Equilibrium Of Game Research Articles

Stationary Markov Equilibrium Strategies in Asynchronous Stochastic Games: Existence and Computation

We study Asynchronous Dynamic games and show that in games with a finite state space and finite action sets, one can obtain the pure strategy Markov perfect equilibrium by using a simple backward induction method when the time period for the game is finite. The equilibrium strategies for games with an infinite horizon are then obtained as the point-wise limit of the equilibrium strategies of a sequence of finite horizon games, where the finite horizon games are truncated versions of the original game with successively longer time periods. We also show that if the game has a fixed K-period cycle, then there is a stationary Markov equilibrium. Using these results, we derive an algorithm to compute the equilibrium strategies. We test the algorithm in three experiments. The first is a two-player asynchronous game with three states and three actions. In the second experiment, we compute the equilibrium of a cybersecurity game in which there are two players, an attacker and a defender. In the third experiment, we compute the stationary equilibrium of a duopoly game with two firms that choose an output in alternate periods.

Bidding Strategy for the Alliance of Prosumer Aggregators in the Distribution Market

Photovoltaic energy storage system (PV-ESS) prosumer aggregators are characterized by a large number but small scale in the distribution system and are not competitive enough to participate in market transactions. For this reason, a prosumer aggregator alliance is proposed to participate in the distribution market bidding strategy. Firstly, based on the framework for prosumer aggregator alliances participating in distribution market trading, a bilevel bidding model is constructed. The upper level represents the optimal decision-making model for the prosumer aggregators, while the lower level constitutes the distribution market-clearing model. Secondly, the additional benefits obtained by the alliance are distributed more fairly using the improved Shapley value based on the PV self-consumption rate. Given the problem that the traditional diagonalization algorithm (DA) has an excessive number of iterations when solving the game equilibrium problem of multiple subjects, the DA is improved by optimizing the initial value of the inputs. Finally, case studies are conducted based on the improved IEEE-33 bus distribution system to validate the feasibility and economic viability of the proposed strategy. The case study results show that forming cooperative alliances to participate in market bidding can significantly increase overall profits. The improved DA reduces the number of bids and computation time by 75% and 80%, respectively. Additionally, the improved Shapley value facilitates compensation for some of the aggregators.

Agricultural product harvest equilibrium with transportation bottleneck and random disasters

The decentralized planting practices of farmers often result in non-cooperative harvesting and transportation of agricultural products. The high concentration of harvesting demand can cause transportation and processing bottlenecks, resulting in queuing congestion in planting, which in turn affects farmers’ decisions on harvest timing. Thus, the fully competed decision-making process leads to a game equilibrium. To solve this harvest equilibrium problem, a nonlinear equation model with transportation bottleneck capacity constraints is developed. Assumptions to be made are that the value of the pre-harvest product is a non-negative continuously differentiable strictly concave function about the harvest time, and that post-harvest losses are a linear increasing function of queuing time. Subsequently, the model is further extended to incorporate the effects of random natural disasters. Additionally, this paper derives optimal harvest schedules for comparison. A numerical example is used to analyze how the related parameters and the probability of disasters influence farmers’ harvest decisions and the unit benefits of agricultural produce. The analysis results show that a larger total volume of agricultural products prompts earlier harvesting and reduces unit benefits in both equilibrium and optimal harvesting scenarios. Expanding bottleneck capacity allows farmers to harvest closer to the optimal time and increases unit benefits, although the marginal increase diminishes at the margin as bottleneck capacity increases. Increasing the queuing penalty within a range leads to an earlier harvest and reduces the unit benefits in equilibrium and optimality until a critical value is reached, beyond which it has no effect. A higher probability of natural disasters will incentivize farmers to harvest earlier, which in turn will reduce unit benefits.

Carbon emission reduction dynamic decision-making in the electricity supply chain with different carbon emission allowance principles

To address the impacts of typical principles for allocating initial carbon emission allowance, namely, the grandfathering and benchmarking principles on long-term emission reduction strategies, this paper presents four differential game models for an electricity supply chain consisting of a single electricity supplier and a single electricity retailer. The equilibrium solutions for different models are obtained and comparatively analyzed. The results show that the power supplier exhibits greater emission reduction efforts and, therefore, that the retailer has higher profits under the benchmarking principle than under the grandfathering principle. Moreover, under the two principles, the level of the power supplier’s emission reduction efforts is related to the decision-making modes of the electricity supply chain. Under specific conditions, the power supplier’s profit is also higher under the benchmarking principle than under the grandfathering principle. In addition, this study extends the models to investigate the constrained differential game models and obtains the game equilibrium solutions, which can be proved as saddle point equilibria.

Rationalizable self-confirming equilibrium in static games with unawareness

Rationalizable self-confirming equilibrium in static games with unawareness

Pairwise Stability in Weighted Network Formation Games: Selection and Computation

This paper is concerned with the selection and computation of pairwise stable networks when agents have differentiable and concave utility functions. We show that a pairwise stable network can be obtained by finding a Nash equilibrium of a noncooperative game played by the nodes and links in the network. Based on this observation, we introduce a logarithmic tracing procedure and a path-following algorithm for network formation games. We apply the algorithm to several models in the literature and make comparisons with two existing algorithms: a path-following algorithm based on the linear tracing procedure (LinTP) and a decompose and exhaustive search method (DaE). Numerical results indicate that the proposed method is more than four times as efficient as LinTP. Although DaE demonstrates exceptional efficiency for small-scale problems, our method outperforms it significantly for large-scale problems, where DaE may fail to find a solution. We also show that the decomposition technique of DaE can be used to further accelerate our algorithm for a special class of problems. History: Accepted by Antonio Frangioni, Area Editor for Design & Analysis of Algorithms–Continuous. Funding: This work was supported in part by the National Natural Science Foundation of China [Grants 12201289, 12122107, and 72394363/72394360], the Natural Science Foundation of Jiangsu Province [Grant BK20220754], the Guangdong Basic and Applied Basic Research Foundation [Grant 2021A1515110207], the Young Elite Scientists Sponsorship Program by CAST [Grant 2023QNRC001], and the Open Research Fund from the Guangdong Provincial Key Laboratory of Big Data Computing [Grant B10120210117-OF05]. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2024.0546 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2024.0546 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

Generalized Nash Equilibrium Seeking for Noncooperative Game With Different Monotonicities by Adaptive Neurodynamic Algorithm.

This article proposes a novel adaptive neurodynamic algorithm (ANA) to seek generalized Nash equilibrium (GNE) of the noncooperative constrained game with different monotone conditions. In the ANA, the adaptive penalty term, which acts as trajectory-dependent penalty parameters, evolves based on the degree of constraints violation until the trajectory enters the action set of noncooperative game. It is shown that the trajectory of the ANA enters the action set in finite time benefited from the adaptive penalty term. Moreover, it is proven that the trajectory exponentially (or polynomially) converges to the unique GNE when the pseudo-gradient of cost function in noncooperative game satisfies strong (or "generalized" strong) monotonicity. To the best of our knowledge, this is the first time to study the polynomial convergence of GNE seeking algorithm. Furthermore, when the pseudo-gradient mentioned above satisfies monotonicity in general, based on Tikhonov regularization method, a new ANA for finding its ε -generalized Nash equilibrium ( ε -GNE) is proposed, and the related exponential convergence of the algorithm is established. Finally, the river basin pollution game and 5G base station location game are given as examples to showcase the algorithm's effectiveness.

Stability of pairwise social dilemma games: Destructive agents, constructive agents, and their joint effects.

Destructive agents, who opt out of the game and indiscriminately harm others, paradoxically foster cooperation, representing an intriguing variant of the voluntary participation strategy. Yet, their impact on cooperation remains inadequately understood, particularly in the context of pairwise social dilemma games and in comparison to their counterparts, constructive agents, who opt out of the game but indiscriminately benefit others. Furthermore, little is known about the combined effects of both agent types on cooperation dynamics. Using replicator dynamics in infinite and well-mixed populations, we find that contrary to their role in facilitating cooperation in multiplayer games, destructive agents fail to encourage cooperation in pairwise social dilemmas. Instead, they replace defection in the prisoners' dilemma and stag-hunt games. Similarly, in the chicken game, they can destabilize or replace the mixed equilibrium of cooperation and defection, undermining cooperation in the harmony (trivial) game. Conversely, constructive agents, when their payoffs exceed their contributions to opponents, can exhibit effects similar to destructive agents. However, if their payoffs are lower, while they destabilize defection in prisoners' dilemma and stag-hunt games, they do not disrupt the cooperation equilibrium in harmony games and have a negligible impact on the coexistence of cooperation in chicken games. The combination of destructive and constructive agents does not facilitate cooperation, but instead generates complex evolutionary dynamics, including bistable, tristable, and quadstable states, with outcomes contingent on their relative payoffs and game types. These results, taken together, enhance our understanding of the impact of the voluntary participation mechanism on cooperation, contributing to a more comprehensive understanding of its influence.

Battery recycling and coordination in information leakage prevention under blockchain technology in a new energy vehicles supply chain

Retired battery recycling and information leakage prevention under blockchain introduction intertwine in the new energy vehicle (NEVs) supply chain, but rare literature has jointly investigated them. Therefore, we characterize the vehicle goodwill under recycling and marketing efforts as well as the automobile demand function with and without blockchain introduction, identify the long-run discounted profits of the manufacturer and retailers in the supply chain, and investigate the Stackelberg differential game equilibrium under four scenarios: no blockchain introduction, manufacturer introduction, manufacturer-retailer cooperative introduction, and cooperative introduction plus wholesale price discounts for preventing information leakage. The results show that the manufacturer can introduce blockchain when costs are below a certain threshold, which increases the recycling rate of decommissioned batteries and improves goodwill, consumer demand, and marginal profits for the manufacturer and retailer; the retailer participation in the introduction of blockchain solves the free-rider problem and reduces the thresholds introduced, but does not affect battery recycling, goodwill, consumer demand, and wholesale and retail prices; wholesale price discounts effectively discourage the retailer from leaking information about the manufacturer and lead to improved profits for the retailer, but results in higher earnings for the manufacturer only when discounts are below a certain threshold.

Two-layer optimal scheduling of distribution network-multi-microgrids based on master–slave game

With the increase in the number of microgrids in the same distribution area usually belong to different subjects of interest, forming a multi-subject game pattern. Considering the interests of distribution networks and microgrids, a distribution network-multi-microgrid master–slave game model is established by selecting distribution networks as game masters and microgrids as game slaves. A master–slave game equilibrium algorithm based on a Kriging metamodel is proposed. The method replaces the microgrid energy internal management model with a proposed Kriging metamodel. In the iterative optimization process, the particle swarm optimization algorithm is used to generate new sampling points and modify the model in a targeted way so as to quickly and accurately obtain the transaction price and output plan of each microgrid. The algorithm does not need all the parameters of the microgrid, which both achieves the purpose of protecting the privacy of the microgrid and avoids a large number of calls to the lower optimization model, effectively reducing the amount of computation and improving the efficiency of the solution. The results show that the overall operating costs of the three microgrids used in the case study are reduced by 1.4%, 4.6%, and 1.6%, respectively, which effectively balances the interests of multiple parties in the microgrid system; the revenue of the distribution network is increased by 50.6%.

Pareto optimal provisions as outcomes of voluntary public good supply

It is shown that a large set of public good supply levels that arise in a Pareto optimal allocation can also be attained as public good provisions in the Nash equilibrium of a non-cooperative provision game. With a not very far-fetched assumption on agents’ preferences this result even holds for all Pareto supply levels.

PPSO and Bayesian game for intrusion detection in WSN from a macro perspective

The security of wireless sensor networks is a hot topic in current research. Game theory can provide the optimal selection strategy for attackers and defenders in the attack-defense confrontation. Aiming at the problem of poor generality of previous game models, we propose a generalized Bayesian game model to analyze the intrusion detection of nodes in wireless sensor networks. Because it is difficult to solve the Nash equilibrium of the Bayesian game by the traditional method, a parallel particle swarm optimization is proposed to solve the Nash equilibrium of the Bayesian game and analyze the optimal action of the defender. The simulation results show the superiority of the parallel particle swarm optimization compared with other heuristic algorithms. This algorithm is proved to be effective in finding optimal defense strategy. The influence of the detection rate and false alarm rate of nodes on the profit of defender is analyzed by simulation experiments. Simulation experiments show that the profit of defender decreases as false alarm rate increases and decreases as detection rate decreases. Using heuristic algorithm to solve Nash equilibrium of Bayesian game provides a new method for the research of attack-defense confrontation. Predicting the actions of attacker and defender through the game model can provide ideas for the defender to take active defense.

Cooperative Path Planning for Multiplayer Reach‐Avoid Games under Imperfect Observation Information

This article investigates a reach‐avoid game and proposes a cooperative path planning algorithm for a target–pursuers (TP) coalition to capture an evader. In the game, the target aims to bait and escape from the evader, and the pursuer aims to capture the evader. Due to imperfect observations, the TP coalition has uncertain information of the evader's state, while the evader is assumed to have perfect observation. The game model is constructed by formulating the optimization problems for each player in a receding horizon fashion. Then, to counter the evader effectively, the TP coalition constructs a virtual evader using the belief information from a Kalman filter. And a chance constraint optimization problem is constructed to predict the virtual evader's trajectory under uncertainties. The TP coalition can capture the actual evader by generating a robust counter‐strategy against the virtual evader with a chance constraint feasible set. Next, to compute the Nash equilibrium of the TP coalition's subjective game, an iterative algorithm is designed that combines the iterative best response and the distributed alternating direction method of multiplier algorithms. Finally, the effectiveness of the algorithm is validated through simulations and experiments.

Sample allocation method for maintainability test based on extended FMECA information

To ensure the accuracy of military equipment maintainability verification results, a sample allocation method for maintainability test based on extended failure mode, effects and criticality analysis (FMECA) information is proposed to address the problem that the influence factors are relatively one-sided and the weighting is subjective in the current test sample allocation method. The influence factors are analyzed and determined from three aspects: the fault occurrence characteristics, fault impact, and test economy. A method for calculating the fault propagation intensity based on the improved LeaderRank is proposed, which is introduced into the FMECA for expansion. It is proposed to use the CRITIC method to determine the objective weights of the influence factors, determine the subjective weights based on the relative importance of the influence factors, introduce game equilibrium to weigh and determine the combination weights, and then complete the sample allocation. Finally, an example showed that the influence factors determined by this method are more comprehensive, the weight results are more scientific, and the sample structure is more reasonable.

The cross-regional settlement methods in hospitals and the treatment-seeking behavior of patients with malignant tumors in China: an evolutionary game model.

Cross-regional settlement management is a key indicator of national health insurance system maturity. Given the significant demand for cross-regional medical treatment among Chinese patients with malignant tumors and the territorially managed health insurance system, further research is necessary to explore the relationship between hospital settlement methods and treatment-seeking behaviors among these patients. This study introduces and validates an evolutionary game model that provides a theoretical foundation for direct settlement policies in cross-regional treatment. An evolutionary game model was constructed with patients and hospitals serving as strategic players within a dynamic system. This model integrates the patients' treatment utility, medical and nonmedical costs, and hospitals' financial and technological advancement benefits. The evolutionary stability analysis revealed seven-game outcomes between hospitals and patients with malignant tumors. The numerical simulations suggest an evolutionary convergence toward strategy (1, 0), indicating a trend where patients with malignant tumors opt for cross-regional treatment, yet hospitals choose not to implement a direct settlement policy. Parameter sensitivity analysis showed that the parameters set in this study affected player behavioral choices and game equilibria. A strong demand for cross-regional medical treatment among Chinese patients with malignant tumors, and some hospitals require more incentives to implement cross-regional settlements. The key factors influencing the willingness of some patients with malignant tumors to resettle include the costs of in-area medical care, costs of cross-regional treatment without direct settlement, and the utility of cross-regional treatment. Technological advancement benefits and input costs influence some hospitals' motivation to adopt cross-regional settlements. Policy adjustments that effectively implement direct settlement policies can facilitate equilibrium, enhance the initiatives of some local health insurance management departments, improve the accessibility and efficiency of medical services, and reduce nonmedical expenses for patients.

Nash equilibrium, dynamics and control of congestion games with resource failures

Nash equilibrium, dynamics and control of congestion games with resource failures

Orbital Impulsive Pursuit–Evasion Game Formulation and Stackelberg Equilibrium Solutions

The spacecraft pursuit–evasion game in orbit with impulsive maneuvers is investigated based on dynamic game theory. First, the game models are developed progressively from the single-impulse game to the multi-impulse game. In the models, the two-sided optimal strategies of both players form a Stackelberg equilibrium in single-impulse games and constitute a multistage Stackelberg equilibrium in multi-impulse games. Then, the optimality necessary conditions for the two equilibriums are derived and simplified as two multipoint boundary value problems (MPBVPs). To handle the complicated MPBVPs, a rolling backward induction based on the Lambert algorithm accompanied by a homotopy shooting method is proposed to solve the equilibrium solutions. Simulations showed the effectiveness of the proposed methods and verified that the obtained solutions are the (multistage) Stackelberg equilibrium. Meanwhile, the differences between the Stackelberg equilibrium and the Nash equilibrium are illustrated in detail. Finally, an analysis of the influence of the observation delay on the game results is provided.

Social vs. individual age-dependent costs of imperfect vaccination

In diseases with long-term immunity, vaccination is known to increase the average age at infection as a result of the decrease in the pathogen circulation. This implies that a vaccination campaign can have negative effects when a disease is more costly (financial or health-related costs) for higher ages. This work considers an age-structured population transmission model with imperfect vaccination. We aim to compare the social and individual costs of vaccination, assuming that disease costs are age-dependent, while the disease’s dynamic is age-independent. A model for pathogen deterministic dynamics in a population consisting of juveniles and adults, assumed to be rational agents, is introduced. The parameter region for which vaccination has a positive social impact is fully characterized and the Nash equilibrium of the vaccination game is obtained. Finally, collective strategies designed to promote voluntary vaccination, without compromising social welfare, are discussed.

Exploiting game equilibrium mechanisms towards social trust-based group consensus reaching

In group decision making (GDM), the real or virtual moderator usually provides the suggestion and compensation based on consensus improvement for the decision maker (DM) to promote group consensus. In order to ensure DMs own interests, there are the game among DMs and the game between DM and moderator to effect the consensus reaching. In this case, we analyze the inherent game reasons and investigate the game-driven consensus model. Firstly, based on the subjective trust of social network and the evaluation consensus of DMs, and inspired by the idea of data envelopment analysis (DEA), this paper explores the quantitative relation between subjective trust and the DMs’ consensus to determine the DM’s weights. Then, under the compensation of consensus improvement, we formulate game strategy between DMs and the moderator to ensure the DM’s consensus, which impacts the cooperation behavior between DM and the moderator. Based on the game strategy, we further establish bi-level game consensus model and investigate Nash equilibrium of group consensus model. Finally, the experiments are used to illustrate the feasibility of the proposed method, and comparative analysis demonstrates its advantages.

A Grid-Wide Comprehensive Evaluation Method of Power Quality Based on Complex Network Theory

To achieve a hierarchical and quantitative evaluation of grid-wide power quality in the distribution network, reflecting the overall power quality level of the distribution network, a comprehensive evaluation method for power quality in a grid-wide system based on complex network theory is proposed. Firstly, based on the propagation characteristics of power quality disturbances, a power quality evaluation index system is constructed. Secondly, to reflect the constraint effect of the local power quality level of nodes on the overall power quality level of the distribution system, corresponding indices such as improved node degree, improved node electrical betweenness, and node self-healing capability are proposed based on complex network theory, and the power quality influence degree of nodes is calculated. Then, the GRA-ANP (Grey Relational Analysis–Analytic Network Process) subjective weight calculation method is improved by introducing grey relational analysis to address the impact of differences in different decision-making results. Based on power quality monitoring data, the entropy weight method is used for objective weighting. To avoid the partiality of a single weight evaluation result, the game equilibrium algorithm is employed to calculate the comprehensive weight of each power quality index. Subsequently, considering the correlation and dependency among indices, the VIKOR (VIseKriterijumska Optimizacija I Kompromisno Resenje) method is used to obtain the power quality grade of each node. Combining this with the calculation of the power quality influence degree of nodes, the overall power quality grade of the distribution network is determined, achieving a hierarchical and quantitative evaluation of power quality in the entire distribution system. Finally, through a case study analysis of an improved 13-node distribution network, it is verified that the proposed method can fully extract data information and produce comprehensive and accurate power quality assessment results by comparing it with other methods. This provides strong support for the safe and stable operation of the distribution system and the subsequent optimization and management of power quality.