  Multi-time scale techniques, such as singular perturbations and averaging
theory, have played an essential role in the development of distributed Nash
equilibrium-seeking algorithms for network systems. Such techniques
intrinsically rely on the uniform asymptotic stability properties of the
dynamics that evolve in each time scale of the closed-loop system. When such
properties are absent, synthesizing multi-time scale Nash equilibrium-seeking
algorithms is more challenging and requires additional regularization
mechanisms. In this paper, we investigate the synthesis and analysis of these
mechanisms in the context of accelerated pseudogradient flows with time-varying
damping in non-cooperative games. Specifically, we introduce a new class of
distributed and hybrid Nash set-seeking (NSS) algorithms that synergistically
combine dynamic momentum-based flows with coordinated discrete-time resets. The
reset mechanisms can be seen as restarting techniques that allow individual
players to choose their own momentum restarting policy to achieve better
transient performance. The resulting closed-loop system is modeled as a hybrid
dynamic inclusion, which is analyzed using tools from hybrid dynamical system's
theory. Our algorithms are developed for potential games, as well as for
monotone games for which a potential function does not exist. They can be
implemented in games where players have access to gradient Oracles with full or
partial information, as well as in games where players have access only to
measurements of their costs. In the latter case, we use tools from hybrid
extremum seeking control to achieve accelerated model-free Nash set seeking.
