Properties Of Nash Equilibria Research Articles

Distributed Adaptive Nash Equilibrium Solution for Differential Graphical Games.

This article investigates differential graphical games for linear multiagent systems with a leader on fixed communication graphs. The objective is to make each agent synchronize to the leader and, meanwhile, optimize a performance index, which depends on the control policies of its own and its neighbors. To this end, a distributed adaptive Nash equilibrium solution is proposed for the differential graphical games. This solution, in contrast to the existing ones, is not only Nash but also fully distributed in the sense that each agent only uses local information of its own and its immediate neighbors without using any global information of the communication graph. Moreover, the asymptotic stability and global Nash equilibrium properties are analyzed for the proposed distributed adaptive Nash equilibrium solution. As an illustrative example, the differential graphical game solution is applied to the microgrid secondary control problem to achieve fully distributed voltage synchronization with optimized performance.

Multivariate systemic optimal risk transfer equilibrium

A Systemic Optimal Risk Transfer Equilibrium (SORTE) was introduced in: “Systemic optimal risk transfer equilibrium”, Mathematics and Financial Economics (2021), for the analysis of the equilibrium among financial institutions or in insurance-reinsurance markets. A SORTE conjugates the classical Bühlmann’s notion of a risk exchange equilibrium with a capital allocation principle based on systemic expected utility optimization. In this paper we extend such a notion to the case when the value function to be optimized is multivariate in a general sense, and it is not simply given by the sum of univariate utility functions. This takes into account the fact that preferences of single agents might depend on the actions of other participants in the game. Technically, the extension of SORTE to the new setup requires developing a theory for multivariate utility functions and selecting at the same time a suitable framework for the duality theory. Conceptually, this more general framework allows us to introduce and study a Nash Equilibrium property of the optimizer. We prove existence, uniqueness, and the Nash Equilibrium property of the newly defined Multivariate Systemic Optimal Risk Transfer Equilibrium.

Nash equilibrium and bang-bang property for the non-zero-sum differential game of multi-player uncertain systems with Hurwicz criterion

How to select the optimisation criterion of the performance function is a pivotal research topic for the uncertain differential game. The Hurwicz criterion is a flexible optimisation measure to allocate some weight between optimism and pessimism to achieve balance for both of them. Based on uncertainty theory, this paper considers the Nash equilibrium and bang-bang property for the multi-player non-zero-sum uncertain differential game model by applying the Hurwicz criterion to optimise the performance function. To coordinate onto the Nash equilibrium, an equilibrium equation is derived. In addition, the application of bang-bang optimal control in the multi-player game problem of the uncertain linear system is investigated. Finally, an example is shown to validate the results obtained.

Dynamic optimization problems for mean-field stochastic large-population systems

This paper considers dynamic optimization problems for a class of control average meanfield stochastic large-population systems. For each agent, the state system is governed by a linear mean-field stochastic differential equation with individual noise and common noise, and the weight coefficients in the corresponding cost functional can be indefinite. The decentralized optimal strategies are characterized by stochastic Hamiltonian system, which turns out to be an algebra equation and a mean-field forward-backward stochastic differential equation. Applying the decoupling method, the feedback representation of decentralized optimal strategies is further obtained through two Riccati equations. The solvability of stochastic Hamiltonian system and Riccati equations under indefinite condition is also derived. The explicit structure of the control average limit and the related mean-field Nash certainty equivalence equation systems are discussed by some separation techniques. Moreover, the decentralized optimal strategies are proved to satisfy the approximate Nash equilibrium property. The good performance of the proposed theoretical results is illustrated by a practical example from the engineering field.

Stability, multi-stability and instability in Cournot duopoly game with knowledge spillover effects and relative profit maximization

A dynamic Cournot duopoly model with isoelastic demand function and knowledge spillover effects is established, in which the goals of two enterprises are relative profit maximization. Since these two enterprises only have incomplete information of the market, then it is assumed that the enterprises are boundedly rational, and they need to adjust their production decision in next period according to the gradient adjustment mechanism. Sufficient conditions for the existence and stable properties of unique Nash equilibrium can be judged with the help of the Jury criterion. The size of stability region is related to the values of parameters and the system has two different paths to chaos, one is via flip bifurcation and another is via Neimark-Sacker bifurcation. Finally, the complex dynamic characteristics also can be further studied by using numerical simulation, including the structures of multi-layer Arnold's tongues and coexistence of attractors.

Individual and mass behavior in large population forward–backward stochastic control problems: Centralized and Nash equilibrium solutions

AbstractThis article studies a class of dynamic optimization of large‐population system in which a large number of negligible agents are coupled via state‐average in their cost functional and state dynamics. The most significant feature in our setup is the dynamics of individual agents are modeled by forward–backward stochastic differential equations. The associated mean‐field game, in its forward–backward sense, is also formulated to seek the decentralized strategies. Unlike the forward case, the consistency conditions of our forward–backward mean‐field games involve five Riccati and force rate equations. Moreover, their initial and terminal conditions are mixed thus some special decoupling technique is applied here. We also verify the ‐Nash equilibrium property of the derived decentralized strategies. To this end, some estimates to backward stochastic system are employed.

Differential Graphical Game With Distributed Global Nash Solution

This article studies multiagent differential graphical games in linear dynamical networks. We point out that in existing graphical game formulations, being global Nash and being distributed are two contradicting properties. In particular, we prove that the best response strategy, most widely used in solving graphical games, leads to Nash but does not provide distributed solutions. On the other hand, the minmax strategy, recently developed in solving graphical games, provides distributed solutions but prevents the agents from reaching a global Nash equilibrium. In this article, we propose a novel differential graphical game formulation, which promises the existence of solutions that 1) lead to global Nash equilibrium and 2) are distributed in the sense that each agent only uses the state information of its own and its direct neighbors. Stability and Nash equilibrium properties of the proposed graphical game are proven, respectively. Simulation studies are conducted to illustrate the theoretical results.

Two Nash-equilibrium-based steering control models for representing a driver’s interaction with vehicle automated steering

ABSTRACT Automated steering technology offers significant benefits to the safety of vehicles, but desire to keep the human driver in the loop requires a better understanding of the interaction between driver and vehicle. An existing noncooperative–game–theoretic framework for modelling such interaction is revisited, leading to the development of two driver steering control models. Both bear Nash–equilibrium properties, but involve different assumptions about driver steering behaviour. A simulation study is performed to demonstrate the difference between the two driver models. An experiment using a fixed–base driving simulator is conducted to measure six test subjects' steering angles in response to the lane–change manoeuvres generated by an automated steering controller. The two driver models' capabilities for representing driver steering behaviour are investigated through fitting them to measured driver steering angles. Key model parameters are identified using a system identification procedure. It is found that the two driver models have equivalent capability in capturing the trend of the six test subjects' measured steering angles, but less good at reproducing the overshoot and oscillation involved in two subjects' steering angles. It is found that the inclusion of an arm neuromuscular system model can improve the performance of the proposed driver models.

Potential difference games and applications

This paper concerns noncooperative difference games with infinite horizon discounted payoffs. More precisely, it is a sequel to a previous paper [A. Fonseca-Morales and O. Hern'andez-Lerma, Potential differential games, Dynamic Games Appl., 8(2) (2018), pp. 254–279.] where the notion of continuous-time potential games was introduced. That is, a noncooperative differential game to which we can associate a continuous-time optimal control problem (OCP) whose solutions are Nash equilibria for the original game. Thus, finding or analysing the properties of Nash equilibria for the game reduces to that of the optimal solution of an OCP. Here, we study difference games, that is, the discrete-time case. First, we give several mild conditions for which a difference game is a potential difference game (PDG). Then, we illustrate our results with several examples and applications.

A Game-Theoretic Approach for Robust Federated Learning

Federated Learning enables aggregating models trained over a large number of clients by sending these models to a central server, while data privacy is preserved since only the models are sent. Federated learning techniques are considerably vulnerable to poisoning attacks. In this paper, we explore the threat of poisoning attacks and introduce a game-based robust federated averaging algorithm to detect and discard bad updates provided by the clients. We model the aggregating process with a mixed-strategy game that is played between the server and each client. The valid actions of the clients are to send good or bad updates while the server can accept or ignore these updates as its valid actions. By employing the Nash Equilibrium property, the server determines the probability of providing good updates by each client. The experimental results show that our proposed game-based aggregation algorithm is significantly more robust to faulty and noisy clients in comparison with the most recently presented methods. According to these results, our algorithm converges after a maximum of 30 iterations and can detect 100% of the bad clients for all the investigated scenarios. In addition, the accuracy of the proposed algorithm is at least 15.8% and 2.3% better than state of the art for flipping and noisy scenarios, respectively.

Linear-quadratic-Gaussian mean-field-game with partial observation and common noise

This paper considers a class of linear-quadratic-Gaussian (LQG) mean-field games (MFGs) with partial observation structure for individual agents. Unlike other literature, there are some special features in our formulation. First, the individual state is driven by some common-noise due to the external factor and the state-average thus becomes a random process instead of a deterministic quantity. Second, the sensor function of individual observation depends on state-average thus the agents are coupled in triple manner: not only in their states and cost functionals, but also through their observation mechanism. The decentralized strategies for individual agents are derived by the Kalman filtering and separation principle. The consistency condition is obtained which is equivalent to the wellposedness of some forward-backward stochastic differential equation (FBSDE) driven by common noise. Finally, the related \begin{document}$ \epsilon $\end{document} -Nash equilibrium property is verified.

Nash equilibrium strategies and survival portfolio rules in evolutionary models of asset markets

We consider a stochastic model of a financial market with one-period assets and endogenous asset prices. The model was initially developed and analyzed in the context of Evolutionary Finance with the main focus on questions of “survival and extinction” of investment strategies (portfolio rules). In this paper we view the model from a different, game-theoretic, perspective and analyze Nash equilibrium properties of survival portfolio rules.

Broadcast Retransmission Algorithm Based on Potential Game in CRNs

The potential game as a special type of game, compared to other types of game, has the finite increment property, the existence and uniqueness of Nash equilibrium and other good properties and can be used to solve the problem of resource allocation in cognitive radio networks (CRNs). Broadcast retransmission is a important function in CRNs, which need to be optimized in order to improve the efficiency and utilization of the networks. When packets failed to reach some receivers in broadcasting, retransmission occurs. Choosing an appropriate channel to delivery packages is critical to the efficiency and quality of data retransmission. In this paper, first, we establish a potential game model for broadcasting retransmission by taking into account of energy consumption and interference in the design of utility function and potential function in spectrum allocation. Then we propose Potential Game-Broadcasting Retransmission Algorithm (PG-BRA) based on potential game model. In PG-BRA, two broadcast retransmission mechanisms are designed to avoid “conflict” problems, named FRM and DRM. FRM is more suitable for static network topology and DRM is more suitable for dynamic network topology. The simulation results show that the PG-BRA proposed in this paper can effectively work in terms of packet reach rate, average retransmission times, average delay and the total transmission power of system compared with existing broadcasting scheme.

Mean-Field Game Strategies for Optimal Execution

ABSTRACTAlgorithmic trading strategies for execution often focus on the individual agent who is liquidating/acquiring shares. When generalized to multiple agents, the resulting stochastic game is notoriously difficult to solve in closed-form. Here, we circumvent the difficulties by investigating a mean-field game framework containing (i) a major agent who is liquidating a large number of shares, (ii) a number of minor agents (high-frequency traders (HFTs)) who detect and trade against the liquidator, and (iii) noise traders who buy and sell for exogenous reasons. Our setup accounts for permanent price impact stemming from all trader types inducing an interaction between major and minor agents. Both optimizing agents trade against noise traders as well as one another. This stochastic dynamic game contains couplings in the price and trade dynamics, and we use a mean-field game approach to solve the problem. We obtain a set of decentralized feedback trading strategies for the major and minor agents, and express the solution explicitly in terms of a deterministic fixed point problem. For a finite population of HFTs, the set of major-minor agent mean-field game strategies is shown to have a -Nash equilibrium property where as .

Mean Field Games with Partial Observation

Subject to reasonable conditions, in large population stochastic dynamics games, where the agents are coupled by the system's mean field (i.e., the state distribution of the generic agent) through their nonlinear dynamics and their nonlinear cost functions, it can be shown that a best response control action for each agent exists which (i) depends only upon the individual agent's state observations and the mean field, and (ii) achieves an $\epsilon$-Nash equilibrium for the system. In this work we formulate a class of problems where each agent has only partial observations on its individual state. We employ nonlinear filtering theory and the separation principle in order to analyze the game in the asymptotically infinite population limit. The main result is that the $\epsilon$-Nash equilibrium property holds where the best response control action of each agent depends upon the conditional density of its own state generated by a nonlinear filter, together with the system's mean field. Finally, comparing this MFG problem with state estimation to that found in the literature with a major agent whose partially observed state process is independent of the control action of any individual agent, it is seen that, in contrast, the partially observed state process of any agent in this work depends upon that agent's control action.

Game-Based Thermal-Delay-Aware Adaptive Routing (GTDAR) for Temperature-Aware 3D Network-on-Chip Systems

The thermal problem of three-dimensional Network-on-Chip (3D NoC) is proven to be severer than 2D NoC due to the stacking dies and heterogeneous thermal conduction between each silicon layers. To control the system temperature under a certain thermal limit, the current Dynamic Thermal Managements ( DTMs ) can be classified into temporal approaches and spatial approaches. The temporal DTM approaches reduce the processing speed of those overheated NoC components. However, for emerging cooling consideration, the full throttling scheme is usually applied as the system temperature reaches the alarming level, which results in significant system performance overhead. On the other hand, the spatial DTM approaches migrate the traffic load away from the overheated components. Although the spatial DTM approaches can mitigate the performance impact during the temperature control, the cooling period is longer than the temporal approaches because of the asynchronous phenomenon of traffic and temperature behavior among the NoC components. To consider the advantages of the temporal and spatial DTM approaches, it is necessary to synchronize the information of traffic and temperature behavior in the NoC systems. In this paper, we apply the Game Theory to propose a Game-based Thermal-Delay-aware Adaptive Routing ( GTDAR ) scheme. The GTDAR first adopts the Thermal-Delay principle to transfer the long-term temperature information to short-term traffic information by allocating the input buffer length of each NoC routers, which can reduce the thermal problem into the traffic problem. Afterward, the GTDAR involves the Nash Equilibrium property to distribute the packet routing to mitigate the thermal problem by considering the traffic and temperature simultaneously. In our experiments, the proposed Game-based Thermal-Delay-aware Adaptive Routing ( GTDAR ) scheme can help to improve 8.7 percent to 130 percent system performance with only 2.4 percent area overhead compared with the previous works.

Linear–Quadratic Mean-Field Game for Stochastic Delayed Systems

This paper studies the linear–quadratic mean-field game (MFG) for a class of stochastic delayed systems. We consider a large-population system, where the dynamics of each agent is modeled by a stochastic differential delayed equation. The consistency condition is derived through an auxiliary system, which is an anticipated forward–backward stochastic differential equation with delay (AFBSDDE). The wellposedness of such an AFBSDDE system can be obtained using a continuation method. Thus, the MFG strategies can be defined on an arbitrary time horizon, not necessary on a small time horizon by a commonly used contraction mapping method. Moreover, the decentralized strategies are verified to satisfy the $\epsilon$ -Nash equilibrium property. For illustration, three special cases of delayed systems are further explored, for which the closed-loop and open-loop MFG strategies are derived, respectively.

Interference-Aware Resource Competition Toward Power-Efficient Ultra-Dense Networks

Ultra-dense networks are envisioned as essential to embrace the skyrocketed traffic for the next-generation wireless networks. In this paper, we consider the uplink transmissions in ultra-dense networks, where the increased interference along with the increased density significantly challenges the efficient utilization of network resources as well as the provisioning of users’ quality of services (QoS). Targeting these issues, we consider the QoS in terms of target signal-to-interference-plus-noise ratio (SINR) and power consumption simultaneously for each user within a multi-objective optimization model. We then investigate the interactions among users by leveraging the non-cooperative game-theoretical framework. By characterizing the properties of Nash equilibrium, we develop the target-SINR oriented resource allocation (TORA) algorithm, which features distributed implementation. Moreover, we obtain the condition to guarantee the convergence of our proposed TORA algorithm and demonstrate that it adapts to different interfering scenarios. Furthermore, considering the heterogeneous service requirements in real practice, we also design the target-SINR constrained resource allocation (TCRA) algorithm, such that TORA and TCRA are able to cope with voice and data services, respectively. Also provided are the simulation results, which demonstrate that, compared with the counterparts, our proposals more effectively guarantee the target-SINR for users with efficient power utilization.

Linear-Quadratic-Gaussian Mixed Mean-Field Games with Heterogeneous Input Constraints

We consider a class of linear-quadratic-Gaussian mean-field games with a major agent and considerable heterogeneous minor agents with mean-field interactions. The individual admissible controls are constrained in closed convex subsets $Γ k$ of full space $R m$. The decentralized strategies for individual agents and consistency condition system are represented in an unified manner via a class of mean-field forward-backward stochastic differential equations involving projection operators on $Γ k$. The well-posedness of consistency system is established in the local and global cases both by the contraction mapping and discounting method respectively. The related $e$−Nash equilibrium property is also verified.

Nash Equilibrium Strategies and Survival Portfolio Rules in Evolutionary Models of Asset Markets

We consider a stochastic model of a financial market with one-period assets and endogenous asset prices. The model was initially developed and analyzed in the context of Evolutionary Finance with the main focus on questions of and extinction of investment strategies (portfolio rules). In this paper we view the model from a different, game-theoretic, perspective and analyze Nash equilibrium properties of survival portfolio rules.