407 research outputs found
Nash equilibrium seeking over digraphs with row-stochastic matrices and network-independent step-sizes
In this paper, we address the challenge of Nash equilibrium (NE) seeking in
non-cooperative convex games with partial-decision information. We propose a
distributed algorithm, where each agent refines its strategy through
projected-gradient steps and an averaging procedure. Each agent uses estimates
of competitors' actions obtained solely from local neighbor interactions, in a
directed communication network. Unlike previous approaches that rely on
(strong) monotonicity assumptions, this work establishes the convergence
towards a NE under a diagonal dominance property of the pseudo-gradient
mapping, that can be checked locally by the agents. Further, this condition is
physically interpretable and of relevance for many applications, as it suggests
that an agent's objective function is primarily influenced by its individual
strategic decisions, rather than by the actions of its competitors. In virtue
of a novel block-infinity norm convergence argument, we provide explicit bounds
for constant step-size that are independent of the communication structure, and
can be computed in a totally decentralized way. Numerical simulations on an
optical network's power control problem validate the algorithm's effectiveness
Distributed Nash Equilibrium Seeking with Limited Cost Function Knowledge via A Consensus-Based Gradient-Free Method
This paper considers a distributed Nash equilibrium seeking problem, where
the players only have partial access to other players' actions, such as their
neighbors' actions. Thus, the players are supposed to communicate with each
other to estimate other players' actions. To solve the problem, a
leader-following consensus gradient-free distributed Nash equilibrium seeking
algorithm is proposed. This algorithm utilizes only the measurements of the
player's local cost function without the knowledge of its explicit expression
or the requirement on its smoothness. Hence, the algorithm is gradient-free
during the entire updating process. Moreover, the analysis on the convergence
of the Nash equilibrium is studied for the algorithm with both diminishing and
constant step-sizes, respectively. Specifically, in the case of diminishing
step-size, it is shown that the players' actions converge to the Nash
equilibrium almost surely, while in the case of fixed step-size, the
convergence to the neighborhood of the Nash equilibrium is achieved. The
performance of the proposed algorithm is verified through 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
Geometric Convergence of Distributed Heavy-Ball Nash Equilibrium Algorithm over Time-Varying Digraphs with Unconstrained Actions
We propose a new distributed algorithm that combines heavy-ball momentum and
a consensus-based gradient method to find a Nash equilibrium (NE) in a class of
non-cooperative convex games with unconstrained action sets. In this approach,
each agent in the game has access to its own smooth local cost function and can
exchange information with its neighbors over a communication network. The
proposed method is designed to work on a general sequence of time-varying
directed graphs and allows for non-identical step-sizes and momentum
parameters. Our work is the first to incorporate heavy-ball momentum in the
context of non-cooperative games, and we provide a rigorous proof of its
geometric convergence to the NE under the common assumptions of strong
convexity and Lipschitz continuity of the agents' cost functions. Moreover, we
establish explicit bounds for the step-size values and momentum parameters
based on the characteristics of the cost functions, mixing matrices, and graph
connectivity structures. To showcase the efficacy of our proposed method, we
perform numerical simulations on a Nash-Cournot game to demonstrate its
accelerated convergence compared to existing methods
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
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