  This paper studies a class of strongly monotone games involving
non-cooperative agents that optimize their own time-varying cost functions. We
assume that the agents can observe other agents' historical actions and choose
actions that best respond to other agents' previous actions; we call this a
best response scheme. We start by analyzing the convergence rate of this best
response scheme for standard time-invariant games. Specifically, we provide a
sufficient condition on the strong monotonicity parameter of the time-invariant
games under which the proposed best response algorithm achieves exponential
convergence to the static Nash equilibrium. We further illustrate that this
best response algorithm may oscillate when the proposed sufficient condition
fails to hold, which indicates that this condition is tight. Next, we analyze
this best response algorithm for time-varying games where the cost functions of
each agent change over time. Under similar conditions as for time-invariant
games, we show that the proposed best response algorithm stays asymptotically
close to the evolving equilibrium. We do so by analyzing both the equilibrium
tracking error and the dynamic regret. Numerical experiments on economic market
problems are presented to validate our analysis.
