Solution Of Game Research Articles

Existence theory for non-separable mean field games in Sobolev spaces

The mean field games system is a coupled pair of nonlinear partial differential equations arising in differential game theory, as a limit as the number of agents tends to infinity. We prove existence and uniqueness of classical solutions for time-dependent mean field games with Sobolev data. Many works in the literature assume additive separability of the Hamiltonian, as well as further structure such as convexity and monotonicity of the resulting components. Problems arising in practice, however, may not have this separable structure; we therefore consider the non-separable problem. For our existence and uniqueness results, we introduce new smallness constraints which simultaneously consider the size of the time horizon, the size of the data, and the strength of the coupling in the system.

Approximating solution of stochastic differential games for distributed control of a water network

In this paper, our objective is to design a distributed optimal control for pumping stations operating in a Water Distribution Network (WDN), where we would like to satisfy consumer demands with minimum energy consumption. The WDN has been modeled using graph theory and stochastic differential equations. This leads to a non-zero sum stochastic differential game. We have approximated the solution of the aforementioned game using Markov chain approximation and combined it with Shapley's algorithm so as to obtain Minimax mixed strategies. Minimax solution can be obtained as a distributed computation at the pumping stations without any knowledge of the costs incurred by the other pumping stations. Simulation results on the water network show convergence to an approximate Minimax solution.

Integration of game theory and response surface method for robust parameter design

Robust parameter design (RPD) is to determine the optimal controllable factors that minimize the variation of quality performance caused by noise factors. The dual response surface approach is one of the most commonly applied approaches in RPD that attempts to simultaneously minimize the process bias (i.e., the deviation of the process mean from the target) as well as process variability (i.e., variance or standard deviation). In order to address this tradeoff issue between the process bias and variability, a number of RPD methods are reported in literature by assigning relative weights or priorities to both the process bias and variability. However, the relative weights or priorities assigned are often subjectively determined by a decision maker (DM) who in some situations may not have enough prior knowledge to determine the relative importance of both the process bias and variability. In order to address this problem, this paper proposes an alternative approach by integrating the bargaining game theory into an RPD model to determine the optimal factor settings. Both the process bias and variability are considered as two rational players that negotiate how the input variable values should be assigned. Then Nash bargaining game solution technique is applied to determine the optimal, fair, and unique solutions (i.e., a balanced agreement point) for this game. This technique may provide a valuable recommendation for the DM to consider before making the final decision. This proposed method may not require any preference information from the DM by considering the interaction between the process bias and variability. To verify the efficiency of the obtained solutions, a lexicographic weighted Tchebycheff method which is often used in bi-objective optimization problems is utilized. Finally, in two numerical examples, the proposed method provides non-dominated tradeoff solutions for particular convex Pareto frontier cases. Furthermore, sensitivity analyses are also conducted for verification purposes associated with the disagreement and agreement points.

Collaborative Resource Sharing Game Based Cloud-Edge Offload Computing Orchestration Scheme

In the foreseeable future, the rapid growth of mobile communications and cloud systems has substantially promoted edge computing paradigm. Although edge computing technology has been attracting much interest, most current research is application-specific without considering a control perspective of cloud providers that provides general-purpose edge services. In this study, we present a new cloud-edge computing orchestration scheme that integrates the different resource sharing problems to maximize benefits for offloading computing services. Specifically, we focus on two cooperative game solutions - <i>Sim bargaining solution</i> and <i>interval Banzhaf Value</i> - to effectively share the computing, communication and cache resources in the cloud-edge system platform. In the proposed scheme, we formalize two different cooperative games, and they work together and act cooperatively to satisfy contradictory requirements. Under widely different service request situations, we aim at accelerating an efficient resource orchestration through dynamic workload balancing. The primary goal of our cooperative game approach is to find optimal control decisions toward the appropriate system operation. Finally, we evaluate the proposed scheme in terms of several performance criteria from the numerical simulations. Experimental results show that we can achieve a mutually desirable solution with a good system performance balance while enhancing the resource efficiency better than the existing cloud-edge control protocols.

A New Game Value Approach for Infinite Interval Matrix Games

The purpose of this paper is to determine when and under which conditions the solution and game value of the infinite interval matrix games will exist. Firstly, the concept of a reasonable solution defined in interval matrix games was extended to infinite interval matrix games. Then, the solution and game value were characterized by using sequences of interval numbers (defined by Chiao, 2002) and their concept of convergence of interval numbers. Considering that each row or column of the payoff matrix is a sequence of interval numbers, we assume that each row converges to the same interval number α ̃=[α_l,α_r] and each column to the same interval number β ̃=[β_l,β_r]. In a conclusion, the existence of the solution of G ̃ is shown.

Assessment of the risks of losing investments aimed at the development of Smart city systems

The article proposes a model for the computational core of the decision support system (DSS) in assessing the risks of investment loss during the dynamic planning (DP) of Smart City development. In contrast to the existing solutions, the proposed model provides specific recommendations when assessing the risks of loss. In case of an unsatisfactory risk forecast, it is possible to flexibly adjust the parameters of the investment process in order for the parties to achieve an acceptable financial result. The scientific novelty of the results is that for the first time it is proposed to apply a new class of bilinear multistep games. This class allowed us to adequately describe the process of assessing the risks of investment loss, using the example of dynamic planning for the placement of financial resources of players in Smart City projects. A distinctive feature of the considered approach is the use of tools based on the solution of a bilinear multistep game of both quality with several terminal surfaces, and a game of degree solved in the class of mixed strategies. Computational experiments were carried out in the Maple mathematical modeling package, and a DSS was developed in which a risk assessment model was implemented. The developed DSS allows to reduce the discrepancies between the data for predicting the risks of investment loss during the Smart City DP and the real return on investment.

Matrix expression of Owen values

AbstractThe Owen value is investigated in this paper. It is a generation of Shapley value as a solution of cooperative games with coalition structures. First, a characteristic function of a cooperative game is expressed as a pseudo‐logical function by using semi‐tensor product of matrices. Then, a matrix formula is given for calculating Owen values. The matrix formula not only makes computing more convenient but also facilitates theoretical analysis. Therefore, an application of Owen value in distributed welfare games is presented. Utility functions of a distributed welfare game with coalition structure are designed based on Owen value. Examples are showed to illustrate the theoretical results.

Strange attractors and nontrivial solutions in games with three players

Two-player games have been sufficiently researched, but three-player games have a much greater wealth of solutions, in particular, nonstationary solutions. However, unsteady solutions of three-player games have not been studied in the literature. On the basis of numerical modeling, the dynamic behavior of a 2 × 2 × 2 game with three players is investigated in the range of payoff matrices from -1 to 1. The system of dynamic equations includes three variables (probabilities of pure strategies) and twelve parameters associated with cubic payoff matrices of players. The phase space of a three-player game is a cube (probabilities vary from 0 to 1). It was found that for some payoff matrices, there are no stable solutions in the system. In addition, the structure of the phase space is complex: the volume of the cube is a fractal repulsive region from which the phase trajectories leave, a strange attractor is located near the edge corresponding to the solution of the "node-center" type, and another strange attractor is a chaotic motion between adjacent vertices of the cube. The obtained dynamic properties of the game can be used for making decisions in economics, negotiations between three persons, and modeling relationships between biological species.

Industrial structure and evasion dynamics, is there any link?

AbstractWe propose a model to describe the link between industrial structure and evasion dynamics in a heterogeneous setting, in which the monitoring efforts put in place by the State to fight evasion of firms depend on the market share of each firm. More precisely, while the convenience to evade taxes is determined as a Nash solution of an incomplete information game, a differentiated monitoring activity related to market concentration represents the original contribution of this work w.r.t. previous studies on the topic. As empirical evidence shows, high dimensional firms are more likely to be monitored, hence, when dynamics are considered, evasion and industrial structure evolution are strictly related. In fact, by combining analytical findings and simulation results, our work shows that: (1) firms with a sufficiently large market share, that is, larger than a certain dimensional threshold, comply with fiscal rules, thus confirming empirical evidence; (2) this dimensional threshold changes with the competition level between firms in a nonlinear way; (3) evasion affects the market structure, so that markets with high (low) competition tend to increase (decrease) their concentration over time; (4) evasion is affected by market structure and is minimum in moderately concentrated markets.

Approximation of value function of differential game with minimal cost

The paper is concerned with the approximation of the value function of the zero-sum differential game with the minimal cost, i.e., the differential game with the payoff functional determined by the minimization of some quantity along the trajectory by the solutions of continuous-time stochastic games with the stopping governed by one player. Notice that the value function of the auxiliary continuous-time stochastic game is described by the Isaacs–Bellman equation with additional inequality constraints. The Isaacs–Bellman equation is a parabolic PDE for the case of stochastic differential game and it takes a form of system of ODEs for the case of continuous-time Markov game. The approximation developed in the paper is based on the concept of the stochastic guide first proposed by Krasovskii and Kotelnikova.

EvoBank: An evolutionary game solution for Bankruptcy problem

There has long been a debate surrounding the bankruptcy problem and how to share bankrupt entity inventory among its claimants. In this paper, we propose a novel inventory sharing algorithm that is based on a population game and evolutionary dynamics. In contrast with most existing sharing rules, In our approach, all claimants are involved in deciding on participating in a game. We set up a population game in which its players corespond to claimants, and available sharing rules are their actions. Then we find the asymptotically stable state of the game or Nash Equilibrium based on pairwise proportional imitation (replicator dynamics). Finally, we propose the founded equilibrium profile in the mentioned game as a novel sharing rule. According to the rationality principle in game theory, there is no profitable individual deviation from any claimants’ equilibrium profile. So, the NE profile is a self-enforcement agreement between all claimants. The implementation of our proposed algorithm in MATLAB confirms its effectiveness completely.

Solution of matrix games with payoffs of single-valued trapezoidal neutrosophic numbers

Solution of matrix games with payoffs of single-valued trapezoidal neutrosophic numbers

An improved predator-prey particle swarm optimization algorithm for Nash equilibrium solution.

Focusing on the problem incurred during particle swarm optimization (PSO) that tends to fall into local optimization when solving Nash equilibrium solutions of games, as well as the problem of slow convergence when solving higher order game pay off matrices, this paper proposes an improved Predator-Prey particle swarm optimization (IPP-PSO) algorithm based on a Predator-Prey particle swarm optimization (PP-PSO) algorithm. First, the convergence of the algorithm is advanced by improving the distribution of the initial predator and prey. By improving the inertia weight of both predator and prey, the problem of “precocity” of the algorithm is improved. By improving the formula used to represent particle velocity, the problems of local optimizations and slowed convergence rates are solved. By increasing pathfinder weight, the diversity of the population is increased, and the global search ability of the algorithm is improved. Then, by solving the Nash equilibrium solution of both a zero-sum game and a non-zero-sum game, the convergence speed and global optimal performance of the original PSO, the PP-PSO and the IPP-PSO are compared. Simulation results demonstrated that the improved Predator-Prey algorithm is convergent and effective. The convergence speed of the IPP-PSO is significantly higher than that of the other two algorithms. In the simulation, the PSO does not converge to the global optimal solution, and PP-PSO approximately converges to the global optimal solution after about 40 iterations, while IPP-PSO approximately converges to the global optimal solution after about 20 iterations. Furthermore, the IPP-PSO is superior to the other two algorithms in terms of global optimization and accuracy.

A Case Study on Stochastic Games on Large Graphs in Mean Field and Sparse Regimes

We study a class of linear-quadratic stochastic differential games in which each player interacts directly only with its nearest neighbors in a given graph. We find a semiexplicit Markovian equilibrium for any transitive graph, in terms of the empirical eigenvalue distribution of the graph’s normalized Laplacian matrix. This facilitates large-population asymptotics for various graph sequences, with several sparse and dense examples discussed in detail. In particular, the mean field game is the correct limit only in the dense graph case, that is, when the degrees diverge in a suitable sense. Although equilibrium strategies are nonlocal, depending on the behavior of all players, we use a correlation decay estimate to prove a propagation of chaos result in both the dense and sparse regimes, with the sparse case owing to the large distances between typical vertices. Without assuming the graphs are transitive, we show also that the mean field game solution can be used to construct decentralized approximate equilibria on any sufficiently dense graph sequence.

Hart–Mas-Colell consistency and the core in convex games

This paper formally introduces Hart–Mas-Colell consistency for general (possibly multi-valued) solutions for cooperative games with transferable utility. This notion is used to axiomatically characterize the core on the domain of convex games. Moreover, we characterize all nonempty solutions satisfying individual rationality, anonymity, scale covariance, superadditivity, weak Hart–Mas-Colell consistency, and converse Hart–Mas-Colell consistency. This family consists of (a) the Shapley value, (b) all homothetic images of the core with the Shapley value as center of homothety and with positive ratios of homothety not larger than one, and (c) their relative interiors.

Global solutions of stochastic Stackelberg differential games under convex control constraint

This paper is concerned with a Stackelberg stochastic differential game, where the systems are driven by stochastic differential equation (SDE for short), in which the control enters the randomly disturbed coefficients (drift and diffusion). The control region is postulated to be convex. By making use of the first-order adjoint equation (backward stochastic differential equation, BSDE for short), we are able to establish the Pontryagin’s maximum principle for the leader’s global Stackelberg solution, within adapted open-loop structure and closed-loop memoryless information one, respectively, where the term global indicates that the leader’s domination over the entire game duration. Since the follower’s adjoint equation turns out to be a BSDE, the leader will be confronted with a control problem where the state equation is a kind of fully coupled forward–backward stochastic differential equation (FBSDE for short). As an application, we study a class of linear–quadratic (LQ for short) Stackelberg games in which the control process is constrained in a closed convex subset Γ of full space R m . The state equations are represented by a class of fully coupled FBSDEs with projection operators on Γ . By means of monotonicity condition method, the existence and uniqueness of such FBSDEs are obtained. When the control domain is full space, we derive the resulting backward stochastic Riccati equations.

Controlled Fractional Order Processes with Distributed Parameters

This work is devoted to the study of the problem of pursuit in systems controlled with distributed parameters of fractional order. Derivatives with respect to spatial variables are ordinary, integer, and even of arbitrary order. Designed and studied sampling schemes. A numerical method is constructed for finding strategies in suitable classes and for constructing the corresponding control laws. Theorems are proved for the possibility of completing the pursuit by the finite difference method, and they can be used to solve finite-difference optimal control problems or numerical solutions of differential games of fractional order. Problems of the type under study are encountered in modeling the processes of economic growth and in problems of stabilizing dynamic systems.

Quantum Game of Dual-channel Supply Chain under Free-riding Behavior

Based on the perspective of the quantum game, this paper explores when the online direct sales channel takes the free-riding behavior after the retail channel provides high-quality experience and services and how the dual-channel supply chain establishes a commodity pricing strategy. The retailer’s selling price follows a decreasing function of the free-riding behavior coefficient. while the online direct selling price does an increasing function of the free-riding behavior coefficient. Under centralized decision-making, there is no quantum entanglement, so the quantum game solution is consistent with the classical game solution. Under decentralized decision-making, the optimal price and profit of the quantum game are higher than those of the classical game when the quantum entanglement degree is greater than zero. When the quantum entanglement tends to be infinite, the optimal price of the quantum game finally remains in convergence. The quantum game theory is a more optimal decision-making method than the classical game theory.

Design of a Digital Game Intervention to Promote Socio-Emotional Skills and Prosocial Behavior in Children

In emergency contexts such as the COVID-19 pandemic, mental health risk factors affect children and may affect behavioral and emotional problems including anxiety, self-blame, emotional disturbance, and stress. Preventive measures are crucial to address these health risks. Research highlights strength-based factors, such as socio-emotional skills and prosocial behaviors, as important for childhood development and socio-emotional wellbeing. Yet, the initial evidence base shows mixed effectiveness and insufficient behavior change theory application into socio-emotional and prosocial digital game interventions. Additionally, few interventions are designed with a clear process to convert theory into practical game solutions and very limited guidance on the digital game development process exists. This study describes the Intervention Mapping Protocol (IMP) and theory application in the design of a digital game intervention which aims to strengthen social-emotional skills development and promote prosocial behavior in 8–11-year-old children. The method systematically describes the steps of the IMP process in detail to guide future theory-based game design. The results explain the final digital game prototype that was co-designed with continuous input and insights provided by stakeholders and feedback from children. This paper contributes to our understanding of theory application in the design of digital game interventions focused on health and behavior change and provides much needed guidance on how theory and stakeholder input can be incorporated into a final game design.

Lattice approximations of the first-order mean field type differential games

The theory of first-order mean field type differential games examines the systems of infinitely many identical agents interacting via some external media under assumption that each agent is controlled by two players. We study the approximations of the value function of the first-order mean field type differential game using solutions of model finite-dimensional differential games. The model game appears as a mean field type continuous-time Markov game, i.e., the game theoretical problem with the infinitely many agents and dynamics of each agent determined by a controlled finite state nonlinear Markov chain. Given a supersolution (resp. subsolution) of the Hamilton–Jacobi equation for the model game, we construct a suboptimal strategy of the first (resp. second) player and evaluate the approximation accuracy using the modulus of continuity of the reward function and the distance between the original and model games. This gives the approximations of the value function of the mean field type differential game by values of the finite-dimensional differential games. Furthermore, we present the way to build a finite-dimensional differential game that approximates the original game with a given accuracy.