Equilibrium Of Game Research Articles

Generalization of the social coalitional equilibrium structure

We generalize the notion of Ichiishi (Econometrica 49(2):369–377, 1981)’s social coalitional equilibrium to a multi-layered coalition structure with parameters, in which agents can incorporate simultaneously multiple coalition structures with multiple independent coalition-deviation opportunities. For each opportunity, agents play a social coalitional equilibrium (SCE) game, called a sub-parametric SCE game, constrained by external environment (parameters and joint decisions of all other sub-parametric SCE games). The generalized social coalitional equilibrium (GSCE) concept is, therefore, considered to be a synthesis of the Nash equilibrium concept and the cooperative solution concept. We provide the definition of GSCE and give the proof of existence theorem. Through some applications to general equilibrium models, the GSCE concept provides a conceptual framework for describing coexisting different industries having independent investment opportunities and their simultaneously determined industrial organizations.

Interest Equilibrium and Path Choice in the Development of Construction Land Decrement: A Theoretical Analysis Based on the Multi-Agent Game Model

High-quality development is a new-era requirement for modern urban construction. The implementation of construction land decrement has become an inevitable choice to solve the urban land use dilemma and achieve high-quality social and economic development. It is a game process for stakeholders. Based on the multi-agent game model, this research analyzes the game relationship and strategy selection between governments, government and farmers, government and developers, and developers and farmers and examines the balance of interests among multiple subjects. From the perspective of space dimension, index dimension, use dimension and time dimension, a game path is constructed for construction land decrement. The research shows the following: There is a clear relationship between the supervision cost of the higher-level government and the intensity of rewards and punishments for the lower-level government and the implementation of the decrement. The district (county) government’s compensation plan and farmers’ perceptions of the success of the boycott will affect the strategic choices of both parties. Governments at all levels play a direct role in restraining development behavior and supervising rent-seeking behavior after developers intervene in volume reduction. It also proposes optimizing the ratio of game factors such as costs, benefits, rewards, verification and punishment to achieve the game equilibrium of the expression of the interests of village residents, developers and governments at all levels. Based on the path planning and coupling of response, the multi-scheme dynamic path selection for the development of construction land decrement is realized.

Edge Resource Pricing and Scheduling for Blockchain: A Stackelberg Game Approach

Blockchain came to prominence as the distributed ledger underneath Bitcoin, which protects the transaction histories in a fully-connected, peer-to-peer network. The blockchain mining process requires high computing power to solve a Proof-of-Work (PoW) puzzle, which is hard to implement on users’ mobile devices. So these miners may leverage the edge/cloud service providers (ESPs/CSP) to calculate the PoW puzzle. The existing edge-assisted blockchain networks assumed that all ESPs have a uniform propagation delay, which is unrealistic. In this article, we consider a more practical scene where ESPs locate in diverse positions of the blockchain network, which causes different propagation delays when supporting the computation of the PoW puzzle. Additionally, these ESPs connect to a remote CSP for resource scheduling when the computing tasks exceed their maximum capacity. The blockchain mining process generally involves complicated competition and games among CSP, ESPs, and miners. Each service provider focuses on how to determine his resource price so that he can maximize his utility. According to the set resource price, each miner concentrates on scheduling his resource requests for each ESP to maximize individual personal utility, which depends on ESPs’ resource price and propagation delays. We first model such a resource pricing and scheduling problem as a three-stage multi-leader multi-follower Stackelberg game and aim at finding the Stackelberg equilibrium. Then, we analyze the subgame optimization problem in each stage and propose an iterative algorithm based on backward induction to achieve the Nash equilibrium of the Stackelberg game. Finally, extensive simulations are conducted to verify the significant performance of the proposed solution.

Nash Equilibrium Seeking for High-Order Multiagent Systems With Unknown Dynamics

In this article, we consider a Nash equilibrium seeking problem for a class of high-order multiagent systems with unknown dynamics. Different from existing results for single integrators, we aim to steer the outputs of this class of uncertain high-order agents to the Nash equilibrium of some noncooperative game in a distributed manner. To overcome the difficulties brought by the high-order structure, unknown nonlinearities, and the regulation requirement, we first introduce a virtual player for each agent and solve an auxiliary noncooperative game for them. Then, we develop a distributed adaptive protocol by embedding this auxiliary game dynamics into some proper tracking controller for the original agent to resolve this problem. We also discuss the parameter convergence issue under certain persistence of excitation conditions. The efficacy of our algorithms is verified by numerical examples.

Differential Game-Based Optimal Control of Autonomous Vehicle Convoy

Group control of connected and autonomous vehicles on automated highways is challenging for the advanced driver assistance systems (ADAS) and the automated driving systems (ADS). This paper investigates the differential game-based approach to autonomous convoy control with the aim of deployment on automated highways. Under the noncooperative differential games, the coupled vehicles make their decisions independently while their states are interdependent. The receding horizon Nash equilibrium of the linear-quadratic differential game provides the convoy a distributed state-feedback control strategy. This approach suffers a fundamental issue that neither a Nash equilibrium’s existence nor the uniqueness is guaranteed. We convert the individual dynamics-based differential game to a relative dynamics-based optimal control problem that carries all the features of the differential game. The existence of a unique Nash control under the differential game corresponds to a unique solution to the optimal control problem. The latter is shown, as well as the asymptotic stability of the closed-loop system. Simulations illustrate the effectiveness of the presented convey control scheme and how it well suits automated highway driving scenarios.

The economics of China’s between-city height competition: A regression discontinuity approach

In this article, I use a regression discontinuity design to quantify the race between Chinese cities to build taller skyscrapers. I begin by characterizing this height competition with a game-theoretic model for multi-stage, two-player, all-pay auctions with carryover. The dominant mixed strategy equilibrium of this dynamic game implies that a city would be more inclined to construct a taller skyscraper if its current tallest building is merely shorter than the tallest in the rival city, and this inclination is discontinuous at which the heights of these two buildings are equal. Utilizing a large data set of China’s skyscrapers constructed between 2004 and 2019, I confirm the presence of this between-city contest by empirically identifying the discontinuity in a city’s probability of constructing a taller building. I also find that this height competition has been more prevalent among the cities with fewer historic amenities or with leaders that have lower promotion likelihoods prior to the start of office.

Modeling Competition over Multiple Variables under Limited Consumer Awareness*

When analyzing firm competition over two strategic variables (e.g., quality and price), it is important to decide whether to model it as a one‐stage or a two‐stage game. Our analysis focuses on markets in which consumers are not aware of all alternatives. We find that, if consumers are sufficiently unaware, both the one‐stage and the two‐stage equilibria of the game that explicitly models limited awareness are close to the one‐stage equilibrium of the standard game, which assumes full awareness. Therefore, markets in which consumers have limited awareness can be studied with standard models, provided that the one‐stage game is analyzed.

An Efficient Load-Balancing Scheme for UAVs in 5G Infrastructure

Deploying caches at the macro base station (MBS), unmanned aerial vehicle (UAV), and mobile user caches can effectively reduce the retransmission of duplicate content in the 5G cellular wireless hotspot network. As the storage capacity of MBS is much higher than UAVs and other hotspot cache nodes, the MBS advertises its vacant storage space so that the participating nodes can rent it. In this article, we proposed an efficient load-balancing scheme by using the Stackelberg equilibrium game model. The proposed scheme sets a unit price ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\xi$</tex-math></inline-formula> ) based on constraints to avoid data traffic uncertainty caused by participation nodes and rent vacant space of MBS. Furthermore, we proposed an efficient scheme for the placement and delivery of hotspot content by using Knapsack and Zipf. Moreover, ensuring the device-to-device link support also minimizes transportation costs. The results validate that considering the above-mentioned techniques significantly improves the overall hotspot network performance.

Cryptocurrency is accounting coordination: Selfish mining and double spending in a simple mining game

The fundamental monetary innovation embedded into cryptocurrencies is accounting coordination. Decentralized management of digital money’s accounting by a network of computers is achieved as a Nash equilibrium of a coordination game among the network’s nodes: the so called miners. Equilibrium analysis demands allowing miners to secretly update their accounting, i.e., to privately build multiple blocks of transactions and to deviate from the longest chain rule. We formalize such reasoning by proposing an accounting coordination game inspired on the Bitcoin design. In particular, by proposing a model that explicitly tells apart mining costs related to energy consumption from those related to computational capacity, we are able to study how symmetric equilibrium existence depends on well known parameters, like the average time for updating accounting records and the rewards collected from mining (accounting) activities. It is shown that the (off-equilibrium) possibility of double spending makes the attractiveness of the equilibrium strategy a decreasing function of the average time for updating accounting records.

Existence and stability results on the unilateral support equilibrium

In this paper, we consider non-cooperative games where each player has an other-regarding behavior. In particular, we study the unilateral support equilibrium introduced by Schouten et al. (2019). A strategy profile is a unilateral support equilibrium if each player’s strategy maximizes the payoff of each other when the opponents do not deviate from their equilibrium strategies. We give necessary and sufficient conditions for the existence of unilateral support equilibria in games where players have discontinuous and non necessarily ordered preferences. Moreover, we study the stability of equilibria when the players partially support the others or perturbations on players’ preferences occur.

Evolutionary game equilibrium in the downstream carbon market: Evidence from the household size in China

AbstractCarbon trading downstream, as a key strategy for decreasing carbon emissions, has the potential to hasten the entire society's decarbonization. Taking the government, household, and carbon trading management center as stakeholders, this paper constructs a tripartite evolutionary game model, examines and stimulates evolutionary stability strategies, and then proposes some policy implications. Findings show that the downstream carbon market faces a complex evolutionary process. The promotion of the policy can be split into two periods: policy promotion and policy stability implementation. And proper household size management is a crucial and effective strategy to encourage downstream carbon trading programs and emission reduction.

The Computation of Approximate Generalized Feedback Nash Equilibria

We present the concept of a generalized feedback Nash equilibrium (GFNE) in dynamic games, extending the feedback Nash equilibrium concept to games in which players are subject to state and input constraints. We formalize necessary and sufficient conditions for (local) GFNE solutions at the trajectory level, which enable the development of efficient numerical methods for their computation. Specifically, we propose a Newton-style method for finding game trajectories which satisfy necessary conditions for an equilibrium, which can then be checked against sufficiency conditions. We show that the evaluation of the necessary conditions in general requires computing a series of nested, implicitly defined derivatives, which quickly becomes intractable. To this end, we introduce an approximation to the necessary conditions which is amenable to efficient evaluation and, in turn, computation of solutions. We call the solutions to the approximate necessary conditions the generalized feedback quasi-Nash equilibria, and we introduce numerical methods for their computation. In particular, we develop a sequential linear-quadratic (LQ) game approach, in which an LQ local approximation of the game is solved at each iteration. The development of this method relies on the ability to compute a GFNE to inequality- and equality-constrained LQ games, and therefore specific methods for the solution of these special cases are developed in detail. We demonstrate the effectiveness of the proposed solution approach on a dynamic game arising in an autonomous driving application.

A linear-quadratic mean-field game of backward stochastic differential equation with partial information and common noise

This paper studies a linear-quadratic (LQ) mean-field game of stochastic large-population system with partial information and common noise, where the large-population system satisfies a class of backward stochastic differential equations (BSDEs), and both coupling structure and ki (a part of solution of BSDE) enter state equations and cost functionals. By virtue of stochastic maximum principle and optimal filter technique, we obtain a Hamiltonian system first, which is a fully-coupled forward-backward stochastic differential equation (FBSDE). Decoupling the Hamiltonian system, we obtain three ordinary differential equations (ODEs), a forward and a backward optimal filtering equations, which enable us to derive an optimal control of an auxiliary limiting control problem. Next, we verify that a decentralized control strategy is an ϵ-Nash equilibrium of the LQ game. Finally, we solve a product pricing problem and a network security control problem in applications. We give a near-optimal price strategy and a near-optimal control strategy with numerical simulations, respectively.

A study on the competitive dissemination of disinformation and knowledge on social media

PurposeThe purpose of this study is to explore the evolutionary path and stable strategy for the competitive dissemination between disinformation and knowledge on social media to provide effective solutions to curb the dissemination of disinformation and promote the spread of knowledge.Design/methodology/approachBased on the social capital (SC) theory, the benefit matrix is constructed and an evolutional game model is established in this paper. Through model solving and Matrix Laboratory (MATLAB) simulation, the factors that influence disinformation-believing users (DUs) and knowledge-believing users (KUs) to choose different strategies are analyzed.FindingsThe initial dissemination willingness, the disinformation infection probability, the knowledge infection probability and the knowledge penetration probability are proved to be crucial factors influencing the game equilibrium in the competitive dissemination process of disinformation and knowledge. Moreover, some countermeasures and recommendations for the governance of disinformation are proposed.Originality/valueCurrently most research interest lies in the disinformation dissemination model but ignores the interaction between disinformation and knowledge in the diffusion process. This study reveals the dynamic mechanism of social media users disseminating disinformation and knowledge and is expected to promote the formation of cleaner cyberspace.

Do we all coordinate in the long run?

Players often fail to coordinate on the efficient equilibrium in laboratory weak-link coordination games. In this paper, we investigate whether such coordination failures can be mitigated by increasing the number of rounds or altering per-period stakes. We find that neither time horizon nor stakes affect equilibrium selection. In contrast to previous findings, players are not more likely to play above the previous period’s minimum choice when the horizon is longer or per-period stakes lower. We also investigate which socio-demographic factors and behavioral traits correlate most strongly with play both in the first round and in subsequent rounds. Cognitive ability as measured by a cognitive reflection test stands out as the characteristic that is most strongly associated with efficient coordination.

Neuro-adaptive control for searching generalized Nash equilibrium of multi-agent games: A two-stage design approach

This paper investigates the generalized Nash equilibrium searching problem of multi-agent games over weight-unbalanced directed networks. In this problem, the feasible action set of each agent is not only constrained by private convex sets and shared coupling equality, but also constrained by its own uncertain dynamics. Moreover, the local cost function is related to both his own actions and others, and each agent is only allowed access to neighbor information. To address this problem, we propose a neuro-adaptive control strategy based on two-stage design. In the first stage of design process, an approximator with adaptive-gain is designed to generate a virtual trajectory that converges to a necessary Nash equilibrium, the compensator based on consensus-tracking is used to obtain non-neighbor information and several auxiliary state variables are used to exchange information with local neighborhoods on a digraphs to deal with equality constraints. In the second stage of the strategy, we further developed a neuro-adaptive tracking controller based on one-step design for tracking the virtual trajectory. In the controller design, the virtual control variables and the actual control laws are obtained in a collective way, without repeating the design process. We introduce a variable called the minimum learning parameter to reduce the number of online learning parameters of the neural network. By using tools from variational inequality theory and Lyapunov stability theory, we prove that the proposed control strategy can drive all agents reach Nash equilibrium. Finally, the effectiveness of control strategy is verified by simulation example.

Approximating Nash equilibrium for anti-UAV jamming Markov game using a novel event-triggered multi-agent reinforcement learning

In the downlink communication, it is currently challenging for ground users to cope with the uncertain interference from aerial intelligent jammers. The cooperation and competition between ground users and unmanned aerial vehicle (UAV) jammers leads to a Markov game problem of anti-UAV jamming. Therefore, a model-free method is adopted based on multi-agent reinforcement learning (MARL) to handle the Markov game. However, the benchmark MARL strategies suffer from dimension explosion and local optimal convergence. To solve these issues, a novel event-triggered multi-agent proximal policy optimization algorithm with Beta strategy (ETMAPPO) is proposed in this paper, which aims to reduce the dimension of information transmission and improve the efficiency of policy convergence. In this event-triggering mechanism, agents can learn to obtain appropriate observation in different moment, thereby reducing the transmission of valueless information. Beta operator is used to optimize the action search. It expands the search scope of policy space. Ablation simulations show that the proposed strategy achieves better global benefits with fewer dimension of information than benchmark algorithms. In addition, the convergence performance verifies that the well-trained ETMAPPO has the capability to achieve stable jamming strategies and stable anti-jamming strategies. This approximately constitutes the Nash equilibrium of the anti-jamming Markov game.

Hierarchical Game for Low-Carbon Energy and Transportation Systems Under Dynamic Hydrogen Pricing

Confronted with the increasing penetration of renewable energy and hydrogen-powered vehicles (HVs), energy and transportation systems need to be optimized coordinately to improve the economy and reduce carbon emissions. However, the effect of dynamic hydrogen pricing on the optimization performance of energy and transportation systems is scarcely investigated. To realize the low-carbon and economic operation of energy and transportation systems, a hierarchical game between the distributed energy station (DES) and HVs under dynamic hydrogen pricing is put forward. First, a novel dynamic hydrogen pricing mechanism related to the proportion of renewable energy in the energy supply is presented and incorporated into the game optimization, which will promote hydrogen production using renewable energy and minimize the DES operation cost. At the game equilibrium, the HVs completing equivalent aggregation realize balanced hydrogen refueling at the guidance of dynamic hydrogen pricing. Meanwhile, the raised routing optimization method can select a minimum cost route for HVs to refuel based on traffic information. Finally, the effectiveness of the proposed hierarchical game optimization strategy and model is verified by simulations.

Zero-sum games involving teams against teams: Existence of equilibria, and comparison and regularity in information

Many emerging problems involve teams of agents taking part in a game. Such problems require a stochastic analysis with regard to the correlation structures among the agents belonging to a given team. In the context of standard Borel state, measurement and action spaces (thus, including real vector spaces and finite spaces), this paper makes the following contributions for two teams of finitely many agents taking part in a zero-sum game: (i) An existence result will be presented for saddle-point equilibria in zero-sum games involving teams against teams when common randomness is assumed to be available in each team with an analysis on conditions for compactness of strategic team measures to be presented. (ii) Blackwell’s ordering of information structures is generalized to n-player teams with standard Borel spaces, where correlated garbling of information structures is introduced as a key attribute; (iii) building on this result Blackwell’s ordering of information structures is established for team-against-team zero-sum game problems. (iv) Finally, continuity of the equilibrium value of team-against-team zero-sum game problems in the space of information structures under total variation is established.

Corrigendum to On the complexity of computing Markov perfect equilibrium in general-sum stochastic games

[This corrects the article DOI: 10.1093/nsr/nwac256.].