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We develop a model for contagion risks and optimal security investment in a directed network of interconnected agents with heterogeneous degrees, loss functions and security profiles. Our model generalizes much of contagion models in the literature; in particular the independent cascade model and the linear threshold model. We state various limit theorems on the final size of infected agents in the case of random networks with given vertex degrees for finite and infinite variance degree distributions. The results allow us to derive a resilience condition of the network to the infection of a large group of agents and quantify how contagion amplifies small shocks to the network. We show that when the degree distribution has infinite variance and, highly correlated in- and out-degrees, then even when agents have high thresholds, a sub-linear fraction of initially infected agents is enough to trigger the infection of a positive fraction. We also show how these results are sensitive to vertex and edge percolation (immunization). We then study the asymptotic Nash equilibrium and socially optimal security investment. In the asymptotic limit, agents' risk depends on all other agents' investment through an aggregate quantity, that we call network vulnerability. The limit theorems allow us to capture the impact of one class of agents' decision on the overall network vulnerability. Based on our results, the vulnerability is semi-analytic allowing for a tractable Nash equilibrium. We give sufficient conditions for investment in equilibrium to be monotone in the network vulnerability. When investment is monotone, we show that the (asymptotic) Nash equilibrium is unique. In the particular example of two types core-periphery agents, we exhibit a strong effect of the cost heterogeneity and in particular non-monotonous investment as a function of costs.

Driven by the increasing demand of real-time mobile application processing, Multi-access Edge Computing (MEC) has been envisioned as a promising paradigm for pushing computational resources to network edges. In this paper, we investigate an MEC network enabled by Unmanned Aerial Vehicles (UAV), and consider both the multi-user computation offloading and edge server deployment to minimize the system-wide computation cost under dynamic environment, where users generate tasks according to time-varying probabilities. We decompose the minimization problem by formulating two stochastic games for multi-user computation offloading and edge server deployment respectively, and prove that each formulated stochastic game has at least one Nash Equilibrium (NE). Two learning algorithms are proposed to reach the NEs with polynomial-time computational complexities. We further incorporate these two algorithms into a chess-like asynchronous updating algorithm to solve the system-wide computation cost minimization problem. Finally, performance evaluations based on real-world data are conducted and analyzed, corroborating that the proposed algorithms can achieve efficient computation offloading coupled with proper server deployment under dynamic environment for multiple users and MEC servers.

Multi-access edge computing (MEC) has been recently considered in challenging environments lacking available terrestrial infrastructures by extending the computing resources to the air for further enhancing the computation capability of the new aerial user equipment (AUE). Additionally, wireless power transfer (WPT) is a promising solution to prolong the battery lifetime of energy-constrained wireless devices like AUEs. In this paper, we investigate the integration of laser-beamed WPT in the high-altitude platform (HAP) aided MEC systems for the HAP-connected AUEs. By discretizing the three-dimensional coverage space of the HAP, we present a multi-tier tile grid-based spatial structure to provide aerial locations for laser charging. With this setup, we identify a new privacy vulnerability caused by the openness during the air-to-air transmission of WPT signaling messages in the presence of a terrestrial adversary. A privacy-aware laser-powered aerial MEC framework is developed that addresses this vulnerability and enhances the location privacy of AUEs for laser WPT. Specifically, the interaction between the HAP as a defender and the adversary in their tile grid allocation as charging locations to AUEs is formulated as a Colonel Blotto game, which models the competition of the players for limited resources over multiple battlefields for a finite time horizon. Moreover, we derive the mixed-strategy Nash equilibria of the tile grid allocation game for both symmetric and asymmetric tile grids between the defender and the adversary. Simulations results show that the proposed framework significantly outperforms the design baselines with a given privacy protection level in terms of system-wide expected total utilities.

Unmanned Aerial Vehicles (UAVs) have been utilized to serve on-ground users with various services, e.g., computing, communication and caching, due to their mobility and flexibility. The main focus of many recent studies on UAVs is to deploy a set of homogeneous UAVs with identical capabilities controlled by one UAV owner/company to provide services. However, little attention has been paid to the issue of how to enable different UAV owners to provide services with differentiated service capabilities in a shared area. To address this issue, we propose a multi-agent imitation learning enabled UAV deployment approach to maximize both profits of UAV owners and utilities of on-ground users. Specially, a Markov game is formulated among UAV owners and we prove that a Nash equilibrium exists based on the full knowledge of the system. For online scheduling with incomplete information, we design agent policies by imitating the behaviors of corresponding experts. A novel neural network model, integrating convolutional neural networks, generative adversarial networks and a gradient-based policy, can be trained and executed in a fully decentralized manner with a guaranteed -Nash equilibrium. Performance results show that our algorithm has significant superiority on average profits, utilities and execution time compared with other representative algorithms.

Learning automata (LA) with artificially absorbing barriers was a completely new horizon of research in the 1980s (Oommen, 1986). These new machines yielded properties that were previously unknown. More recently, absorbing barriers have been introduced in continuous estimator algorithms so that the proofs could follow a martingale property, as opposed to monotonicity (Zhang et al., 2014), (Zhang et al., 2015). However, the applications of LA with artificial barriers are almost nonexistent. In that regard, this article is pioneering in that it provides effective and accurate solutions to an extremely complex application domain, namely that of solving two-person zero-sum stochastic games that are provided with incomplete information. LA have been previously used (Sastry et al., 1994) to design algorithms capable of converging to the game's Nash equilibrium under limited information. Those algorithms have focused on the case where the saddle point of the game exists in a pure strategy. However, the majority of the LA algorithms used for games are absorbing in the probability simplex space, and thus, they converge to an exclusive choice of a single action. These LA are thus unable to converge to other mixed Nash equilibria when the game possesses no saddle point for a pure strategy. The pioneering contribution of this article is that we propose an LA solution that is able to converge to an optimal mixed Nash equilibrium even though there may be no saddle point when a pure strategy is invoked. The scheme, being of the linear reward-inaction ( $L_{R-I}$ ) paradigm, is in and of itself, absorbing. However, by incorporating artificial barriers, we prevent it from being ``stuck'' or getting absorbed in pure strategies. Unlike the linear reward-εpenalty ( $L_{R-ε P}$ ) scheme proposed by Lakshmivarahan and Narendra almost four decades ago, our new scheme achieves the same goal with much less parameter tuning and in a more elegant manner. This article includes the nontrial proofs of the theoretical results characterizing our scheme and also contains experimental verification that confirms our theoretical findings.

In this paper, a novel data offloading decision-making framework is proposed, where users have the option to partially offload their data to a complex Multi-access Edge Computing (MEC) environment, consisting of both ground and UAV-mounted MEC servers. The problem is treated under the perspective of risk-aware user behavior as captured via prospect-theoretic utility functions, while accounting for the inherent computing environment uncertainties. The UAV-mounted MEC servers act as a common pool of resources with potentially superior but uncertain payoff for the users, while the local computation and ground server alternatives constitute safe and guaranteed options, respectively. The optimal user task offloading to the available computing choices is formulated as a maximization problem of each user's satisfaction, and confronted as a non-cooperative game. The existence and uniqueness of a Pure Nash Equilibrium (PNE) are proven, and convergence to the PNE is shown. Detailed numerical results highlight the convergence of the system to the PNE in few only iterations, while the impact of user behavior heterogeneity is evaluated. The introduced framework's consideration of the user risk-aware characteristics and computing uncertainties, results to a sophisticated exploitation of the system resources, which in turn leads to superior users' experienced performance compared to alternative approaches.

We study an optimal targeting problem for super-modular games with binary actions and finitely many players. The considered problem consists in the selection of a subset of players of minimum size such that, when the actions of these players are forced to a controlled value while the others are left to repeatedly play a best response action, the system will converge to the greatest Nash equilibrium of the game. Our main contributions consist in showing that the problem is NP-complete and in proposing an efficient iterative algorithm with provable convergence properties for its solution. We discuss in detail the special case of network coordination games and its relation with the notion of cohesiveness. Finally, we show with simulations the strength of our approach with respect to naive heuristics based on classical network centrality measures.

This paper 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.

We formally assume that players in a game consider Nash Equilibrium (NE) the behavioral norm. In finite games of perfect information this leads to a refinement of NE: Faithful Nash Equilibrium (FNE). FNE is outcome equivalent to NE of the trimmed game, obtained by restricting the original tree to its NE paths. Thus, it always exists but it need not be unique. Iterating the norm ensures uniqueness of outcome. FNE may violate backward induction when subgame perfection requires play according to the SPE following a deviation from it. We thus provide an alternative view of tenable threats in equilibrium analysis.

In a simple two-country Ricardian economy with public infrastructures, we consider a simultaneous and non-cooperate game between governments with respect to public infrastructure supply. Then it is shown that a country with larger (smaller) factor endowment exports a good whose production is more (less) dependent on public infrastructures, and both countries will gain from trade as long as factor endowment differs between countries. However, the following special features appear. (i) Any incompletely specialising country produces two goods at an inner point of the production possibility set. (ii) If factor endowment is the same between countries, the trading equilibrium is attained by the pattern of specialisation such that each country specialises in one good different from each other and both countries become better off. Which country specialises in which good is indeterminate. The result shows a typical case of symmetric breaking. JEL codes: F11, H41

In this second part of our work, we study the steady state of the population and the social utility for the three dynamics SSD, NBRD and NRPM; which were introduced in the first part. We provide sufficient conditions on the network based on a maximum payoff density parameter of each node under which there exists a unique Nash equilibrium. We then utilize positive correlation properties of the dynamics to reduce the flow graph in order to provide an upper bound on the steady state social utility. Finally we extend the idea behind the sufficient condition for the existence of a unique Nash equilibrium to partition the graph appropriately in order to provide a lower bound on the steady state social utility. We also illustrate interesting cases as well as our results using simulations.

We consider the distributed scheduling problem on parallel machines with the central objective of maximizing the number of on-time jobs. Jobs are self-interested utility-maximizers that can choose the machines they are processed on and are exclusively interested in reducing their own private objective function. Each machine processes the jobs according to a local policy. We discuss Nash equilibria in the resulting schedules and perform a thorough analysis of the resulting (absolute) prices of anarchy for various parallel machine environments, utilities of the agents, and local policies of the machines. We show that local policies that are based on simple sorting-based procedures like shortest processing time first (SPT) and earliest due date first (EDD) lead to big losses in welfare compared to the global optimum. However, when employing Moore-Hodgson’s algorithm as a local policy, we can prove a price of anarchy of ( 2 m − 1 ) / m for identical machines and a price of anarchy of 2 for related and unrelated parallel machines. Moreover, we show how these results can be used to prove approximation ratios for greedy scheduling algorithms. This paper is the first to prove approximation ratios for two greedy scheduling procedures, turning them from simple heuristics into actual approximation ratios with a provable approximation ratio.

A generalized game is a strategic situation in which agents’ behavior restricts their opponents’ available action choices, giving rise to interdependencies in terms of what strategy profiles remain mutually feasible. This paper proposes a novel example of a simple generalized game in which the well-known convexity, compactness, continuity, and concavity assumptions are satisfied, but no Nash equilibrium exists. This finding reinforces that certain additional conditions appearing in the literature (primarily, the lower hemicontinuity of feasibility correspondences) are indispensable for equilibrium existence and must be considered as supplemental desiderata beyond the usual regularity conditions. Implications for institutional design are discussed.

Due to the nature of high mobility and dynamic network topology, Intrusion Detection Systems (IDSs) in Vehicular Ad-hoc Networks (VANETs) face lots of challenges, especially in balancing the accuracy and efficiency of detection. Current researches about the deployment of IDSs in VANETs mainly focus on a tradeoff between the effectiveness and efficiency, but few efforts have been done about the adaptability of the tradeoff in the changeable networks. Thus, we address two crucial problems: 1) how to perceive the environmental change in the perspective of an IDS? 2) how to make the IDS adaptive in different scenarios? In this paper, a Bayesian Game theory and Deep Q-learning Network-based IDS is proposed for VANETs, called GaDQN-IDS. The interactions between an IDS and attackers are formulated as a dynamic intrusion detection game, in which the IDS decides either to just adjust the tradeoff between the accuracy and efficiency or to be retrained completely when its detection capacity has declined. The Nash Equilibria (NE) of the game is derived to reveal how the optimal decision of the IDS depends on the detection performance and road conditions. Moreover, a Deep Q-learning Network (DQN)-Adjustment is proposed to realize the self-adaptation of the IDS in the dynamic game, while an Error Priority Learning (EPL) is further designed for IDS retraining in changing VANETs. Simulation results show that the GaDQN-IDS has better performance than other existing IDSs with higher detection rate as well as lower detection time and overhead.

Human-machine shared control has become an effective cutting-edge approach to enhance driving safety and assist in the transition from manual to autonomous driving. However, driving authority allocation when conflict occurs between a human driver and a machine still represents an intractable problem. In order to solve this problem, this paper proposes an adaptive steering torque coupling framework to achieve the human-machine shared control with conflict resolution. Additionally, a driver trajectory prediction method considering human-machine interaction is established based on the extended state observer and long short-term memory network. In addition, the human-machine shared steering system is modeled based on a non-cooperative dynamic game by the prediction model method for path-tracking, and the obtained solution is given according to the Nash equilibrium. Particularly, a dynamic load allocation approach is designed to resolve human-machine conflict and relax the driver. In order to verify the strategy formed based on the proposed framework, a driver-in-the-loop experiment is conducted. The experimental results reveal that the proposed strategy can effectively reduce the human-machine conflict torque and ensure a driver has absolute control authority. Furthermore, the proposed strategy can also transfer the driving workload between the driver and machine, provide driving experience, or improve driving comfort when there is no conflict.

Renewable resources are starting to constitute a growing portion of the total generation mix of the power system. A key difference between renewables and traditional generators is that many renewable resources are managed by individuals, especially in the distribution system. In this paper, we study the capacity investment and pricing problem, where multiple renewable producers compete in a decentralized market. It is known that most deterministic capacity games tend to result in very inefficient equilibria, even when there are a large number of similar players. In contrast, we show that due to the inherent randomness of renewable resources, the equilibria in our capacity game becomes efficient as the number of players grows and coincides with the centralized decision from the social planner's problem. This result provides a new perspective on how to look at the positive influence of randomness in a game framework as well as its contribution to resource planning, scheduling, and bidding. We validate our results by simulation studies using real world data., Appears in IEEE Transactions on Sustainable Computing

Switch-based adaptive dynamic programming (ADP) is an optimal control problem in which a cost must be minimized by switching among a family of dynamical modes. When the system dimension increases, the solution to switch-based ADP is made prohibitive by the exponentially increasing structure of the value function approximator and by the exponentially increasing modes. This technical correspondence proposes a distributed computational method for solving switch-based ADP. The method relies on partitioning the system into agents, each one dealing with a lower dimensional state and a few local modes. Each agent aims to minimize a local version of the global cost while avoiding that its local switching strategy has conflicts with the switching strategies of the neighboring agents. A heuristic algorithm based on the consensus dynamics and Nash equilibrium is proposed to avoid such conflicts. The effectiveness of the proposed method is verified via traffic and building test cases.

Edge computing, as an emerging and prospective computing paradigm, allows a service provider to serve its users by allocating them to nearby edge servers delivering services with low latency. From the service provider's perspective, a cost-effective service user allocation aims to allocate maximum service users to minimum edge servers. Such an allocation leverages multi-tenancy to reduce the resources hired by the service provider for serving the service users. However, the allocation of excessive service users to an edge server may result in severe interference and consequently impact their data rates. There is a trade-off between multi-tenancy and interference in the pursuit of a cost-effective service user allocation. In this paper, we formally model this service user allocation (SUA) problem. To solve the SUA problem effectively and efficiently, we propose a game-theoretic approach, namely MI-SUAGame, to formulate the SUA problem as a potential game. We analyze the game and prove its admission to a Nash equilibrium. Then, a novel decentralized algorithm is designed for finding a Nash equilibrium in the game as the solution to the SUA problem. The performance of MI-SUAGame is theoretically analyzed and experimentally evaluated against the state-of-the-art approach.

When a set of self-interested users shares multiple resources in a distributed system, we face the problem of allocating resources, called the load balancing problem. In particular, load balancing is defined as allocating the load to the servers of the distributed system such that jobs' response time is minimized, and the utilization of servers is improved. In this paper, the load balancing problem in a distributed system consists of a finite set of servers, and a finite set of users is studied. The load balancing problem considered here is a bi-objective problem with two highly probable conflicting objectives: (i) minimizing jobs' response time (ii) providing the fair utilization of servers. In order to satisfy these two objectives simultaneously, both the objectives are considered in an integrated manner. Next, the load balancing problem is formulated as a noncooperative game; and to solve the game (i.e., to find the Nash equilibrium), a distributed load balancing algorithm (DLBA) is proposed. An experimental study is carried out to ascertain the efficacy of the proposed DLBA. Further, we compare DBLA with three existing load balancing approaches to evaluate its comparative effectiveness. The experimental results validate the effectiveness of the DLBA over the existing approaches.

In this article, we study a variation of the active target-attacker-defender (ATAD) differential game involving multiple targets, an attacker, and a defender. Our model allows for 1) a capability of the defender to switch roles from rescuer (rendezvous with all the targets) to interceptor (intercepts the attacker) and vice versa and 2) the attacker to continuously pursue the closest target (which can change during the course of the game). We assume that the mode of the defender (rescue or interception) defines the mode of the game itself. Using the framework of Games of a Degree, we first analyze the game within each mode. More specifically, the objectives of the players are taken as a combination of weighted Euclidean distances and penalties on their control efforts. We model the interaction of the players within each mode as a linear quadratic differential game (LQDG) and obtain the open-loop Nash equilibrium strategies. We then use the receding horizon approach to enable switching between the modes to obtain switching strategies for the players. By partitioning the matrices associated with the Riccati differential equations we obtain geometric characterization of the trajectories of the players. Furthermore, under mild restrictions on the problem parameters and for a particular choice of the defender's switching function we show that interception mode is invariant. We illustrate our results with numerical simulations. Experimental results involving multiple autonomous differential drive mobile robots are presented.

In this article, the problem of distributed generalized Nash equilibrium (GNE) seeking in noncooperative games is investigated via multiagent networks, where each player aims to minimize his or her own cost function with a nonsmooth term. Each player's cost function and feasible action set in the noncooperative game are both determined by actions of others who may not be neighbors, as well as his/her own action. Particularly, feasible action sets are constrained by private convex inequalities and shared linear equations. Each player can only have access to his or her own cost function, private constraint, and a local block of shared constraints, and can only communicate with his or her neighbours via a digraph. To address this problem, a novel continuous-time distributed primal-dual algorithm involving Clarke's generalized gradient is proposed based on consensus algorithms and the primal-dual algorithm. Under mild assumptions on cost functions and graph, we prove that players' actions asymptotically converge to a GNE. Finally, a simulation is presented to demonstrate the effectiveness of our theoretical results.

Mobile edge computing is emerging to provide cloud-computing capabilities to mobile users, so that they can offload computation intensive tasks to close proximity for execution. However, most existing works imply that a transmission-finished task still occupies the channel until all users on the same channel finish the transmission, leading to severe channel resource waste. To solve this problem, we propose an efficient computation offloading mechanism which releases the channel resources of transmission-finished tasks for transmission-unfinished tasks, and aims to minimize the response time and energy consumption for each user. Specifically, we formulate the computation offloading problem as a game, analyze its structural properties and show how it possesses a Nash equilibrium and admits the finite improvement property, in the cases of elastic cloud and non-elastic cloud respectively. We then propose a Distributed Multi-channel Computation Offloading (DMCO) algorithm, which can converge to a Nash equilibrium, and find the upper bound of the convergence time. We further evaluate the performance of DMCO using the price of anarchy. Numerical results show that DMCO scales well with the number of users, and outperforms existing works, for example, benefits 13.3% more users and reduces cost by 23.7% than CO, one of the best existing works.

Edge computing (EC) is an emerging paradigm that extends cloud computing by pushing computing resources onto edge servers that are attached to base stations or access points at the edge of the cloud in close proximity with end-users. Due to edge servers' geographic distribution, the EC paradigm is challenged by many new security threats, including the notorious distributed Denial-of-Service (DDoS) attack. In the EC environment, edge servers usually have constrained processing capacities due to their limited sizes. Thus, they are particularly vulnerable to DDoS attacks. DDoS attacks in the EC environment render existing DDoS mitigation approaches obsolete with its new characteristics. In this paper, we make the first attempt to tackle the edge DDoS mitigation (EDM) problem. We model it as a constraint optimization problem and prove its NP-hardness. To solve this problem, we propose an optimal approach named EDMOpti and a novel game-theoretical approach named EDMGame for mitigating edge DDoS attacks. EDMGame formulates the EDM problem as a potential EDM Game that admits a Nash equilibrium and employs a decentralized algorithm to find the Nash equilibrium as the solution. Through theoretical analysis and experimental evaluation, we demonstrate that our approaches can solve the EDM problem effectively and efficiently.

Edge Computing, extending cloud computing, has emerged as a prospective computing paradigm. It allows a SaaS (Software-as-a-Service) vendor to allocate its users to nearby edge servers to minimize network latency and energy consumption on their devices. From the SaaS vendor's perspective, a cost-effective SaaS user allocation (SUA) aims to allocate maximum SaaS users on minimum edge servers. However, the allocation of excessive SaaS users to an edge server may result in severe interference and consequently impact SaaS users data rates. In this paper, we formally model this problem and prove that finding the optimal solution to this problem is NP-hard. Thus, we propose ISUAGame, a game-theoretic approach that formulates the interference-aware SUA (ISUA) problem as a potential game. We analyze the game and show that it admits a Nash equilibrium. Then, we design a novel decentralized algorithm for finding a Nash equilibrium in the game as a solution to the ISUA problem. The performance of this algorithm is theoretically analyzed and experimentally evaluated. The results show that the ISUA problem can be solved effectively and efficiently.

To support the increasing computation-intensive applications in the Internet of Things (IoT), edge computing is introduced to provide mobile devices computing resources for performing low-latency tasks. Therefore, how to design an effective and secure computing resource allocation mechanism is attracting increasing attention. A lot of works have been done to design an effective computational resource market for IoT, but the problems of vulnerability and inefficiency still exist. In this paper, we propose a two-level Stackelberg game-based computing resource trading mechanism for mobile IoT devices with a credit-based payment approach, which is implemented by smart contracts on blockchain. In our model, the Stackelberg game consists of two levels, i.e., leader-level and user-level. In the leader-level, the computing service provider (CSP) and its agent constitute a composite leader. The agent purchases computing resource from CSP on credit and acts as a broker among leader-level and user-level reselling these computing resources to users. In the user-level, every user experiences social externality, which means users are interdependent. The leader-level subgame makes credit payment easier by making loaning and trading become a joint credit payment. The user-level subgame makes the market more active and closer to reality by introducing social externality. Besides, smart contracts can prevent malicious behaviors such as delay payment. We also conduct equilibrium analysis and prove the existence and uniqueness of the Nash equilibrium in our Stackelberg game-based model. Finally, we conduct numerical experiments to evaluate the cost of smart contracts and the performance of each entity with the proposed pricing mechanism.

We propose a game attack-defense graph (GADG) approach that integrates the attack-defense graph and the game theory to model and analyze cyberattacks and defenses in the local metering system (LMS). Different from previous studies concentrating on static analyses of cybervulnerabilities, the GADG method considers correlations among these vulnerabilities. Besides, to avoid the uncertainty brought by unilateral analysis, we introduce the game theory for the interaction analysis between the attacker and the defender. The mixed-strategy Nash equilibrium that they finally reach can serve as the input for the inference of system states. We also propose two eliminating algorithms to reduce the complexity of solving mixed-strategy Nash equilibrium on large-scale LMS. In the case study, our GADG model has been conducted on a real LMS, and the results prove its efficiency. This research aims to assist the power company in optimizing the allocation of limited defensive resources, especially specific in attacking steps.

This paper studies a setting in which multiple suppliers compete for a buyer's procurement business. The buyer faces uncertain demand and there is a requirement to reserve capacity in advance of knowing the demand. Each supplier has costs that are two dimensional, with some costs incurred before demand is realized in order to reserve capacity and some costs incurred after demand is realized at the time of delivery. A distinctive feature of our model is that the marginal costs may not be constants, and this naturally leads us to a supply function competition framework in which each supplier offers a schedule of prices and quantities. We treat this problem as an example of a general class of capacity games and show that, when the optimal supply chain profit is submodular, in equilibrium the buyer makes a reservation choice that maximizes the overall supply chain profit, each supplier makes a profit equal to their marginal contribution to the supply chain, and the buyer takes the remaining profit. We further prove that this submodularity property holds under two commonly studied settings: (1) there are only two suppliers; and (2) in the case of more than two suppliers, the marginal two-dimension costs of each supplier are non-decreasing and constant, respectively., Comment: 35 pages

In this article, we investigate the distributed Nash equilibrium (NE) seeking problem for the multi-cluster game under switching communication topologies. Specifically, the communication topology switches between a group of jointly connected digraphs. First, a new distributed NE seeking algorithm for the multi-cluster games is designed by the consensus protocol and gradient play rule under the switching communication topologies. Furthermore, in order to make the algorithm still applicable when the agent only knows part of the decision information, the leader-following consensus protocol is used to generate the estimates for all agents action in the cluster under the assumption that the switching topology between clusters is directed and strongly connected. A more general NE seeking algorithm for the multi-cluster games is designed. For these two algorithms, the results of local convergence and non-local convergence are given, respectively. Two examples verify the validity of the theoretical results.

In this paper, we investigate the behavior of agents with bounded rationality, attacking a set of stochastic sensors measuring the state of a binary event. The coordination problem between the attackers is formulated as a multi-player non-zero-sum one-shot game. Each attacker aims to maximize the probability that a certain detector will produce an erroneous estimate of the true event, while they remain stealthy. To better predict the outcome of this game, we categorize the players based on the number of strategic thinking steps they will take. Each level-k attacker behaves based on subjective beliefs of the others' behaviors, which are quantified via a Poisson distribution over the lower levels. The expected best responses for the type of each attacker are derived. The limiting conditions as the cognitive levels increase, as well as when the attackers fully coordinate, are shown to converge to the Nash equilibrium.

We consider a queueing system which opens at a given point in time and serves a finite number of users according to the last-come first-served discipline with preemptive-resume (LCFS-PR). Each user must decide individually when to join the queue. We allow for general classes of user preferences and service time distributions and show existence and uniqueness of a symmetric Nash equilibrium. Furthermore, we show that no continuous asymmetric equilibrium exists, if the population consists of only two users, or if arrival strategies satisfy a mild regularity condition. For an illustrative example, we implement a numerical procedure for computing the symmetric equilibrium strategy based on our constructive existence proof for the symmetric equilibrium. We then compare its social efficiency to that obtained if users are instead served on a first-come first-served (FCFS) basis.

The increasing popularity of ride-hailing applications has given rise to a new channel in which ride-hailing platforms are bundled into aggregation platforms to earn additional orders by charging commissions and slotting fees. Such bundled channels, unlike traditional reseller electronic ones, may flutter prices, service levels, market demands, and then further affect their profits. These divergent attitudes raise an interesting and key question about whether and under what conditions bundled channels should be introduced to ride-hailing platforms. In this paper, we provide an analytical framework for ride-hailing and aggregation platforms in unbundled and bundled scenarios, respectively. We build a Stackelberg game model in which ride-hailing and aggregation platforms as leaders obtain prices by constructing Nash equilibria, while drivers as followers determine service levels given to two platforms. Drivers’ best responses in terms of service levels for two platforms, as well as platforms’ optimal pricing strategies and profits are achieved. To capture access conditions of the ride-hailing platform and the profit contention between two platforms, we further conduct sensitivity analysis on cost coefficients of service levels, price and cross-price substitutions, service level and cross-service level substitutions, revenue-sharing ratio, cost, as well as commission and slotting fee. Based on numerical examples and analysis of the results, some interesting managerial insights about bundling strategies are gained for ride-hailing platforms.

Efficient grant-free uplink transmission is critical in minimizing Age of Information (AoI) in multi-channel Internet of Things (IoT) networks. But less attention has been paid to this topic especially when dynamic channel access attacks (DCAAs) exist. To bridge this gap, this paper formulates the distributed channel access problem in AoI-oriented IoT networks, and then a reinforcement learning-based solution is put forward based on the theoretical results of the game theory. Firstly, a utility maximization problem is formulated for each sensor node based on its average AoI under DCAAs with probabilistic ACK feedback. Secondly, the problem is transformed into two ordinary potential game (OPG) models, which are both proved to have at least one Nash Equilibrium (NE); and a distributed learning algorithm is proposed to reach the NE. Finally, extensive simulations are conducted to evaluate the proposal’s performance. Simulation results verify the effectiveness of the proposed algorithm in various parameters settings.

We aim to address the Nash equilibrium (NE) seeking problem for multiple players over Markovian switching communication networks in this article, where a new type of distributed synchronous discrete-time algorithm is proposed and utilized. Specifically, each player in the present game model is assumed to employ a gradient-like projection algorithm to choose its action based upon the estimated ones for all the others. Under the mild condition that the union network of all communication network candidates is connected, we show that the players' actions could converge to an arbitrarily small neighborhood of the NE in the mean-square sense by adjusting the algorithm parameters. It is further found that the unique NE is mean-square stable when it is not restricted by any constraint set. In addition, we show that the proposed distributed discrete-time NE seeking algorithm can be utilized to deal with the energy trading problem in microgrids where each microgrid is modeled as a rational player using a purchase price as its action to buy energy from other microgrids with surplus supplies. The energy market allocates the excess energy according to the principle of proportional distribution. Some numerical simulations are finally presented to verify the validity of the present discrete-time NE seeking algorithm in solving the energy trading problem.

We study the production/inventory competition problem between two manufacturers each adopting the co-production system to process a raw material into two products in fixed proportions that are substitutable between the manufacturers. We first consider the one-period problem where the demands for the two manufacturers’ products are uncertain and independent. The unmet demand of one manufacturer's product can be met by the other manufacturer's leftover stock of the same product, if available, and is lost otherwise. The manufacturers compete for the substitute demands by choosing their own purchase/processing quantities. We show the existence of the unique Nash equilibrium processing quantity, and examine the impact of competition by comparing the total equilibrium quantity of the game with that of the centralized-decision case. We then study the multi-period production/inventory competition problem in which the manufacturers make decisions in each period according to the initial inventory levels of the products and the rivals’ decisions. We show the existence and uniqueness of the Nash equilibrium strategy and that the equilibrium strategy has the simple base-stock structure. We then consider the case where the manufacturers’ demands are correlated. In addition, studying the case where the manufacturers are capacitated, we show that the unique Nash equilibrium exists and has a modified base-stock structure. We further generalize the results to the infinite-period case and the case with more than two products. Finally, numerical studies are conducted to investigate the impacts of the model parameters on the equilibrium outcomes and get some managerial insights from the analytical findings.

In the multi-access edge computing (MEC) environment, app vendors’ data can be cached on edge servers to ensure low-latency data retrieval. Massive users can simultaneously access edge servers with high data rates through flexible allocations of transmit power. The ability to manage networking resources offers unique opportunities to app vendors but also raises unprecedented challenges. To ensure fast data retrieval for users in the MEC environment, edge data caching must take into account the allocations of data, users, and transmit power jointly. We make the first attempt to study the Data, User, and Power Allocation (DUPA $^3$ 3 ) problem, aiming to serve the most users and maximize their overall data rate. First, we formulate the DUPA $^3$ 3 problem and prove its $\mathcal {NP}$ NP -completeness. Then, we model the DUPA $^3$ 3 problem as a potential DUPA $^3$ 3 game admitting at least one Nash equilibrium and propose a two-phase game-theoretic decentralized algorithm named DUPA $^3$ 3 Game to achieve the Nash equilibrium as the solution to the DUPA $^3$ 3 problem. To evaluate DUPA $^3$ 3 Game, we analyze its theoretical performance and conduct extensive experiments to demonstrate its effectiveness and efficiency.

In this paper, we present a resource allocation mechanism for the problem of incentivizing filtering among a finite number of strategic social media platforms. We consider the presence of a strategic government and private knowledge of how misinformation affects the users of the social media platforms. Our proposed mechanism incentivizes social media platforms to filter misleading information efficiently, and thus indirectly prevents the spread of fake news. In particular, we design an economically inspired mechanism that strongly implements all generalized Nash equilibria for efficient filtering of misleading information in the induced game. We show that our mechanism is individually rational, budget balanced, while it has at least one equilibrium. Finally, we show that for quasi-concave utilities and constraints, our mechanism admits a generalized Nash equilibrium and implements a Pareto efficient solution.

The multiplayer stochastic noncooperative tracking game (NTG) with conflicting target strategy and cooperative tracking game (CTG) with a common target strategy of the mean-field stochastic jump-diffusion (MFSJD) system with external disturbance is investigated in this study. Due to the mean (collective) behavior in the system dynamic and cost function, the designs of the NTG strategy and CTG strategy for target tracking of the MFSJD system are more difficult than the conventional stochastic system. By the proposed indirect method, the NTG and CTG strategy design problems are transformed into linear matrix inequalities (LMIs)-constrained multiobjective optimization problem (MOP) and LMIs-constrained single-objective optimization problem (SOP), respectively. The LMIs-constrained MOP could be solved effectively for all Nash equilibrium solutions of NTG at the Pareto front by the proposed LMIs-constrained multiobjective evolutionary algorithm (MOEA). Two simulation examples, including the share market allocation and network security strategies in cyber-social systems, are given to illustrate the design procedure and validate the effectiveness of the proposed LMI-constrained MOEA for all Nash equilibrium solutions of NTG strategies of the MFSJD system.

The stability property of the loss-aversion-based noncooperative switched systems with quadratic payoffs is investigated. In this system, each agent adopts the lower sensitivity parameter in the myopic pseudo-gradient dynamics for the case of losing utility than gaining utility, and both systems' dynamics and switching events (conditions) are depending on agents' payoff functions. Sufficient conditions under which agents' state converges towards the Nash equilibrium are derived in accordance with the location of the Nash equilibrium. In the analysis, the mode transition sequence and interesting phenomena which we call flash switching instants are characterized. Finally, we present several numerical examples to illustrate the properties of our results.

The ongoing quest for transportation electrification with the massive proliferation of EV charging stations (EVCSs) will deepen the interaction and require the further coordination of coupled power and transportation networks (PTN). The individually-owned EVCSs located in an urban transportation network (UTN) will compete using price signals to maximize their respective payoffs. In this paper, a multi-agent deep reinforcement learning (MA-DRL) method is proposed to model the pricing game in UTN and determine the optimal charging prices for a single EVCS. The EVCS charging demand is first analyzed using a modified user equilibrium traffic assignment problem (UE-TAP) with elastic traveling demands and different charging prices. The price competition problem is then formulated as a game with incomplete information in which the market environment is very complex due to nonlinear traffic assignments. Thus, the MA-DRL approach is proposed to learn the charging pricing strategies of multiple EVCSs and approximate the Nash Equilibrium (NE) of the pricing game using the incomplete information. The proposed solution will determine the optimal pricing strategies for an EVCS in UTN. The case studies on a 24-node Sioux-Falls network and real-world Xian city are conducted to verify the effectiveness of the proposed approach.

Evolutionary games and population dynamics are finding increasing applications in design learning and control protocols for a variety of resource allocation problems. The implicit requirement for full communication has been the main limitation of the evolutionary game dynamic approach in engineering tasks with various information constraints. This article intends to build population games and dynamics with both static and dynamical graphical communication structures. To this end, we formulate a population game model with graphical strategy interactions and derive its corresponding population dynamics. In particular, we first introduce the concept of generalized Nash equilibria for population games with graphical strategy interactions, and establish the equivalence between the set of generalized Nash equilibria and the set of rest points of its distributed population dynamics. Furthermore, the conditions for convergence to generalized Nash equilibrium and particularly to Nash equilibrium are obtained for the distributed population dynamics with both static and dynamical graphical structures. These results provide a new approach to design distributed Nash equilibrium seeking algorithms for population games with both static and dynamical communication networks, and hence, expand the applicability of the population game dynamics in the design of learning and control protocols under distributed circumstances.

This paper considers the privacy-preserving distributed Nash equilibrium seeking strategy design for average aggregative games, in which the players' objective functions are considered to be sensitive information to be protected. In particular, we consider that the game is free of central node and the aggregate information is not directly available to the players. As there is no central authority to provide the aggregate information required by each player to update their actions, a dynamic average consensus protocol is employed to estimate it. To protect the players' privacy, we perturb the transmitted information among the players by independent random noises drawn from Laplace distributions. By synthesizing the perturbed average consensus protocol with a gradient algorithm, a distributed privacy-preserving Nash equilibrium seeking strategy is established for the aggregative games under both fixed and time-varying communication topologies. With explicit quantifications of the mean square errors, the convergence results of the proposed methods are presented. Moreover, it is analytically proven that the proposed algorithm is μ-differentially private. The presented results indicate that there is a tradeoff between the convergence accuracy and the privacy level. Lastly, a numerical example is provided for the verification of the proposed methods.

Nowadays, with the explosive growth of sensor-based devices connected to Internet of Thing (IoT), massive amount of data are generated every day with potential tremendous value. We argue that the value of those data can be extracted through monetize data platform in IoT-Edge-Cloud ecosystems for many parts of the business. In such monetize data platform, the data can be computed and transformed into services in IoT-Edge-Cloud ecosystems and provide Data-As-A-Service (DAAS) for applications. The key to implement such a monetize data platform is to evenly distribute DAAS computing tasks to network devices to maximize the benefits of the system. So, in this paper, we study the Task Type-based Computation Offloading algorithm (TTCO) to implement such platform. We use the "IoT-Edge-Cloud" three-layer multi-hop model, which is closer to the complex scene in monetize data platform. We divide tasks into data-intensive tasks and CPU-intensive tasks, and then combine the cost model of computation offloading with task type to make data-intensive tasks prefer local computing and CPU-intensive tasks prefer offload computing, thereby reducing the monetize data platform transmission volume and improving the overall quality of computation offloading. We then use a hierarchical game model combined with fictitious play to solve the Nash Equilibrium (NE) of the system and obtain the mixed strategies of the devices. Finally, we propose a TTL-constrained flood strategy transmission mechanism to make the algorithm apply to practice. The experimental results prove that our algorithm has a large performance gain in various scenarios, which can be severed as a monetize data platform for IoT-Edge-Cloud ecosystems.

Aeronautical Ad hoc Networking (AANET) of the air vehicles is envisioned to support future enhanced applications, such as free flight and in-flight Internet, and AANET serves as the middle layer of the air-space-ground integrated network, bridging the space and the ground components. However, intermittent connectivity is the greatest challenge to an AANET. To deal with the intermittence, the air vehicle acts as a relay in an AANET using the buffer onboard to temporally cache the data when an interruption occurs, known as opportunistic transmission. For the highly dynamic topology of an AANET, we use a sampled dynamic graph to capture significant variations while ignoring trivial changes for avoiding extra complexity. And thus we formulate a coalitional game incorporating with the dynamic graph for the AANET to obtain the optimal transmission schedule in terms of the effective throughput with limited transmission delay. The corresponding coalitional dynamic graph game algorithm will then generate an approximately optimal AANET formation, which converges to Nash equilibrium within finite iterations. The simulations conducted with the real flight data show that 700 Mbit of buffer size onboard and 1400 Mbit of buffer in the Internet Gateway Station (IGS) are the optimal settings for the opportunistic transmission, and the coalitional dynamic graph game algorithm outperforms the geographic location based greedy perimeter stateless routing algorithm in terms of the total received data amounts.

In recent years internet auctions have attracted much attention. This paper discusses the "last minute bidding" (sniping) phenomenon, first investigated by Roth and Ockenfels (American Economic Review 92, 1093-1103) in eBay, fixed-end, auctions. Unlike standard auctions where each offer is processed by the auctioneer due to system traffic, and connection time, in eBay auctions very late bids may be left unprocessed. In the paper we consider a simple two-players, two-periods sealed-bid auction model, inspired by eBay auctions, with private values and complete information. The main difference with respect to e-Bay auctions is given by the first period. Indeed, players can submit at most one bid per period. In the first stage bids are processed with certainty by the system while in the second stage, with positive probability, bids may not be processed due to system congestion. Unlike eBay we show how in this model last minute, that is second stage only, bidding may have low plausibility, as it can be a Nash equilibrium only under very specific circumstances and is never a unique best reply. Intuitively this is because the first stage is sealed-bid, with at most one offer per bidder, and players have no guarantee that they could outbid the opponent in the second period, once after the first round they realise that their bid is not the best.

In this paper, we study Nash equilibria for real R&D options in a context of non-cooperative oligopoly under fuzzy uncertainty in market shares and revealed information. In particular, we consider a duopoly fuzzy game in which each firm has two available strategies, that is to invest or to defer its research investment. The deferring option gives the benefit of the information revelation about market reaction but, on the other hand, the disadvantage of a lower market share. Since these two effects are not easily to quantify, we describe the revealed information and the market shares as fuzzy numbers. As shown by numerical examples, the proposed fuzzy framework allows to reach different firms' strategic behaviours and new Nash equilibria with respect to the crisp case.

Area coverage and network lifetime are two contradictory issues to the architecture development of a wireless sensor network (WSN). A satisfactory balance could be achieved by deploying abundant sensor nodes randomly and dividing them into k exclusive cover sets. Toward self-organized partition with higher efficiency, we address the problem from the perspective of networked potential games and propose a memorial mixed-response algorithm (MMRA), which is implemented in a distributed and synchronous manner. Being viewed as a game player, each sensor node first updates its memory using a temporary action, which is generated by following a mixed response rule. After this, the coordination evolves into the next iteration by each player randomly drawing an action from its memory with equal probabilities. We prove that our algorithm converges with probability 1 to a convention of Nash equilibria, with the worst approximation ratio strictly larger than 0.5. Moreover, it is also found that a tradeoff between solution efficiency and computation time could be achieved via the adjustment of the amount of randomness introduced via the memory length m as well as the probability pₘ, where better partition results are more likely to be generated using a larger m and smaller pₘ. Comparisons with existing distributed methods demonstrate the superiority of our method in terms of solution refinement as well as convergence speed.

This paper investigates the resilient control problem of networked switched systems against denial-of-service (DoS) attack. The resilient control strategies include the security game of the network layer and the optimal control for the physical layer. Firstly, an N-coalition noncooperative game is utilized to characterise the network security game between an intrusion detection system (IDS) and the attacker, where their strategy sets are obtained from the Nash equilibrium. Then, by the dynamic programming principles, the optimization strategy is developed to obtain the optimal controller gains for the subsystems. Meanwhile, for ensuring the optimality of the switched system, a genetic algorithm is adopted to search for the optimal switching signal. Finally, the proposed methods are applied to a networked continuous stirred tank reactors (CSTR) system which demonstrates its effectiveness.

In this paper, the generalized Nash equilibrium problem over a fuzzy domain is considered. An existence result of the solution is given. Then, on the basis of the variational inequality approach, an algorithm is developed to solve the fuzzy Nash equilibrium problem. In addition, a stability analysis of the solution set is provided while the membership function is perturbed by a parameter. Finally, the fuzzy Nash equilibrium formulation is applied to the study of the conflicts faced by the fashion and textile industry: production and pollution.

Multi-agent policy evaluation and seeking are longstanding challenges in developing theories for multi-agent reinforcement learning (MARL), due to multi-dimensional learning goals, non-stationary environment and scalability issues in the joint policy space. This paper 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 (SBRD) 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 RL than -rank which relies on weakly better responses. We first consider settings where the difference between largest and second largest underlying 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