  We investigate asynchronous distributed algorithms with delayed information
for seeking generalized Nash equilibrium over multi-agent networks. The
considered game model has all players' local decisions coupled with a shared
affine constraint. We assume each player can only access its local objective
function, local constraint, and a local block matrix of the affine constraint.
We first give the algorithm for the case when each agent is able to fully
access all other players' decisions. With the help of auxiliary edge variables
and edge Laplacian matrix, each player can carry on its local iteration in an
asynchronous manner, using only local data and possibly delayed neighbour
information. And then we investigate the case when the agents cannot know all
other players' decisions, which is called a partial decision case. We introduce
a local estimation of the overall decisions for each agent in the partial
decision case, and develop another asynchronous algorithm by incorporating
consensus dynamics on the local estimations of the overall decisions. Since
both algorithms do not need any centralized clock coordination, the algorithms
fully exploit the local computation resource of each player, and remove the
idle time due to waiting for the "slowest" agent. Both algorithms are developed
by a preconditioned forward-backward operator splitting method, while
convergence is shown with the help of asynchronous fixed-point iterations under
proper assumptions and fixed step-size choices. Numerical studies verify both
algorithms' convergence and efficiency.
