PDE games, such as a duopoly with one parabolic and one hyperbolic agent, are considered. Several engineering applications are touched upon for illustration, including flow-traffic control for urban mobility, oil-drilling systems, deep-sea cable-actuated source seeking, additive manufacturing modeled by the Stefan PDE, biological reactors, light-source seeking with flexible-beam structures, and neuromuscular electrical stimulation., Comment: Preprint submitted to IEEE Control Systems Magazine (Special Issue: Into the Second Century of Extremum Seeking Control, 38 pages and 34 figures)

We consider generalized Nash equilibrium (GNE) problems in games with strongly monotone pseudo-gradients and jointly linear coupling constraints. We establish the convergence rate of a payoff-based approach intended to learn a variational GNE (v-GNE) in such games. While convergent algorithms have recently been proposed in this setting given full or partial information of the gradients, rate of convergence in the payoff-based information setting has been an open problem. Leveraging properties of a game extended from the original one by a dual player, we establish a convergence rate of $O(\frac{1}{t^{4/7}})$ to a v-GNE of the game.

This paper investigates a fully distributed adaptive Nash equilibrium (NE) seeking algorithm for constrained noncooperative games with prescribed-time stability. On the one hand, prescribed-time stability for the proposed NE seeking algorithm is obtained by using an adaptive penalty technique, a time-varying control gain and a cosine-related time conversion function, which extends the prior asymptotic stability result. On the other hand, uncoordinated integral adaptive gains are incorporated in order to achieve the fully distribution of the algorithm. Finally, the theoretical result is validated through a numerical simulation based on a standard power market scenario.

Recent advancements in artificial intelligence (AI) have leveraged large-scale games as benchmarks to gauge progress, with AI now frequently outperforming human capabilities. Traditionally, this success has largely relied on solving Nash equilibrium (NE) using variations of the counterfactual regret minimization (CFR) method in games with incomplete information. However, the variety of Nash equilibria has been largely overlooked in previous research, limiting the adaptability of AI to meet diverse human preferences. To address this challenge, where AI is powerful but struggles to meet customization needs, we introduce a novel approach: Preference-CFR, which incorporates two new parameters: preference degree and vulnerability degree. These parameters allow for greater flexibility in AI strategy development without compromising convergence. Our method significantly alters the distribution of final strategies, enabling the creation of customized AI models that better align with individual user needs. Using Texas Hold'em as a case study, our experiments demonstrate how Preference CFR can be adjusted to either emphasize customization, prioritizing user preferences, or to enhance performance, striking a balance between the depth of customization and strategic optimality.

Inspired by the recent work by Shapiro et al. [45], we propose a Bayesian distributionally robust Nash equilibrium (BDRNE) model where each player lacks complete information on the true probability distribution of the underlying uncertainty represented by a random variable and subsequently determines the optimal decision by solving a Bayesian distributionally robust optimization (BDRO) problem under the Nash conjecture. Unlike most of the DRO models in the literature, the BDRO model assumes (a) the true unknown distribution of the random variable can be approximated by a randomized parametric family of distributions, (b) the average of the worst-case expected value of the objective function with respect to the posterior distribution of the parameter, instead of the worst-case expected value of the objective function is considered in each player's decision making, and (c) the posterior distribution of the parameter is updated as more and more sampling information of the random variable is gathered. Under some moderate conditions, we demonstrate the existence of a BDRNE and derive asymptotic convergence of the equilibrium as the sample size increases. Moreover, we propose to solve the BDRNE problem by Gauss-Seidel-type iterative method in the case when the ambiguity set of each player is constructed via Kullback-Leibler (KL) divergence. Finally, we apply the BDRNE model to a price competition problem under multinomial logit demand. The preliminary numerical test results show that the proposed model and computational scheme perform well.

This paper presents a pioneering investigation into discrete-time two-person nonzero-sum linear quadratic (LQ) stochastic games characterized by random coefficients. We derive necessary and sufficient conditions for the existence of open-loop Nash equilibria using convex variational calculus. To obtain explicit expressions for the Nash equilibria, we introduce fully coupled forward-backward stochastic difference equations (stochastic Hamiltonian systems), which provide a dual characterization of these Nash equilibria. Additionally, we develop non-symmetric stochastic Riccati equations that decouple the stochastic Hamiltonian system for each player, enabling the derivation of closed-loop feedback forms for open-loop Nash equilibrium strategies. A notable aspect of this research is the complete randomness of the coefficients, which results in the corresponding Riccati equations becoming fully nonlinear higher-order backward stochastic difference equations. It distinguishes our nonzero-sum difference game from the deterministic case, where the Riccati equations reduce to algebraic forms.

The vaccination against ongoing epidemics is seldom compulsory but remains one of the most classical means to fight epidemic propagation. However recent debates concerning the innocuity of vaccines and their risk with respect to the risk of the epidemic itself lead to severe vaccination campaign failures and new mass behaviors appeared driven by individual self-interest. Prompted by this context we analyze, in a Susceptible-Infected-Recovered (SIR) model, whether egocentric individuals can reach an equilibrium with the rest of the society. Using techniques from the "Mean Field Games" theory, we extend previous results and show that an equilibrium exists and characterizes completely the individual best vaccination strategy (with or without discounting). We also compare with a strategy based only on overall societal optimization and exhibit a situation with non-negative price of anarchy. Finally, we apply the theory to the 2009-2010 Influenza A (H1N1) vaccination campaign in France and hint that a group of individuals stopped vaccinating at levels that indicated a pessimistic perception of the risk of the vaccine.

The equilibrium selection problem in the generalized Nash equilibrium problem (GNEP) has recently been studied as an optimization problem, defined over the set of all variational equilibria achievable first through a non-cooperative game among players. However, to make such a selection fairly for all players, we have to rely on an unrealistic assumption, that is, the availability of a reliable center not possible to cause any bias for all players. In this paper, we propose a new equilibrium selection achievable by solving a further GNEP, named the hierarchical Nash equilibrium problem (HNEP), within only the players. The HNEP covers existing optimization-based equilibrium selections as its simplest cases, while the general style of the HNEP can ensure a fair equilibrium selection without assuming any trusted center or randomness. We also propose an iterative algorithm for the HNEP as an application of the hybrid steepest descent method to a variational inequality newly defined over the fixed point set of a quasi-nonexpansive operator. Numerical experiments show the effectiveness of the proposed equilibrium selection via the HNEP.

One key in real-life Nash equilibrium applications is to calibrate players' cost functions. To leverage the approximation ability of neural networks, we proposed a general framework for optimizing and learning Nash equilibrium using neural networks to estimate players' cost functions. Depending on the availability of data, we propose two approaches (a) the two-stage approach: we need the data pair of players' strategy and relevant function value to first learn the players' cost functions by monotonic neural networks or graph neural networks, and then solve the Nash equilibrium with the learned neural networks; (b) the joint approach: we use the data of partial true observation of the equilibrium and contextual information (e.g., weather) to optimize and learn Nash equilibrium simultaneously. The problem is formulated as an optimization problem with equilibrium constraints and solved using a modified Backpropagation Algorithm. The proposed methods are validated in numerical experiments., Comment: This is an incomplete draft, we need to make more modifications

In this paper, we first provide a simple variational proof of the existence of Nash equilibrium in Hilbert spaces by using optimality conditions in convex minimization and Schauder's fixed-point theorem. Then applications of convex analysis and generalized differentiation are given to the existence of Nash equilibrium and extended versions of von Neumann's minimax theorem in locally convex topological vector spaces. Our analysis in this part combines generalized differential tools of convex analysis with elements of fixed point theory revolving around Kakutani's fixed-point theorem and related issues.

Learning in zero-sum games studies a situation where multiple agents competitively learn their strategy. In such multi-agent learning, we often see that the strategies cycle around their optimum, i.e., Nash equilibrium. When a game periodically varies (called a ``periodic'' game), however, the Nash equilibrium moves generically. How learning dynamics behave in such periodic games is of interest but still unclear. Interestingly, we discover that the behavior is highly dependent on the relationship between the two speeds at which the game changes and at which players learn. We observe that when these two speeds synchronize, the learning dynamics diverge, and their time-average does not converge. Otherwise, the learning dynamics draw complicated cycles, but their time-average converges. Under some assumptions introduced for the dynamical systems analysis, we prove that this behavior occurs. Furthermore, our experiments observe this behavior even if removing these assumptions. This study discovers a novel phenomenon, i.e., synchronization, and gains insight widely applicable to learning in periodic games., Comment: 8 pages, 5 figures (main); 7 pages, 1 figure (appendix)

Games with incomplete preferences are an important model for studying rational decision-making in scenarios where players face incomplete information about their preferences and must contend with incomparable outcomes. We study the problem of computing Nash equilibrium in a subclass of two-player games played on graphs where each player seeks to maximally satisfy their (possibly incomplete) preferences over a set of temporal goals. We characterize the Nash equilibrium and prove its existence in scenarios where player preferences are fully aligned, partially aligned, and completely opposite, in terms of the well-known solution concepts of sure winning and Pareto efficiency. When preferences are partially aligned, we derive conditions under which a player needs cooperation and demonstrate that the Nash equilibria depend not only on the preference alignment but also on whether the players need cooperation to achieve a better outcome and whether they are willing to cooperate.We illustrate the theoretical results by solving a mechanism design problem for a drone delivery scenario., Comment: 14 page, 6 figure, under development

We use system-theoretic passivity methods to study evolutionary Nash equilibria learning in large populations of agents engaged in strategic, non-cooperative interactions. The agents follow learning rules (rules for short) that capture their strategic preferences and a payoff mechanism ascribes payoffs to the available strategies. The population's aggregate strategic profile is the state of an associated evolutionary dynamical system. Evolutionary Nash equilibrium learning refers to the convergence of this state to the Nash equilibria set of the payoff mechanism. Most approaches consider memoryless payoff mechanisms, such as potential games. Recently, methods using $\delta$-passivity and equilibrium independent passivity (EIP) have introduced dynamic payoff mechanisms. However, $\delta$-passivity does not hold when agents follow rules exhibiting ``imitation" behavior, such as in replicator dynamics. Conversely, EIP applies to the replicator dynamics but not to $\delta$-passive rules. We address this gap using counterclockwise dissipativity (CCW). First, we prove that continuous memoryless payoff mechanisms are CCW if and only if they are potential games. Subsequently, under (possibly dynamic) CCW payoff mechanisms, we establish evolutionary Nash equilibrium learning for any rule within a convex cone spanned by imitation rules and continuous $\delta$-passive rules., Comment: 8 pages, 2 figures

We study the perfect information Nash equilibrium between a broker and her clients -- an informed trader and an uniformed trader. In our model, the broker trades in the lit exchange where trades have instantaneous and transient price impact with exponential resilience, while both clients trade with the broker. The informed trader and the broker maximise expected wealth subject to inventory penalties, while the uninformed trader is not strategic and sends the broker random buy and sell orders. We characterise the Nash equilibrium of the trading strategies with the solution to a coupled system of forward-backward stochastic differential equations (FBSDEs). We solve this system explicitly and study the effect of information, profitability, and inventory control in the trading strategies of the broker and the informed trader., Comment: 24 pages, 3 figures

In socio-technical multi-agent systems, deception exploits privileged information to induce false beliefs in "victims," keeping them oblivious and leading to outcomes detrimental to them or advantageous to the deceiver. We consider model-free Nash-equilibrium-seeking for non-cooperative games with asymmetric information and introduce model-free deceptive algorithms with stability guarantees. In the simplest algorithm, the deceiver includes in his action policy the victim's exploration signal, with an amplitude tuned by an integrator of the regulation error between the deceiver's actual and desired payoff. The integral feedback drives the deceiver's payoff to the payoff's reference value, while the victim is led to adopt a suboptimal action, at which the pseudogradient of the deceiver's payoff is zero. The deceiver's and victim's actions turn out to constitute a "deceptive" Nash equilibrium of a different game, whose structure is managed - in real time - by the deceiver. We examine quadratic, aggregative, and more general games and provide conditions for a successful deception, mutual and benevolent deception, and immunity to deception. Stability results are established using techniques based on averaging and singular perturbations. Among the examples in the paper is a microeconomic duopoly in which the deceiver induces in the victim a belief that the buyers disfavor the deceiver more than they actually do, leading the victim to increase the price above the Nash price, and resulting in an increased profit for the deceiver and a decreased profit for the victim. A study of the deceiver's integral feedback for the desired profit reveals that, in duopolies with equal marginal costs, a deceiver that is greedy for very high profit can attain any such profit, and pursue this with arbitrarily high integral gain (impatiently), irrespective of the market preference for the victim.

This paper mainly investigates a class of distributed generalized Nash equilibrium (GNE) seeking problems for online nonmonotone game with time-varying coupling inequality constraints. Based on a time-varying control gain, a novel continuous-time distributed GNE seeking algorithm is proposed, which realizes the constant regret bound and sublinear fit bound, matching those of the criteria for online optimization problems. Furthermore, to reduce unnecessary communication among players, a dynamic event-triggered mechanism involving internal variables is introduced into the distributed GNE seeking algorithm, while the constant regret bound and sublinear fit bound are still achieved. Also, the Zeno behavior is strictly prohibited. Finally, a numerical example is given to demonstrate the validity of the theoretical results.

This paper introduces a new integer programming game (IPG) named the Edge-weighted Budgeted Maximum Coverage (EBMC) game and proposes a new algorithm, the Best Response Plus (BR-plus) algorithm, for finding the best Pure Nash Equilibrium (PNE). We demonstrate this methodology by optimizing county-level decisions to prevent aquatic invasive species (AIS) in Minnesota lakes, where each county-level decision makers has self-serving objectives while AIS is an interconnected issue that crosses county borders. Specifically, we develop EBMC games to model the strategic interactions among county-level decision-makers with two variations in utility functions. We also study and prove the existence of a PNE in these models under specified conditions. We advance the current state-of-the-art, which is limited to only a few players, by presenting the BR-plus algorithm that can handle a large set of players via utilizing the best response dynamics for finding PNE in normal-form games. Experimental results show that our BR-plus algorithm offers computational advantages over the ZR algorithm, especially in larger games, on both random and real-world networks.

In the domain of machine learning and game theory, the quest for Nash Equilibrium (NE) in extensive-form games with incomplete information is challenging yet crucial for enhancing AI's decision-making support under varied scenarios. Traditional Counterfactual Regret Minimization (CFR) techniques excel in navigating towards NE, focusing on scenarios where opponents deploy optimal strategies. However, the essence of machine learning in strategic game play extends beyond reacting to optimal moves; it encompasses aiding human decision-making in all circumstances. This includes not only crafting responses to optimal strategies but also recovering from suboptimal decisions and capitalizing on opponents' errors. Herein lies the significance of transitioning from NE to Bayesian Perfect Equilibrium (BPE), which accounts for every possible condition, including the irrationality of opponents. To bridge this gap, we propose Belief Update Fictitious Play (BUFP), which innovatively blends fictitious play with belief to target BPE, a more comprehensive equilibrium concept than NE. Specifically, through adjusting iteration stepsizes, BUFP allows for strategic convergence to both NE and BPE. For instance, in our experiments, BUFP(EF) leverages the stepsize of Extensive Form Fictitious Play (EFFP) to achieve BPE, outperforming traditional CFR by securing a 48.53\% increase in benefits in scenarios characterized by dominated strategies.

The use of reinforcement learning algorithms in financial trading is becoming increasingly prevalent. However, the autonomous nature of these algorithms can lead to unexpected outcomes that deviate from traditional game-theoretical predictions and may even destabilize markets. In this study, we examine a scenario in which two autonomous agents, modeled with Double Deep Q-Learning, learn to liquidate the same asset optimally in the presence of market impact, using the Almgren-Chriss (2000) framework. Our results show that the strategies learned by the agents deviate significantly from the Nash equilibrium of the corresponding market impact game. Notably, the learned strategies exhibit tacit collusion, closely aligning with the Pareto-optimal solution. We further explore how different levels of market volatility influence the agents' performance and the equilibria they discover, including scenarios where volatility differs between the training and testing phases.

The concept of Nash equilibrium (NE), pivotal within game theory, has garnered widespread attention across numerous industries. Recent advancements introduced several quantum Nash solvers aimed at identifying pure strategy NE solutions (i.e., binary solutions) by integrating slack terms into the objective function, commonly referred to as slack-quadratic unconstrained binary optimization (S-QUBO). However, incorporation of slack terms into the quadratic optimization results in changes of the objective function, which may cause incorrect solutions. Furthermore, these quantum solvers only identify a limited subset of pure strategy NE solutions, and fail to address mixed strategy NE (i.e., decimal solutions), leaving many solutions undiscovered. In this work, we propose C-Nash, a novel ferroelectric computing-in-memory (CiM) architecture that can efficiently handle both pure and mixed strategy NE solutions. The proposed architecture consists of (i) a transformation method that converts quadratic optimization into a MAX-QUBO form without introducing additional slack variables, thereby avoiding objective function changes; (ii) a ferroelectric FET (FeFET) based bi-crossbar structure for storing payoff matrices and accelerating the core vector-matrix-vector (VMV) multiplications of QUBO form; (iii) A winner-takes-all (WTA) tree implementing the MAX form and a two-phase based simulated annealing (SA) logic for searching NE solutions. Evaluations show that C-Nash has up to 68.6% increase in the success rate for identifying NE solutions, finding all pure and mixed NE solutions rather than only a portion of pure NE solutions, compared to D-Wave based quantum approaches. Moreover, C-Nash boasts a reduction up to 157.9X/79.0X in time-to-solutions compared to D-Wave 2000 Q6 and D-Wave Advantage 4.1, respectively.

This paper addresses the problem of distributed online generalized Nash equilibrium (GNE) learning for multi-cluster games with delayed feedback information. Specifically, each agent in the game is assumed to be informed a sequence of local cost functions and constraint functions, which are known to the agent with time-varying delays subsequent to decision-making at each round. The objective of each agent within a cluster is to collaboratively optimize the cluster's cost function, subject to time-varying coupled inequality constraints and local feasible set constraints over time. Additionally, it is assumed that each agent is required to estimate the decisions of all other agents through interactions with its neighbors, rather than directly accessing the decisions of all agents, i.e., each agent needs to make decision under partial-decision information. To solve such a challenging problem, a novel distributed online delay-tolerant GNE learning algorithm is developed based upon the primal-dual algorithm with an aggregation gradient mechanism. The system-wise regret and the constraint violation are formulated to measure the performance of the algorithm, demonstrating sublinear growth with respect to time under certain conditions. Finally, numerical results are presented to verify the effectiveness of the proposed algorithm.

We consider seeking generalized Nash equilibria (GNE) for noncooperative games with coupled nonlinear constraints over networks. We first revisit a well-known gradientplay dynamics from a passivity-based perspective, and address that the strict monotonicity on pseudo-gradients is a critical assumption to ensure the exact convergence of the dynamics. Then we propose two novel passivity-based gradient-play dynamics by introducing parallel feedforward compensators (PFCs) and output feedback compensators (OFCs). We show that the proposed dynamics can reach exact GNEs in merely monotone regimes if the PFCs are strictly passive or the OFCs are output strictly passive. Following that, resorting to passivity, we develop a unifying framework to generalize the gradient-play dynamics, and moreover, design a class of explicit passive-based dynamics with convergence guarantees. In addition, we explore the relation between the proposed dynamics and some existing methods, and extend our results to partial-decision information settings.

In this paper we solve the general problem, already formulated in Awerkin and Vargiolu (Decis. Econ. Finance 44(2), 2021) of finding a Nash equilibrium between two agents who can install irreversibly photovoltaic panels in order to maximize their profits of selling the produced electricity net of installation costs, in the case that their cumulative installations have an impact on power prices. Starting from a static version of the game, we find out that there exists a region in the state space where Nash equilibrium dictates that both players install, and in some cases this optimal installation strategy is non-unique. We then come back to the original continuous time problem, which needs a generalization of the Verification Theorem present in Awerkin and Vargiolu (Decis. Econ. Finance 44(2), 2021), taking into account a lack of smoothness of the value functions and the possible non-uniqueness of optimal strategies seen above. Finally, we explicitly construct the equilibrium strategies and the value functions, which depend on the two free boundaries which separate the region where each player installs or waits.

Generative artificial intelligence (Generative AI), and in particular Large Language Models (LLMs) have gained significant popularity among researchers and industrial communities, paving the way for integrating LLMs in different domains, such as robotics, telecom, and healthcare. In this paper, we study the intersection of game theory and generative artificial intelligence, focusing on the capabilities of LLMs to find the Nash equilibrium in games with a mixed strategy Nash equilibrium and no pure strategy Nash equilibrium (that we denote mixed strategy Nash equilibrium games). The study reveals a significant enhancement in the performance of LLMs when they are equipped with the possibility to run code and are provided with a specific prompt to incentivize them to do so. However, our research also highlights the limitations of LLMs when the randomization strategy of the game is not easy to deduce. It is evident that while LLMs exhibit remarkable proficiency in well-known standard games, their performance dwindles when faced with slight modifications of the same games. This paper aims to contribute to the growing body of knowledge on the intersection of game theory and generative artificial intelligence while providing valuable insights into LLMs strengths and weaknesses. It also underscores the need for further research to overcome the limitations of LLMs, particularly in dealing with even slightly more complex scenarios, to harness their full potential.

Bayesian game is a strategic decision-making model where each player's type parameter characterizing its own objective is private information: each player knows its own type but not its rivals' types, and Bayesian Nash equilibrium (BNE) is an outcome of this game where each player makes a strategic optimal decision according to its own type under the Nash conjecture. In this paper, we advance the literature by considering a generalized Bayesian game where each player's action space depends on its own type parameter and the rivals' actions. This reflects the fact that in practical applications, a firm's feasible action is often related to its own type (e.g. marginal cost) and the rivals' actions (e.g. common resource constraints in a competitive market). Under some moderate conditions, we demonstrate existence of continuous generalized Bayesian Nash equilibria (GBNE) and uniqueness of such an equilibrium when each player's action space is only dependent on its type. In the case that each player's action space is also dependent on rivals' actions, we give a simple example to show that uniqueness of GBNE is not guaranteed under standard monotone conditions. To compute an approximate GBNE, we restrict each player's response function to the space of polynomial functions of its type parameter and consequently convert the GBNE problem to a stochastic generalized Nash equilibrium problem (SGNE). To justify the approximation, we discuss convergence of the approximation scheme. Some preliminary numerical test results show that the approximation scheme works well.

This paper introduces a new method to achieve stable convergence to Nash equilibrium in duopoly noncooperative games. Inspired by the recent fixed-time Nash Equilibrium seeking (NES) as well as prescribed-time extremum seeking (ES) and source seeking schemes, our approach employs a distributed sliding mode control (SMC) scheme, integrating extremum seeking with sinusoidal perturbation signals to estimate the pseudogradients of quadratic payoff functions. Notably, this is the first attempt to address noncooperative games without relying on models, combining classical extremum seeking with relay components instead of proportional control laws. We prove finite-time convergence of the closed-loop average system to Nash equilibrium using stability analysis techniques such as time-scaling, Lyapunov's direct method, and averaging theory for discontinuous systems. Additionally, we quantify the size of residual sets around the Nash equilibrium and validate our theoretical results through simulations., Comment: 8 pages and 2 figures. arXiv admin note: substantial text overlap with arXiv:2404.07287

This paper investigates the challenge of learning in black-box games, where the underlying utility function is unknown to any of the agents. While there is an extensive body of literature on the theoretical analysis of algorithms for computing the Nash equilibrium with complete information about the game, studies on Nash equilibrium in black-box games are less common. In this paper, we focus on learning the Nash equilibrium when the only available information about an agent's payoff comes in the form of empirical queries. We provide a no-regret learning algorithm that utilizes Gaussian processes to identify the equilibrium in such games. Our approach not only ensures a theoretical convergence rate but also demonstrates effectiveness across a variety collection of games through experimental validation., Comment: 40th Conference on Uncertainty in Artificial Intelligence (UAI 2024)

The problem of the distributed Nash equilibrium seeking for aggregative games has been studied over strongly connected and weight-balanced static networks and every time strongly connected and weight-balanced switching networks. In this paper, we further study the same problem over jointly connected and weight-balanced networks. The existing approaches critically rely on the connectedness of the network for constructing a Lyapunov function for their algorithms and theses approaches fail if the network is not connected. To overcome this difficulty, we propose an approach to show the exponential convergence of the output of the closed-loop system to the unknown Nash equilibrium (NE) point under a set of mild conditions.

Nash Equilibrium (NE) is the canonical solution concept of game theory, which provides an elegant tool to understand the rationalities. Though mixed strategy NE exists in any game with finite players and actions, computing NE in two- or multi-player general-sum games is PPAD-Complete. Various alternative solutions, e.g., Correlated Equilibrium (CE), and learning methods, e.g., fictitious play (FP), are proposed to approximate NE. For convenience, we call these methods as "inexact solvers", or "solvers" for short. However, the alternative solutions differ from NE and the learning methods generally fail to converge to NE. Therefore, in this work, we propose REinforcement Nash Equilibrium Solver (RENES), which trains a single policy to modify the games with different sizes and applies the solvers on the modified games where the obtained solution is evaluated on the original games. Specifically, our contributions are threefold. i) We represent the games as $\alpha$-rank response graphs and leverage graph neural network (GNN) to handle the games with different sizes as inputs; ii) We use tensor decomposition, e.g., canonical polyadic (CP), to make the dimension of modifying actions fixed for games with different sizes; iii) We train the modifying strategy for games with the widely-used proximal policy optimization (PPO) and apply the solvers to solve the modified games, where the obtained solution is evaluated on original games. Extensive experiments on large-scale normal-form games show that our method can further improve the approximation of NE of different solvers, i.e., $\alpha$-rank, CE, FP and PRD, and can be generalized to unseen games., Comment: IJCAI 2024

In Evolutionary Game Theory (EGT), a population reaches a Nash equilibrium when none of the agents can improve its objective by solely changing its strategy on its own. Roughly speaking, this equilibrium is a protection against betrayal. Generalized Nash Equilibrium (GNE) is a more complex version of this idea with important implications in real-life problems in economics, wireless communication, the electricity market, or engineering among other areas. In this paper, we propose a first approach to GNE with Membrane Computing techniques and show how GNE problems can be modeled with P systems, bridging both areas and opening a door for a flow of problems and solutions in both directions.

This paper proposes a novel approach for locally stable convergence to Nash equilibrium in duopoly noncooperative games based on a distributed event-triggered control scheme. The proposed approach employs extremum seeking, with sinusoidal perturbation signals applied to estimate the Gradient (first derivative) of unknown quadratic payoff functions. This is the first instance of noncooperative games being tackled in a model-free fashion integrated with the event-triggered methodology. Each player evaluates independently the deviation between the corresponding current state variable and its last broadcasted value to update the player action, while they preserve control performance under limited bandwidth of the actuation paths and still guarantee stability for the closed-loop dynamics. In particular, the stability analysis is carried out using time-scaling technique, Lyapunov's direct method and averaging theory for discontinuous systems. We quantify the size of the ultimate small residual sets around the Nash equilibrium and illustrate the theoretical results numerically on an example.