  Weakly acyclic games generalize potential games and are fundamental to the
study of game theoretic control. In this paper, we present a generalization of
weakly acyclic games, and we observe its importance in multi-agent learning
when agents employ experimental strategy updates in periods where they fail to
best respond. While weak acyclicity is defined in terms of path connectivity
properties of a game's better response graph, our generalization is defined
using a generalized better response graph. We provide sufficient conditions for
this notion of generalized weak acyclicity in both two-player games and
$n$-player games. To demonstrate that our generalization is not trivial, we
provide examples of games admitting a pure Nash equilibrium that are not
generalized weakly acyclic. The generalization presented in this work is
closely related to the recent theory of satisficing paths, and the
counterexamples presented here constitute the first negative results in that
theory.
