A complete solution for two-person, single-stage, deterministic, stackelberg game

We study the problem of optimizing the performance of a system shared by selfish, noncooperative users. We consider the concrete setting of scheduling jobs on a set of shared machines with load-dependent latency functions specifying the length of time necessary to complete a job; we measure system performance by the total latency of the system.Assigning jobs according to the selfish interests of individual users (who wish to minimize only the latency that their own jobs experience) typically results in suboptimal system performance. However, in many systems of this type there is a mixture of “selfishly controlled” and “centrally controlled” jobs; as the assignment of centrally controlled jobs will influence the subsequent actions by selfish users, we aspire to contain the degradation in system performance due to selfish behavior by scheduling the centrally controlled jobs in the best possible way.We formulate this goal as an optimization problem via Stackelberg games, games in which one player acts a leader (here, the centralized authority interested in optimizing system performance) and the rest as followers (the selfish users). The problem is then to compute a strategy for the leader (a em Stackelberg strategy) that induces the followers to react in a way that (at least approximately) minimizes the total latency in the system.In this paper, we prove that it is NP-hard to compute the optimal Stackelberg strategy and present simple strategies with provable performance guarantees. More precisely, we give a simple algorithm that computes a strategy inducing a job assignment with total latency no more than a constant times that of the optimal assignment of all of the jobs; in the absence of centrally controlled jobs and a Stackelberg strategy, no result of this type is possible.

Cancer treatment faces significant challenges due to tumor adaptation, heterogeneity, and rapid resistance development, where sequential therapies often fail against the evolving dynamics of cancer cells. Traditional methods, such as fixed-dose regimens, overlook the adaptive, game-like interactions between oncologists and tumors, leading to suboptimal outcomes and limited personalization in precision medicine. To address these limitations, we introduce Nash-RL, an innovative framework that integrates game theory and reinforcement learning (RL) to enable optimized, evolutionarily informed cancer therapy. By modeling the oncologist-tumor interaction as a Stackelberg game, where the oncologist acts as an RL-trained leader optimizing strategies and the tumor as an evolutionary follower adapting to pressures, our approach anticipates resistance mechanisms and captures the sequential decision-making process inherent in adaptive treatments. Nash-RL combines key game-theoretic concepts, including Nash equilibrium, Stackelberg strategies, and evolutionary games, with multi-agent RL algorithms to develop robust, patient-specific treatment protocols. This integration not only enhances therapy efficacy by dynamically adjusting doses but also delays resistance development, advancing fields like adaptive therapy and precision oncology. We formulate the mathematical foundations, incorporating state-action-reward models and equilibrium computations, and outline computational implementation using deep Q-networks and policy gradient methods. Theoretical advantages over existing RL or game-theoretic approaches are analyzed, with proof-of-concept simulations demonstrating practical feasibility. Overall, Nash-RL provides a scalable foundation for sustainable strategies that collaborate with, rather than combat, cancer's adaptive nature, paving the way for intelligent, personalized oncology and potentially transforming clinical practice.

In this article, an output‐feedback Q‐learning algorithm is proposed for the discrete‐time linear system to deal with the tracking control problem. The problem is formulated as a zero‐sum game in the Stackelberg game framework with a discount factor to make the value function bounded. According to the principle of optimality, the game algebraic Riccati equation (GARE) is derived and solved by the Q‐learning algorithm to get the optimal solution of the Stackelberg game without requiring the knowledge of system dynamics and state. It is proved that the solution of the algorithm converges to the optimal control input and the worst‐case disturbance with excitation noises during training, and the Stackelberg strategy can achieve a lower disturbance attenuation level than the Nash one. Moreover, the impacts of the discount factor on the stability of the closed‐loop system and solvability of the GARE are analyzed to provide some criteria for the choice of the discount factor. Simulation examples are provided to validate the effectiveness of the algorithm.

Solutions and properties of multi-stage stackelberg games

Closed-Loop Strategy for Stackelberg Game Problem with Incomplete Information Structures

Wireless virtualization is becoming a popular solution to resource sharing problems integral in providing access to future wireless networking applications by enabling access to the same physical wireless infrastructure by multiple virtual networks concurrently. This paper investigates cyber deception using game theory model for defining the deceptive interactions between a cyber attacker and the defender. We consider both the naive attacker who is not aware of the defender's strategy, and the rational (powerful) attacker who knows the defense strategy. In the Stackelberg game, the defender controls the observable configuration of the network when probed by an attacker. We show that the process for selecting an optimal defense strategy against a naive attacker is NP-hard. We also proposed an algorithm for computing optimal strategy for a naive attacker and a greedy based algorithm for selecting the optimal strategy against a powerful attacker. Results from extensive simulation runs show the scalability and effectiveness of the proposed approaches based on the run time in finding the optimum defense strategy as well as minimizing the attacker's utility are better than the approaches in state of the art literature.

The derivation is given of the closed-loop Stackelberg strategy for a class of continuous three-player non-zero-sum differential games using the idea of the team optimal Stackelberg strategy. The game systems are described by linear state dynamics and quadratic objective functions. First, a definition of the three-player hierarchical equilibrium is given, then a general theorem which was developed by Basar (1981) is examined and applied to the game under consideration to deduce some sufficient conditions for the solution of the game to exist. A simple example is given.

According to the IEEE 802.15.6 standard, interference within each wireless body area network (WBAN) can be well addressed by the time division multiple access (TDMA)-based media access control (MAC) protocol. However, the inter-WBAN interference will be caused after multiple WBANs are gathered together. This paper proposes a priority-aware price-based power control (PPPC) scheme for mitigating the inter-WBAN interference. Specifically, to maximize the transmission data rate of sensors and control the aggregate interference suffered by coordinators, a Stackelberg game is established, in which the coordinators issue interference prices and the active sensors adjust their transmission power accordingly. On the other hand, since the information about the identities of the active sensors in a specific time slot is kept private, a Bayesian game is designed to model the interaction among sensors. Moreover, the timeliness and reliability of data transmission are guaranteed by designing the sensors’ priority factors and setting a priority-related active probability for each sensor. At last, a power control algorithm is designed to obtain optimal strategies of game players. Simulation results show that compared with other existing schemes, the proposed scheme achieves better fairness with a comparable network sum data rate and is more energy efficient.

Real time Demand-Response algorithm is a real time price (RTP) based demand response (DR) algorithm which helps in obtaining optimal control of load devices in a facility. We form a virtual electricity trading process, where energy management center (EMC) calculates the virtual retail prices and allots to each device, from which devices purchases energy. Here EMC acts as the leader and devices in the facility are followers. So the Stackelberg game between one-leader and M-followers is formulated to extract interactions between them, and optimization problems are formed for each player of the game to assist in finding the optimal strategy. We also demonstrated the unique Stackelberg equilibrium of the Stackelberg game which gives the optimal strategy or optimal energy demands for each device. The results of the simulation showed that Stackelberg game based DR algorithm is successful to get the optimal load control of device at all RTP throughout the day.

Closed-loop supply chain (CLSC), especially the reverse logistics system with the third-party reverse logistics providers (3PRLP) is very important for materials return, remanufacturing, repair, recovery, recycling, and reusing. In this paper, based on a one-leader and multi-follower Stackelberg game with the manufacturer being the leader, we model a multi-echelon CLSC with 3PRLP under the consideration of impacts of environmental legislation on scrap recycling. Furthermore, in order to encourage the 3PRLP to exert him to return activities we design a target rebate-punish contract between the manufacturer and the 3PRLP under both stochastic price-dependent demands and stochastic effort-dependent returns. We analyze the closed-form analytic expressions for both united optimization strategies in centralized closed-loop system and the Stackelberg strategies in decentralized system, and conclude that the wholesale price contract may not coordinate the forward supply chain and the target rebate-punish may coordinate the reverse chain under certain conditions. Finally, according to the practice of Chinese steel enterprises we carry out some simulation experiments to validate our theoretical analyses and gain some insights into the impacts of reverse logistics activities on the practitioners operations and decisions.

Idle-parked vehicles assisted collaborative resource allocation in VEC based on Stackelberg game

Optimal encryption strategy for cyber-physical systems against stealthy attacks with energy constraints: A Stackelberg game approach

UAV-assisted ISAC network physical layer security based on Stackelberg game

With the development of the peak load regulation ancillary service market and the improvement of the rules, the role of thermal power unit changed from the passive participant in peak load regulation for duty to the active bidder. The power grid dispatching agency called the depth peak load regulation (DPLR) resources on demand according to the price. In order to research the optimal bidding strategy of thermal power units in the depth peak load regulation ancillary service market, a bi-Ievel model of multiple thermal power manufacturers and single market dispatching institution is established based on stackelberg game. The upper level is the bidding strategy of thermal power manufacturers, including the income and cost of thermal power manufacturers. The lower level is the optimal scheduling scheme of the market dispatching institution. The nonlinear bi-Ievel model is solved by the particle swarm optimization algorithm. The cases show the importance and validity of the proposed model

When the system contains slow and fast modes, the Stackelberg game is numerically stiff and ill-conditioned. It is necessary to find some methods to alleviate this ill-conditioning and to reduce the amount of computation. In this paper, we consider the linear-quadratic infinite-time Stackelberg game for the discrete-time systems with slow and fast modes. A team-near optimal incentive strategy is constructed by solving a series of well-conditioned and reduced-order problems. Well-posedness of this near-optimal incentive strategy is investigated. By using this near-optimal incentive strategy to the full-order Stackelberg game, we can assure that the resulting value of the cost function for the leader has the same limit with the full-order team cost value as a small positive parameter ? tends to zero.