Search

We consider an environmentwhere players need to decide whether to buy a certain product (or adopt a technology) or not. The product is either good or bad, but its true value is unknown to the players. Instead, each player has her own private information on its quality. Each player can observe the previous actions of other players and estimate the quality of the product. A classic result in the literature shows that in similar settings, informational cascades occur, where learning stops for the whole network and players repeat the actions of their predecessors. In contrast to this literature, in this paper, players get more than one opportunity to act. In each turn, a player is chosen uniformly at random from all the players and can decide to buy the product and leave the market or wait. Her utility is the total expected discounted reward, and thus, myopic strategies may not constitute equilibria. We provide a characterization of perfect Bayesian equilibria (PBEs) with forward-looking strategies through a fixed-point equation of dimensionality that grows only quadratically with the number of players. Using this tractable fixed-point equation, we show the existence of a PBE and characterize PBEs with threshold strategies. Based on this characterization, we study informational cascades in two regimes. First, we show that for a discount factor $\delta$ strictly smaller than 1, informational cascades happen with high probability as the number of players $N$ increases. Furthermore, only a small portion of the total information in the system is revealed before a cascade occurs. Second, and more surprisingly, we show that for a fixed $N$ , and for a sufficiently large $\delta < 1$ , when the product is bad, there exists an equilibrium where an informational cascade can happen only after at least half of the players revealed their private information, and consequently, the probability for a “bad cascade” where all the players buy the product vanishes exponentially with $N$. Finally, when $\delta =1$ and the product is bad, there exists an equilibrium where informational cascades do not happen at all. [ABSTRACT FROM AUTHOR]

We consider a setting where two noncooperative players optimally influence the evolution of an initial spatial probability in a game-theoretic hierarchical fashion (Stackelberg differential game), so that at a specific final time the distribution of the state matches a given final target measure. We provide a sufficient condition for the existence and uniqueness of an optimal transport map and prove that it can be characterized as the gradient of some convex function. An important by-product of our formulation is that it provides a means to study a class of Stackelberg differential games where the initial and final states of the underlying system are uncertain, but drawn randomly from some probability measures. [ABSTRACT FROM AUTHOR]

This article considers remote state estimation in cyber-physical systems (CPSs) with multiple sensors. Each plant is modeled by a discrete-time stochastic linear system with measurements of each sensor transmitted to the corresponding remote estimator over a shared communication network when their securities are interdependent due to network-induced risks. A dynamic nonzero-sum game with asymmetric information is formulated in which each sensor subject to a resource budget constraint needs to decide whether to invest in security for sending data packets, taking the behaviors of other sensors into account. To overcome the difficulty in characterizing or computing the Nash equilibria (NE), the game with asymmetric information is transformed into another game with symmetric information such that the equilibrium of the original game can be obtained by solving the equilibrium of the new game. Under certain conditions, we devise a backward induction algorithm to obtain a subclass of NE of the original game, known as common information-based Markov perfect equilibria (CIBMPE). Finally, a numerical example is provided to illustrate the results obtained. [ABSTRACT FROM AUTHOR]

This article investigates the multiplayer pursuit-evasion (MPE) differential games in the presence of malicious pursuers. First, three types of malicious pursuers, namely greedy, lazy, and traitorous pursuers are formulated by using two subjectively malicious factors in the performance index. Then, an N-player nonzero-sum game framework is established to study the relationships among healthy pursuers, malicious pursuers, and evaders. The capturability and Nash equilibrium of such a new MPE differential game are rigorously analyzed. Finally, the effects of malicious pursuers on the capture time are explicitly quantified by the minimal pursuit coefficient, whose explicit form is obtained by the solutions from coupled Riccati differential equations. [ABSTRACT FROM AUTHOR]

We consider the problem of two-player zero-sum games. This problem is formulated as a min–max Markov game in this article. The solution of this game, which is the min–max payoff, starting from a given state is called the min–max value of the state. In this article, we compute the solution of the two-player zero-sum game, utilizing the technique of successive relaxation that has been successfully applied in this article to compute a faster value iteration algorithm in the context of Markov decision processes. We extend the concept of successive relaxation to the setting of two-player zero-sum games. We show that, under a special structure on the game, this technique facilitates faster computation of the min–max value of the states. We then derive a generalized minimax Q-learning algorithm, which computes the optimal policy when the model information is not known. Finally, we prove the convergence of the proposed generalized minimax Q-learning algorithm utilizing stochastic approximation techniques, under an assumption on the boundedness of iterates. Through experiments, we demonstrate the [ABSTRACT FROM AUTHOR]

In this article, we consider the misspecified optimization problem of minimizing a convex function $f(x;\theta ^*)$ in $x$ over a conic constraint set represented by $h(x;\theta ^*) \in \mathcal {K}$ , where $\theta ^*$ is an unknown (or misspecified) vector of parameters, $\mathcal {K}$ is a closed convex cone, and $h$ is affine in $x$. Suppose that $\theta ^*$ is unavailable but may be learnt by a separate process that generates a sequence of estimators $\theta _k$ , each of which is an increasingly accurate approximation of $\theta ^*$. We develop a first-order inexact augmented Lagrangian (AL) scheme for computing an optimal solution $x^*$ corresponding to $\theta ^*$ while simultaneously learning $\theta ^*$. In particular, we derive rate statements for such schemes when the penalty parameter sequence is either constant or increasing and derive bounds on the overall complexity in terms of proximal gradient steps when AL subproblems are inexactly solved via an accelerated proximal gradient scheme. Numerical results for a portfolio optimization problem with a misspecified covariance matrix suggest that these schemes perform well in practice, while naive sequential schemes may perform poorly in comparison. [ABSTRACT FROM AUTHOR]

We solve the stochastic generalized Nash equilibrium (SGNE) problem in merely monotone games with expected value cost functions. Specifically, we present the first distributed SGNE-seeking algorithm for monotone games that require one proximal computation (e.g., one projection step) and one pseudogradient evaluation per iteration. Our main contribution is to extend the relaxed forward–backward operator splitting by the Malitsky (Mathematical Programming, 2019) to the stochastic case and in turn to show almost sure convergence to an SGNE when the expected value of the pseudogradient is approximated by the average over a number of random samples. [ABSTRACT FROM AUTHOR]

This article studies the sensor placement problem in a leader–follower networked control system for improving its security against cyber-physical attacks. In a zero-sum game, the attacker selects $f$ nodes of the network to attack, and the detector places $f$ sensors to detect the presence of the attack signals. In our formulation, the attacker’s objective is to have a large impact on a target node in the network while being as little visible as possible to the detector. The detector, however, seeks to maximize the visibility of the attack signals. The effects of the attack signals on both the target node and the detector node are captured via the system $L_2$ gain from the attack signals to the target node and deployed sensors’ outputs, respectively. The equilibrium strategy of the game determines the optimal locations of the sensors. The existence of Nash equilibrium for the single-attack single-sensor case is studied when the underlying connectivity graph is a directed or an undirected tree. We show that, under the optimal sensor placement strategy, an undirected topology provides a higher security level for a networked control system compared to its corresponding directed topology. For the multiple-attack multiple-sensor case, we show that the game does not necessarily admit a Nash equilibrium and introduce a Stackelberg game approach, where the detector acts as the leader. Finally, these results are used to study the sensor placement problem in a vehicle platooning application in the presence of bias injectionattacks. [ABSTRACT FROM AUTHOR]

In many dynamical state estimation problems, not all the values that the state can take have the same importance; hence, missing to deliver an appropriate estimate has more severe consequences for certain state values than for others. In many applications, such important state values correspond to events that have low a priori probability to happen (e.g., unsafe situations or conditions that one tries to avoid by design). Provably, Kalman filtering techniques are inadequate to correctly estimate such rare events. In this article, a new state estimation paradigm is introduced to build confidence regions that contain the true state value, whatever this value is, with a user-chosen probability. Among regions having this property, an algorithm is introduced that generates in a Gaussian setup the region that satisfies a minimum-volume condition. [ABSTRACT FROM AUTHOR]

This article considers a discrete-time robust optimal control problem on matrix Lie groups. The underlying system is assumed to be perturbed by exogenous unmeasured bounded disturbances, and the control problem is posed as a min–max optimal control wherein the disturbance is the adversary and tries to maximize a cost that the control tries to minimize. Assuming the existence of a saddle point in the problem, we present a version of the Pontryagin maximum principle (PMP) that encapsulates first-order necessary conditions that the optimal control and disturbance trajectories must satisfy. This PMP features a saddle point condition on the Hamiltonian and a set of backward difference equations for the adjoint dynamics. We also present a special case of our result on Euclidean spaces. We conclude with applying the PMP to robust version of single axis rotation of a rigid body. [ABSTRACT FROM AUTHOR]

This article formalizes two types of modeling uncertainties in a stochastic Stackelberg linear-quadratic (LQ) differential game and then discusses the associated robust Stackelberg strategy design for either the leader or follower. Both uncertainties are primarily motivated by practical applications in engineering and management. The first uncertainty is connected to a disturbance unknown to the follower but known to the leader. A soft-constraint min-max control is applied by the follower to determine the optimal response, and then an augmented LQ forward-backward stochastic differential equation control is solved by the leader to ensure a robust strategy design. The second uncertainty involves a disturbance, the realization of which can be completely observed by the follower, but only its distribution can be accessed by the leader. Thus, a hard-constraint min-max control on an affine-equality-constraint is studied by the leader to address the exact-optimal robust design. Moreover, based on a weak convergence technique, a minimizing sequence of near-optimal robust designs is constructed, which is more tractable in computation. Some numerical results of the abovementioned robust strategies are also presented. [ABSTRACT FROM AUTHOR]

We analyze a tree search problem with an underlying Markov decision process, in which the goal is to identify the best action at the root that achieves the highest cumulative reward. We present a new tree policy that optimally allocates a limited computing budget to maximize a lower bound on the probability of correctly selecting the best action at each node. Compared to widely used upper confidence bound (UCB) tree policies, the new tree policy presents a more balanced approach to manage the exploration and exploitation tradeoff when the sampling budget is limited. Furthermore, UCB assumes that the support of reward distribution is known, whereas our algorithm relaxes this assumption. Numerical experiments demonstrate the efficiency of our algorithm in selecting the best action at the root. [ABSTRACT FROM AUTHOR]

Are rewards or penalties more effective in influencing user behavior? This article compares the effectiveness of subsidies and tolls in incentivizing user behavior in congestion games. The predominantly studied method of influencing user behavior in network routing problems is to institute taxes which alter users’ observed costs in a manner that causes their self-interested choices to more closely align with a system-level objective. Another conceivable method to accomplish the same goal is to subsidize the users’ actions that are preferable from a system-level perspective. We show that, when users behave similarly and predictably, subsidies offer superior performance guarantees to tolls under similar budgetary constraints; however, in the presence of unknown player heterogeneity, subsidies fail to offer the same robustness as tolls. [ABSTRACT FROM AUTHOR]

Consider a network of $N$ providers that each has a certain supply of energy and $B$ consumers that each has a certain demand. The efficiency of transmitting energy between providers and consumers is modeled using a weighted bipartite graph $G$. Our goal is to maximize the amount of utilized energy using a distributed algorithm that each provider runs locally. We propose a noncooperative energy-allocation game and adopt the best-response dynamics for this game as our distributed algorithm. We prove that the best-response dynamics converge in no more than $N$ steps to one of at most $N!$ pure Nash equilibria (NE) of our game. However, we show that these NE can be very inefficient. Remarkably, our algorithm avoids the inefficient NE and achieves asymptotically (in $B$) optimal performance in “almost all” games. The traditional game-theoretic analysis using a potential function does not explain this encouraging finding. To fill this gap, we analyze the best-response dynamics in a random game, where the network is generated using a random model for the graph $G$. We prove that the ratio between the utilized energy of our algorithm and that of the optimal solution converges to one in probability as $B$ increases (and $N$ is any function of $B$). Numerical simulations demonstrate that our asymptotic analysis is valid even for $B=10$ consumers. Our novel random games approach analytically explains why the performance of our algorithm is asymptotically optimal almost always despite the fact that bad NE may exist. [ABSTRACT FROM AUTHOR]

This article investigates quantitative supervisory control with local mean payoff objectives on discrete event systems modeled as weighted automata. Weight flows are generated as new events occur, which are required to satisfy some quantitative conditions. We focus on mean weights (payoffs) over a finite number of events, which serve as a measure for the stability or robustness of weight flows. The range of events to evaluate the mean payoff is termed a window, which slides as new events occur. Qualitative requirements such as safety and liveness are also necessary along with quantitative requirements. Supervisory control is employed to manipulate the operation of the system so that the requirements are satisfied. We consider two different scenarios based on whether the window size is fixed or not. Correspondingly, we formulate two supervisory control problems, both of which are solved sequentially by first tackling the qualitative issues and then the quantitative ones. The automaton model is then transformed to a two-player game between the supervisor and the environment, where safety and liveness are enforced. Based on the intermediate results, several quantitative objectives are defined to formulate two games, which correspond to the two proposed supervisory control problems. Finally, we synthesize provably correct supervisors by solving the games and completely resolve both problems. [ABSTRACT FROM AUTHOR]

In this article, we study the problem of information sharing among rational self-interested agents as a dynamic game of asymmetric information. We assume that the agents imperfectly observe a Markov chain, and they are called to decide whether they will share their noisy observations or not at each time instant. We utilize the notion of conditional mutual information to evaluate the information being shared among the agents. The challenges that arise due to the interdependence of agents’ information structure and decision making are exhibited. For the finite horizon game, we prove that agents do not have incentive to share information. In contrast, we show that cooperation can be sustained in the infinite-horizon case by devising appropriate punishment strategies, which are defined over the agents’ beliefs on the system state. We show that these strategies are closed under the best-response mapping and that cooperation can be the optimal choice in some subsets of the state belief simplex. We characterize these equilibrium regions, prove uniqueness of a maximal equilibrium region, and devise an algorithm for its approximate computation. [ABSTRACT FROM AUTHOR]

Multiagent policy evaluation and seeking are long-standing challenges in developing theories for multiagent reinforcement learning (MARL), due to multidimensional learning goals, nonstationary environment, and scalability issues in the joint policy space. This article introduces two metrics grounded on a game-theoretic solution concept called sink equilibrium, for the evaluation, ranking, and computation of policies in multiagent learning. We adopt strict best response dynamics (SBRDs) to model selfish behaviors at a meta-level for MARL. Our approach can deal with dynamical cyclical behaviors (unlike approaches based on Nash equilibria and Elo ratings), and is more compatible with single-agent reinforcement learning than $\alpha$ -rank, which relies on weakly better responses. We first consider settings where the difference between the largest and second largest equilibrium metric has a known lower bound. With this knowledge, we propose a class of perturbed SBRD with the following property: only policies with maximum metric are observed with nonzero probability for a broad class of stochastic games with finite memory. We then consider settings where the lower bound for the difference is unknown. For this setting, we propose a class of perturbed SBRD such that the metrics of the policies observed with nonzero probability differ from the optimal by any given tolerance. The proposed perturbed SBRD addresses the scalability issue and opponent-induced nonstationarity by fixing the strategies of others for the learning agent, and uses empirical game-theoretic analysis to estimate payoffs for each strategy profile obtained due to the perturbation. [ABSTRACT FROM AUTHOR]

In this article, the framework of network aggregative game (NAG) is extended by considering the interaction of players in a dynamic environment whose model is not known to them. The cost function of each player depends on its own strategy, the aggregate strategy of its neighbor players in the network, and also the state of the environment. The state of the environment evolves with a dynamics that is affected by the strategies of players as an input to the model. The players of NAG are modeled as myopic selfish agents who observe the decision value of their neighbors and the state of the environment with some time delay and, then, adjust their decisions using a projected subgradient rule. The main contribution of this article is to provide conditions in which the strategies of myopic agents converge to the unique fixed-point Nash equilibrium point of the proposed dynamic NAG. Furthermore, as a case study, the framework is applied to a wireless network by modeling the service providers as the agents and evolution of service selection by users as the dynamics of the environment. [ABSTRACT FROM AUTHOR]

The price of anarchy and price of stability are two well-studied performance metrics that seek to characterize the inefficiency of equilibria in distributed systems. The distinction between these two performance metrics centers on the equilibria that they focus on: the price of anarchy characterizes the quality of the worst-performing equilibria, while the price of the stability characterizes the quality of the best-performing equilibria. While much of the literature focuses on these metrics from an analysis perspective, in this article, we consider these performance metrics from a design perspective. Specifically, we focus on the setting where a system operator is tasked with designing local agent utility functions to optimize these performance metrics in a class of games termed covering games. Our main result characterizes a fundamental tradeoff between the price of anarchy and price of the stability in the form of a fully explicit Pareto frontier. Within this setup, we observe that optimizing the price of anarchy comes directly at the expense of the price of stability (and vice versa). Our second result demonstrates how a system operator could incorporate an additional piece of system-level information into the design of the agents’ utility functions to breach these limitations and improve the efficiency guarantees associated with the resulting equilibria. Informally, this valuable piece of system-level information pertains to the value of the largest uncovered resource in our covering game. [ABSTRACT FROM AUTHOR]

We propose a two-layer, semidecentralized algorithm to compute a local solution to the Stackelberg equilibrium problem in aggregative games with coupling constraints. Specifically, we focus on a single-leader, multiple-follower problem, and after equivalently recasting the Stackelberg game as a mathematical program with complementarity constraints (MPCC), we iteratively convexify a regularized version of the MPCC as the inner problem, whose solution generates a sequence of feasible descent directions for the original MPCC. Thus, by pursuing a descent direction at every outer iteration, we establish convergence to a local Stackelberg equilibrium. Finally, the proposed algorithm is tested on a numerical case study, a hierarchical instance of the charging coordination problem of plug-in electric vehicles. [ABSTRACT FROM AUTHOR]

A fundamental component of the game-theoretic approach to distributed control is the design of local utility functions. Relative to resource allocation problems that are additive over the resources, Part I showed how to design local utilities so as to maximize the associated performance guarantees (Paccagnan, et al., 2020), which we measure by the price of anarchy. The purpose of this article is to specialize these results to the case of submodular, covering, and supermodular problems. In all these cases, we obtain tight expressions for the price of anarchy that often match or improve the guarantees associated with state-of-the-art approximation algorithms. Two applications and corresponding numerics are presented: the vehicle-target assignment problem and a coverage problem arising in wireless data caching. [ABSTRACT FROM AUTHOR]

Despite the increasing interest in multiagent reinforcement learning (MARL) in multiple communities, understanding its theoretical foundation has long been recognized as a challenging problem. In this article, we address this problem by providing a finite-sample analysis for decentralized batch MARL. Specifically, we consider a type of mixed MARL setting with both cooperative and competitive agents, where two teams of agents compete in a zero-sum game setting, while the agents within each team collaborate by communicating over a time-varying network. This setting covers many conventional MARL settings in the literature. We then develop batch MARL algorithms that can be implemented in a decentralized fashion, and quantify the finite-sample errors of the estimated action-value functions. Our error analysis captures how the function class, the number of samples within each iteration, and the number of iterations determine the statistical accuracy of the proposed algorithms. Our results, compared to the finite-sample bounds for single-agent reinforcement learning, involve additional error terms caused by decentralized computation, which is inherent in our decentralized MARL setting. This article provides the first finite-sample analysis for batch MARL, a step toward rigorous theoretical understanding of general MARL algorithms in the finite-sample regime. [ABSTRACT FROM AUTHOR]

We consider a game-theoretic model where individuals compete over a shared failure-prone system or resource. We investigate the effectiveness of a taxation mechanism in controlling the utilization of the resource at the Nash equilibrium when the decision-makers have behavioral risk preferences, captured by prospect theory. We first observe that heterogeneous prospect-theoretic risk preferences can lead to counter-intuitive outcomes. In particular, for resources that exhibit network effects, utilization can increase under taxation and there may not exist a tax rate that achieves the socially optimal level of utilization. We identify conditions under which utilization is monotone and continuous, and then characterize the range of utilizations that can be achieved by a suitable choice of tax rate. We further show that resource utilization is higher when players are charged differentiated tax rates compared to the case when all players are charged an identical tax rate, under suitable assumptions. [ABSTRACT FROM AUTHOR]

This articleproposes an extension to a family of evolutionary game dynamics called escort replicator dynamics (ERD), defined by (Harper, 2011) based on information-geometric concepts. Specifically, the objective of this article is to propose three types of convenient so-called escort functionsand evaluate the stability of ERD under the use of these functions. Since it is proven that ERD is stable using the proposed definitions, ERD is able to confine trajectories of the state vector in regions defined by upper and lower boundaries which can be, but are not required to be, nonnegative. From an engineering application perspective, these proposed escort functions are convenient since they allow to extend replicatorlike dynamics applications to engineering problems with feasible regions more general than the standard simplex or the nonnegative orthant. From a population dynamics perspective the proposed escort functions are useful to represent upper and lower limits on the share of a population that is allowed to play a given pure strategy of the underlying symmetric game. [ABSTRACT FROM AUTHOR]

We study distributed algorithms for seeking a Nash equilibrium in a class of convex networked Nash games with strongly monotone mappings. Each player has access to her own smooth local cost function and can communicate to her neighbors in some undirected graph. To deal with fast distributed learning of Nash equilibria under such settings, we introduce a so called augmented game mapping and provide conditions under which this mapping is strongly monotone. We consider a distributed gradient play algorithm for determining a Nash equilibrium (GRANE). The algorithm involves every player performing a gradient step to minimize her own cost function while sharing and retrieving information locally among her neighbors in the network. Using the reformulation of the Nash equilibrium problem based on the strong monotone augmented game mapping, we prove the convergence of this algorithm to a Nash equilibrium with a geometric rate. Furthermore, we introduce the Nesterov type acceleration for the gradient play algorithm. We demonstrate that, similarly to the accelerated algorithms in centralized optimization and variational inequality problems, our accelerated algorithm outperforms GRANE in the convergence rate. Moreover, to relax assumptions required to guarantee the strongly monotone augmented mapping, we analyze the restricted strongly monotone property of this mapping and prove geometric convergence of the distributed gradient play under milder assumptions. [ABSTRACT FROM AUTHOR]

We consider concave continuous-kernel games characterized by monotonicity properties and propose discounted mirror descent type dynamics. We introduce two classes of dynamics whereby the associated mirror map is constructed based on a strongly convex or a Legendre regularizer. Depending on the properties of the regularizer, we show that these new dynamics can converge asymptotically in concave games with merely monotone (negative) pseudogradient. Furthermore, we show that when the regularizer enjoys strong convexity, the resulting dynamics can converge even in games with hypomonotone (negative) pseudogradient, which corresponds to a shortage of monotonicity. [ABSTRACT FROM AUTHOR]

Competitive noncooperative online decision-making agents whose actions increase congestion of scarce resources constitute a model for widespread modern large-scale applications. To ensure sustainable resource behavior, we introduce a novel method to steer the agents toward a stable population state, fulfilling the given coupled resource constraints. The proposed method is a decentralized resource pricing method based on the resource loads resulting from the augmentation of the game's Lagrangian. Assuming that the online learning agents have only noisy first-order utility feedback, we show that for a polynomially decaying agents step size/learning rate, the population's dynamic will almost surely converge to generalized Nash equilibrium. A particular consequence of the latter is the fulfillment of resource constraints in the asymptotic limit. Moreover, we investigate the finite-time quality of the proposed algorithm by giving a nonasymptotic time decaying bound for the expected amount of resource constraint violation. [ABSTRACT FROM AUTHOR]

In this article, we propose centralized and distributed continuous-time penalty methods to find a Nash equilibrium for a generalized noncooperative game with shared inequality and equality constraints and private inequality constraints that depend on the player itself. By using the $\ell _{1}$ penalty function, we prove that the equilibrium of a differential inclusion is a normalized Nash equilibrium of the original generalized noncooperative game, and the centralized differential inclusion exponentially converges to the unique normalized Nash equilibrium of a strongly monotone game. Suppose that the players can communicate with their neighboring players only and the communication topology can be represented by a connected undirected graph. Based on a leader-following consensus scheme and singular perturbation techniques, we propose distributed algorithms by using the exact $\ell _{1}$ penalty function and the continuously differentiable squared $\ell _{2}$ penalty function, respectively. The squared $\ell _{2}$ penalty function method works for games with smooth constraints and the exact $\ell _{1}$ penalty function works for certain scenarios. The proposed two distributed algorithms converge to an $\eta$ -neighborhood of the unique normalized Nash equilibrium and an $\eta$ -neighborhood of an approximated Nash equilibrium, respectively, with $\eta$ being a positive constant. For each $\eta >0$ and each initial condition, there exists an $\varepsilon ^*$ such that for each $0< \varepsilon < \varepsilon ^*$ , the convergence can be guaranteed where $\varepsilon$ is a parameter in the algorithm. [ABSTRACT FROM AUTHOR]

Distributed bandit online convex optimization with time-varying coupled inequality constraints is considered, motivated by a repeated game between a group of learners and an adversary. The learners attempt to minimize a sequence of global loss functions and at the same time satisfy a sequence of coupled constraint functions, where the constraints are coupled across the distributed learners at each round. The global loss and the coupled constraint functions are the sum of local convex loss and constraint functions, respectively, which are adaptively generated by the adversary. The local loss and constraint functions are revealed in a bandit manner, i.e., only the values of loss and constraint functions are revealed to the learners at the sampling instance, and the revealed function values are held privately by each learner. Both one- and two-point bandit feedback are studied with the two corresponding distributed bandit online algorithms used by the learners. We show that sublinear expected regret and constraint violation are achieved by these two algorithms, if the accumulated variation of the comparator sequence also grows sublinearly. In particular, we show that $\mathcal {O}(T^{\theta })$ expected static regret and $\mathcal {O}(T^{7/4-\theta })$ constraint violation are achieved in the one-point bandit feedback setting, and $\mathcal {O}(T^{\max \lbrace \kappa,1-\kappa \rbrace })$ expected static regret and $\mathcal {O}(T^{1-\kappa /2})$ constraint violation in the two-point bandit feedback setting, where $\theta \in (3/4,5/6]$ and $\kappa \in (0,1)$ are user-defined tradeoff parameters. Finally, the tightness of the theoretical results is illustrated by numerical simulations of a simple power grid example, which also compares the proposed algorithms to algorithms existing in the literature. [ABSTRACT FROM AUTHOR]