103 research outputs found
On the linear convergence of distributed Nash equilibrium seeking for multi-cluster games under partial-decision information
This paper considers the distributed strategy design for Nash equilibrium
(NE) seeking in multi-cluster games under a partial-decision information
scenario. In the considered game, there are multiple clusters and each cluster
consists of a group of agents. A cluster is viewed as a virtual noncooperative
player that aims to minimize its local payoff function and the agents in a
cluster are the actual players that cooperate within the cluster to optimize
the payoff function of the cluster through communication via a connected graph.
In our setting, agents have only partial-decision information, that is, they
only know local information and cannot have full access to opponents'
decisions. To solve the NE seeking problem of this formulated game, a
discrete-time distributed algorithm, called distributed gradient tracking
algorithm (DGT), is devised based on the inter- and intra-communication of
clusters. In the designed algorithm, each agent is equipped with strategy
variables including its own strategy and estimates of other clusters'
strategies. With the help of a weighted Fronbenius norm and a weighted
Euclidean norm, theoretical analysis is presented to rigorously show the linear
convergence of the algorithm. Finally, a numerical example is given to
illustrate the proposed algorithm
Nash Equilibrium Seeking in N-Coalition Games via a Gradient-Free Method
This paper studies an -coalition non-cooperative game problem, where the
players in the same coalition cooperatively minimize the sum of their local
cost functions under a directed communication graph, while collectively acting
as a virtual player to play a non-cooperative game with other coalitions.
Moreover, it is assumed that the players have no access to the explicit
functional form but only the function value of their local costs. To solve the
problem, a discrete-time gradient-free Nash equilibrium seeking strategy, based
on the gradient tracking method, is proposed. Specifically, a gradient
estimator is developed locally based on Gaussian smoothing to estimate the
partial gradients, and a gradient tracker is constructed locally to trace the
average sum of the partial gradients among the players within the coalition.
With a sufficiently small constant step-size, we show that all players' actions
approximately converge to the Nash equilibrium at a geometric rate under a
strongly monotone game mapping condition. Numerical simulations are conducted
to verify the effectiveness of the proposed algorithm
Generalized Nash Equilibrium Seeking Algorithm Design for Distributed Constrained Multi-Cluster Games
The multi-cluster games are addressed in this paper, where all players team
up with the players in the cluster that they belong to, and compete against the
players in other clusters to minimize the cost function of their own cluster.
The decision of every player is constrained by coupling inequality constraints,
local inequality constraints and local convex set constraints. Our problem
extends well-known noncooperative game problems and resource allocation
problems by considering the competition between clusters and the cooperation
within clusters at the same time. Besides, without involving the resource
allocation within clusters, the noncooperative game between clusters, and the
aforementioned constraints, existing game algorithms as well as resource
allocation algorithms cannot solve the problem. In order to seek the
variational generalized Nash equilibrium (GNE) of the multi-cluster games, we
design a distributed algorithm via gradient descent and projections. Moreover,
we analyze the convergence of the algorithm with the help of variational
analysis and Lyapunov stability theory. Under the algorithm, all players
asymptotically converge to the variational GNE of the multi-cluster game.
Simulation examples are presented to verify the effectiveness of the algorithm
Gradient-Free Nash Equilibrium Seeking in N-Cluster Games with Uncoordinated Constant Step-Sizes
In this paper, we consider a problem of simultaneous global cost minimization
and Nash equilibrium seeking, which commonly exists in -cluster
non-cooperative games. Specifically, the agents in the same cluster collaborate
to minimize a global cost function, being a summation of their individual cost
functions, and jointly play a non-cooperative game with other clusters as
players. For the problem settings, we suppose that the explicit analytical
expressions of the agents' local cost functions are unknown, but the function
values can be measured. We propose a gradient-free Nash equilibrium seeking
algorithm by a synthesis of Gaussian smoothing techniques and gradient
tracking. Furthermore, instead of using the uniform coordinated step-size, we
allow the agents across different clusters to choose different constant
step-sizes. When the largest step-size is sufficiently small, we prove a linear
convergence of the agents' actions to a neighborhood of the unique Nash
equilibrium under a strongly monotone game mapping condition, with the error
gap being propotional to the largest step-size and the smoothing parameter. The
performance of the proposed algorithm is validated by numerical simulations
Fully Distributed Nash Equilibrium Seeking in N-Cluster Games
Distributed optimization and Nash equilibrium (NE) seeking problems have
drawn much attention in the control community recently. This paper studies a
class of non-cooperative games, known as -cluster game, which subsumes both
cooperative and non-cooperative nature among multiple agents in the two
problems: solving distributed optimization problem within the cluster, while
playing a non-cooperative game across the clusters. Moreover, we consider a
partial-decision information game setup, i.e., the agents do not have direct
access to other agents' decisions, and hence need to communicate with each
other through a directed graph whose associated adjacency matrix is assumed to
be non-doubly stochastic. To solve the -cluster game problem, we propose a
fully distributed NE seeking algorithm by a synthesis of leader-following
consensus and gradient tracking, where the leader-following consensus protocol
is adopted to estimate the other agents' decisions and the gradient tracking
method is employed to trace some weighted average of the gradient. Furthermore,
the algorithm is equipped with uncoordinated constant step-sizes, which allows
the agents to choose their own preferred step-sizes, instead of a uniform
coordinated step-size. We prove that all agents' decisions converge linearly to
their corresponding NE so long as the largest step-size and the heterogeneity
of the step-size are small. We verify the derived results through a numerical
example in a Cournot competition game
Game Theory Relaunched
The game is on. Do you know how to play? Game theory sets out to explore what can be said about making decisions which go beyond accepting the rules of a game. Since 1942, a well elaborated mathematical apparatus has been developed to do so; but there is more. During the last three decades game theoretic reasoning has popped up in many other fields as well - from engineering to biology and psychology. New simulation tools and network analysis have made game theory omnipresent these days. This book collects recent research papers in game theory, which come from diverse scientific communities all across the world; they combine many different fields like economics, politics, history, engineering, mathematics, physics, and psychology. All of them have as a common denominator some method of game theory. Enjoy
Coalitional Game Theory for Communication Networks: A Tutorial
Game theoretical techniques have recently become prevalent in many
engineering applications, notably in communications. With the emergence of
cooperation as a new communication paradigm, and the need for self-organizing,
decentralized, and autonomic networks, it has become imperative to seek
suitable game theoretical tools that allow to analyze and study the behavior
and interactions of the nodes in future communication networks. In this
context, this tutorial introduces the concepts of cooperative game theory,
namely coalitional games, and their potential applications in communication and
wireless networks. For this purpose, we classify coalitional games into three
categories: Canonical coalitional games, coalition formation games, and
coalitional graph games. This new classification represents an
application-oriented approach for understanding and analyzing coalitional
games. For each class of coalitional games, we present the fundamental
components, introduce the key properties, mathematical techniques, and solution
concepts, and describe the methodologies for applying these games in several
applications drawn from the state-of-the-art research in communications. In a
nutshell, this article constitutes a unified treatment of coalitional game
theory tailored to the demands of communications and network engineers.Comment: IEEE Signal Processing Magazine, Special Issue on Game Theory, to
appear, 2009. IEEE Signal Processing Magazine, Special Issue on Game Theory,
to appear, 200
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