2 research outputs found
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
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