Nash Equilibrium Research Articles

Nash–Cournot Equilibrium and Its Impact on Network Transmission Congestion

This paper evaluates the impact of congestion on transmission lines when the operation cost is minimized using economic dispatch (ED), comparing the results obtained with the Nash–Cournot Equilibrium (NCE). A methodology is developed for the optimal power flow solution through the NCE, considering the network topology (upper and lower generation limits, upper and lower limits of the transmission lines, and power balance) for a nine-node system without and considering two bilateral power transactions. The results show that the operation cost is higher when the NCE is implemented than ED. However, the problem of congestion in the transmission lines is reduced due to the equilibrium obtained in the power dispatch against minimizing the operation cost in the dispatch; the transmission lines with the most significant participation tend to become congested when additional bilateral transactions occur. Finally, the above is verified by obtaining the mean and median of the transmission line percentages used in the two simulations.

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/ .

Anti-jamming for cognitive radio networks with Stackelberg game-assisted DSSS approach

The proposed study introduces a novel anti-jamming approach for cognitive radio networks (CRNs) by integrating the Stackelberg game model with direct sequence spread spectrum (DSSS) techniques. This innovative combination enhances the security and performance of CRNs by optimizing resource allocation and fortifying network resilience against jamming attacks. The Stackelberg game model provides a strategic framework where the Defender and Adversary dynamically adjust their strategies to achieve Nash equilibrium, ensuring strategic stability. The application of DSSS further improves signal robustness, mitigating interference from jamming attempts. Simulation results demonstrate significant improvements in network security, resource utilization, and overall performance, validating the efficacy and advantages of the proposed scheme in maintaining reliable communication in the presence of adversarial threats.

Coordination of a socially responsible two-stage supply chain under random yield and demand

This study integrates corporate social responsibility (CSR) and channel coordination in a supply chain (SC) under random yield and random demand. In the SC, a supplier with random yield determines wholesale price and production input, and a producer with CSR decides order quantity and CSR investment facing CSR-related random demand. For centralized SCs with and without CSR, we prove the unimodality of the expected profits of the SCs for generic random yield and demand, and show the uniqueness of optimal ratio of order quantity to production input quantity. We disclose that the optimal expected profit of the SC with CSR is larger than that of the SC without CSR, and the profit difference increases with CSR effort and expected yield. We also find that the optimal CSR investment increases with the expected yield. Subsequently, the Nash equilibrium solutions of decentralized SCs under wholesale price and revenue-sharing contracts are analyzed. Next, we present a revenue and CSR sharing contract to realize channel coordination and win-win of SC members and related stakeholders. Lastly, we verify theoretical statements of centralized and decentralized SCs versus three numerical examples, and present managerial insights for the effect of yield uncertainty and CSR effort.

Stochastic Differential Games and a Unified Forward–Backward Coupled Stochastic Partial Differential Equation with Lévy Jumps

We establish a relationship between stochastic differential games (SDGs) and a unified forward–backward coupled stochastic partial differential equation (SPDE) with discontinuous Lévy Jumps. The SDGs have q players and are driven by a general-dimensional vector Lévy process. By establishing a vector-form Ito-Ventzell formula and a 4-tuple vector-field solution to the unified SPDE, we obtain a Pareto optimal Nash equilibrium policy process or a saddle point policy process to the SDG in a non-zero-sum or zero-sum sense. The unified SPDE is in both a general-dimensional vector form and forward–backward coupling manner. The partial differential operators in its drift, diffusion, and jump coefficients are in time-variable and position parameters over a domain. Since the unified SPDE is of general nonlinearity and a general high order, we extend our recent study from the existing Brownian motion (BM)-driven backward case to a general Lévy-driven forward–backward coupled case. In doing so, we construct a new topological space to support the proof of the existence and uniqueness of an adapted solution of the unified SPDE, which is in a 4-tuple strong sense. The construction of the topological space is through constructing a set of topological spaces associated with a set of exponents {γ1,γ2,…} under a set of general localized conditions, which is significantly different from the construction of the single exponent case. Furthermore, due to the coupling from the forward SPDE and the involvement of the discontinuous Lévy jumps, our study is also significantly different from the BM-driven backward case. The coupling between forward and backward SPDEs essentially corresponds to the interaction between noise encoding and noise decoding in the current hot diffusion transformer model for generative AI.

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.

Cooperative coevolution for non-separable large-scale black-box optimization: Convergence analyses and distributed accelerations

Given the ubiquity of non-separable optimization problems in real worlds, in this paper we analyze and extend the large-scale version of the well-known cooperative coevolution (CC), a divide-and-conquer black-box optimization framework, on non-separable functions. First, we reveal empirical reasons of when decomposition-based methods are preferred or not in practice on some non-separable large-scale problems, which have not been clearly pointed out in many previous CC papers. Then, we formalize CC to a continuous-game model via simplification, but without losing its essential property. Different from previous evolutionary game theory for CC, our new model provides a much simpler but useful viewpoint to analyze its convergence, since only the pure Nash equilibrium concept is needed and more general fitness landscapes can be explicitly considered. Based on convergence analyses, we propose a hierarchical decomposition strategy for better generalization, as for any decomposition, there is a risk of getting trapped into a suboptimal Nash equilibrium. Finally, we use powerful distributed computing to accelerate it under the recent multi-level learning framework, which combines the fine-tuning ability from decomposition with the invariance property of CMA-ES. Experiments on a set of high-dimensional test functions validate both its search performance and scalability (w.r.t. CPU cores) on a clustering computing platform with 400 CPU cores.

Defensive resource allocation in terrorism conflict management based on graph model with relative preferences

In real-world counterterrorism activities, it is usually difficult for the defender and the attacker to accurately know the private information of the each other such as valuations of targets. Instead, players may only know the relative preference on the target valuations from the adversary. In the conflict analysis, graph model is a powerful tool for dealing with relative preferences. This paper studies the defensive resource allocation in terrorism conflict management with incomplete information by establishing a graph model. To solve the model, we divide the conflict states into two types and discuss the conditions under which these two types of states are at equilibrium. Furthermore, we study how the defender should optimally allocate the resource to achieve two goals: (i) achieving a certain Nash equilibrium state desired by the defender; and (ii) minimizing the total loss from an attack in equilibrium. Subsequently, we conduct several numerical analyses: (i) analyzing the effects of both players' investment effectiveness on the optimal defense loss; (ii) comparing our model's results with those obtained using three classical decision methods, revealing that the defense loss in our model is lower; and (iii) presenting a case study to illustrate the applicability of the proposed model. This paper provides novel insights on how to efficiently allocate defensive resource when the defender and attacker know only the relative preference of the adversary on target valuations.

Stability of open-loop Nash equilibria for [formula omitted]-person nonzero-sum bounded rationality generalized differential games

The problem of determining the existence of Nash equilibria in n-person nonzero-sum generalized differential games is highly intricate and constrained by the advancement of partial differential equations theory. There is limited existing research literature on this subject. This paper presents an existence theorem for open-loop Nash equilibria employing the Fan-Glicksberg fixed point theorem. The n-person nonzero-sum bounded rationality generalized differential game model is formulated by introducing a bounded rationality function, and its structural stability and robustness are studied. The conclusions indicate that in the sense of Baire classification, most n-person nonzero-sum bounded rationality generalized differential games are structurally stable and robust in the set of ɛ-open-loop Nash equilibria, and we can approximate the equilibrium set obtained with full rationality generalized differential games by the ɛk-open-loop Nash equilibria set obtained with bounded rationality generalized differential games.

Necessary and sufficient optimality conditions for nonsmooth nonconvex generalized Nash equilibrium problems

The generalized Nash equilibrium problem (GNEP) is an extension of the Nash equilibrium problem (NEP), which allows the strategies and the objective function of each player to depend on the decision variables of all other players. In this paper, we consider a GNEP where the objective and constraint functions are not necessarily differentiable or convex. By proposing an alternative approach based on the equivalence with a single constrained optimization problem, we then present new sharp KKT-type necessary optimality conditions for a solution of this problem under a weak constraint qualification. We also offer an example to show that these conditions are more efficient for such problems than the previous ones. Further under weak convexity assumptions, we prove that these necessary conditions are also sufficient for optimality. Finally, we extend the definitions of some constraint qualifications from classical optimization to GNEP and study their interrelations. Throughout the paper, several examples are provided to illustrate the results.

The ascending auction with flexible reporting

In the assignment problem where multiple heterogeneous indivisible items are assigned to unit-demand bidders, we introduce a novel ascending auction called the Flexible Reporting Ascending Auction to balance the privacy preservation and the speed of the ascending auction. Assuming bidders behave truthfully, it always results in a Walrasian equilibrium, along with the minimum Walrasian equilibrium price vector. If each bidder reports without a contradiction, bidders’ truthful behavior forms a Nash equilibrium. Our auction generalizes some well-known ascending auctions in literature, and we also provide simulation results to compare these auctions.

3-Player 3-Strategy Nash Equilibruim of Party’s Presidential Primaries in the Build-Up to 2023 General Elections in Nigeria

The research formulated a political game model of All Progressive Congress (APC) Party Contestants for the Presidential Primaries in order to determine the best possible candidate who would be presented by the Party for winning the 2023 General Election in Nigeria. The method of solution involved Nash equilibrium which yielded a pure Nash result of Propaganda, Consultation and Youth mobilization Strategies for Yahaya Bello, Ahmed Bola Tinubu and Yemi Osinbajo respectively. The study concluded that, the outcome of the Party’s primaries which produced Ahmed Bola Tinubu as its flag bearer was consistent with the model’s result. Because, Ahmed Bola Tinubu did a lot of deep consultation with stakeholders while Yahaya Bello’s best response was propaganda as Osibanjo was preferred by the Youth of the Party.

Performance evaluation of marine ecological compensation in coastal cities of China via a novel two-stage bargaining game DEA with imprecise data

To alleviate the pressure of economic development on the marine environment, the Marine Ecological Compensation (MEC) has become a major policy tool for the Chinese government to reconcile the contradiction between economic development and the marine environment. In this paper, we propose a novel two-stage bargaining game data envelopment analysis (DEA) model to evaluate the performance of MEC under the cooperative structure. The proposed model considers the link between marine economic development (MED) and marine environmental protection (MEP). Meanwhile, the equivalent form, Nash equilibrium solution, sensitivity and stability of the model are as well documented to further analyze MED and MEP. Eventually, a case study of 30 coastal cities in China serves to verify the practicable effectiveness of the foregoing model combined with numerical simulation and support key insights as below: (i) According to the results evaluated by the decentralized DEA model, we find that if one party of MED and MEP takes priority, the efficiency score of the other party will be severely affected; (ii) the evaluation results of sensitivity and stability demonstrate that inputs and outputs have different impact degrees on the efficiency scores of MED and MEP, which provide directions for improving the efficiency of both systems; (iii) The proposed model addresses the limitation of the conventional two-stage DEA model that cannot handle uncertain variables, thus revealing the influence of uncertainty on MEC efficiency. The compelling evidence presented in the case study solidifies the effectiveness of the proposed model, establishing its promising prospects for application in evaluating the performance of DMUs with a two-stage structure.

Tullock contest with reference‐dependent preferences

AbstractWe study the Tullock contest model with loss aversion and endogenously formed reference points. In a contest with n possibly heterogeneous players and convex effort costs, we establish sufficient conditions for a unique Nash equilibrium in pure strategies. Subsequently, we analyze the impact of loss aversion on players' spending behavior, probability of winning, and rent dissipation.

Dynamic environment UAV deployment algorithm based on potential game theory

AbstractTo address the issue of low coverage resulting from the challenge of acquiring the optimal deployment position in commonly used distributed deployment algorithms, this study presents a three‐dimensional deployment algorithm for Unmanned Aerial Vehicles (UAVs) based on potential games. First, a local mutually beneficial game model is designed to demonstrate the existence of exact potential games and Nash equilibrium. The Nash equilibrium solution corresponds to the maximum coverage. Next, drawing inspiration from exploration, a solution method called Exploration Spatial Adaptive Play is proposed. It utilizes the maximum utility function value from multiple step sizes in the exploration direction to update the action selection probability, thereby ensuring the optimal deployment position in each decision cycle. To address the issue of sensor position error, a method for processing sensor position errors is proposed. The simulation results demonstrate that the proposed distributed deployment algorithm achieves higher coverage compared to commonly used methods.

Inefficiency of pure Nash equilibria in network congestion games: the impact of symmetry and network structure

Inefficiency of pure Nash equilibria in network congestion games: the impact of symmetry and network structure

Should the pricing or advertising decision come first in a supply chain with a network externality?

PurposeThis paper aims to study how the timing of these decisions affects the total profit and the individual profits of the two agents.Design/methodology/approachThis paper study a supply chain for a network good where there is a manufacturer and a retailer. The manufacturer determines its wholesale price and its share in the retailer’s advertising cost while the retailer decides the retail price and the advertising cost.FindingsThis paper finds that a stronger network externality leads to higher prices and higher advertising efforts. This increases the profits of both manufacturer and retailer, but the manufacturer’s share of advertising costs depends on the order in which the supply chain enterprise make their decisions, the strength of network externality and the effect of advertising determines which decision timeline results in a higher price and greater advertising effort. The manufacturer prefers the price decision to be made before the advertising decision, while the retailer prefers these decisions to be made simultaneously.Research limitations/implicationsAlthough this paper studies the price and advertising decision-making order preferences of channel members based on network externalities, this research can also be expanded from the following aspects based on network effects. First, network externality affects advertising cooperation between both parties in the situation such that the pricing power of retail prices is transferred from the retailer to the manufacturer and the retailer relies on revenue sharing (revenue sharing contract, nonwholesale price contract. Second, the manufacturer dominates the issues in the supply chain, but in reality, a retailer can also be the dominator or there are no dominators (Nash equilibrium). Finally, it is possible to consider pricing and advertising decisions in situations where two manufacturers or retailers compete.Practical implicationsWhen the price is reasonable, advertising investment is the main determinant of product sales. The greater the intensity of network externalities the more retailers will be willing to invest in advertising. An increase in the intensity of network externalities may not necessarily enhance manufacturers’ motivation or cooperative advertising, but it depends on the decision-making sequence. The strength of network externalities determines the decision-making sequence preferences of supply chain channel members whose preferences vary leading to conflicts of interest.Originality/valueThe impact of cooperative advertising or decision sequence on corporate decision-making has not been considered. To fill this gap, the paper integrates network externality and supply chain cooperative advertising models, focusing on the impact of network externality on pricing and advertising decisions, as well as on the sequence of decisions.

Predefined-time convergence strategies for multi-cluster games in hybrid heterogeneous systems

This paper explores the problem of (generalized) Nash equilibrium search in multi-cluster games with heterogeneous dynamics and multiple constraints. Within this research framework, each agent acquires information solely through local interactions with its neighbors and forms clusters based on similarity of interests. These clusters manifest dual relationships of cooperation and competition: agents within the same cluster enhance decision-making capabilities through cooperation, while different clusters compete to maximize their respective benefits. To delve into these complex interactions among clusters and the learning and evolution processes among agents, four distributed control algorithms suitable for various scenario requirements are designed and implemented. These algorithms ensure that each agent converges to a Nash equilibrium (NE) or generalized Nash equilibrium (GNE) of the multi-cluster system within predefined time points. Finally, we apply these algorithms to the connectivity control problem of unmanned aerial vehicle swarms with diverse dynamics, validating the theoretical results through comprehensive simulations.

Equilibria in abstract economies with a continuum of agents with discontinuous and non-ordered preferences

This paper focus on the problem of the existence of an equilibrium in abstract economies and exchange economies. Spanning over the literature we have managed to extend and generalize some previous results. In particular, we generalize the main theorem of Yannelis (1987) on the existence of an equilibrium in an abstract economy with a continuum of agents, by allowing for discontinuous preferences. As a corollary of this result, we extend the finite agent Cournot–Nash equilibrium existence theorems with discontinuous preferences (e.g., Reny, 1999; Bareli and Meneghel, 2013; He and Yannelis, 2016; among others), to a continuum of agents. We also obtain an existence theorem for an abstract economy which allows for a convexifying effect on aggregation and nonconvex strategy and constraint sets. Furthermore, our new main theorem is used to prove the existence of a Walrasian equilibrium with a continuum of agents with discontinuous, non-ordered, interdependent and price-dependent preferences and thus extending the results of Aumman (1966) and Schmeidler (1969).

Event-Triggered Mechanism-Based Discrete-Time Nash Equilibrium Seeking for Graphic Game With Outlier-Resistant ESO.

In this article, we consider a discrete-time Nash equilibrium (NE) seeking problem for graphic game subject to disturbances. For the first-order dynamics, the discrete-time outlier-resistant extended state observer (ESO)-based game strategy is proposed to enable the players to estimate the disturbances under effect of anomaly measurements and then compensate them. An event-triggered mechanism is applied between adjacent players to reduce the frequency of communication. The convergence of the outlier-resistant ESO and control strategy is presented. Moreover, the upper bound of ϵ -NE solution deviating from the unique point of nominal system is given analytically. Then, the addressed issues are extended to high-order game systems. The NE seeking-based control strategy for each player is designed such that the equilibrium point converges to the ϵ -NE which is also analytically calculated. Finally, in order to verify the effectiveness of the proposed game strategy, an example of satellite system is given.