Motivated by theoretical and experimental economics, we propose novel evolutionary dynamics for games on networks, called the h-Relative Best Response (h-RBR) dynamics, that mixes the relative performance considerations of imitation dynamics with the rationality of best responses. Under such a class of dynamics, the players optimize their payoffs over the set of strategies employed by a time-varying subset of their neighbors. As such, the h-RBR dynamics share the defining non-innovative characteristic of imitation based dynamics and can lead to equilibria that differ from classic Nash equilibria. We study the asymptotic behavior of the h-RBR dynamics for both finite and convex games in which the strategy spaces are discrete and compact, respectively, and provide preliminary sufficient conditions for finite-time convergence to a generalized Nash equilibrium.