Abstract
In this paper, we deal with the problem of finding a Nash equilibrium for a generalized convex game. Each player is associated with a convex cost function and multiple shared constraints. Supposing that each player can exchange information with its neighbors via a connected undirected graph, the objective of this paper is to design a Nash equilibrium seeking law such that each agent minimizes its objective function in a distributed way. Consensus and singular perturbation theories are used to prove the stability of the system. A numerical example is given to show the effectiveness of the proposed algorithms.
Highlights
How to find a Nash equilibrium is an interesting and important problem for non-cooperative games [1]
A generalized convex game usually has continuous strategy spaces and the actions are coupled through both objective functions and constraints
Compared with [19], the objective functions and constraints can be coupled arbitrarily; 2) The convergence of the players actions to the normalized Nash equilibrium is analyzed, by using singular perturbation based techniques, it is proven that the proposed algorithms converge into a neighborhood of the Nash equilibrium and the error bound can be arbitrarily small by selecting the control parameters
Summary
How to find a Nash equilibrium is an interesting and important problem for non-cooperative games [1]. The recently proposed algorithm in [18] solved a distributed Nash equilibrium seeking problem for unconstrained non-cooperative games based on average consensus and singular perturbation theory. In [19], discrete-time adaptive algorithms were presented to solve Nash equilibrium seeking problems, the objectives and constraints are required to be neighbor-coupled. We present a continuous-time distributed algorithm to seek a Nash equilibrium for a generalized convex game with shared constraints. Compared with [19], the objective functions and constraints can be coupled arbitrarily; 2) The convergence of the players actions to the normalized Nash equilibrium is analyzed, by using singular perturbation based techniques, it is proven that the proposed algorithms converge into a neighborhood of the Nash equilibrium and the error bound can be arbitrarily small by selecting the control parameters.
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