  The class of weakly acyclic games, which includes potential games and
dominance-solvable games, captures many practical application domains. In a
weakly acyclic game, from any starting state, there is a sequence of
better-response moves that leads to a pure Nash equilibrium; informally, these
are games in which natural distributed dynamics, such as better-response
dynamics, cannot enter inescapable oscillations. We establish a novel link
between such games and the existence of pure Nash equilibria in subgames.
Specifically, we show that the existence of a unique pure Nash equilibrium in
every subgame implies the weak acyclicity of a game. In contrast, the possible
existence of multiple pure Nash equilibria in every subgame is insufficient for
weak acyclicity in general; here, we also systematically identify the special
cases (in terms of the number of players and strategies) for which this is
sufficient to guarantee weak acyclicity.
