We discretize a continuous-time game of strategic experimentation with Poisson bandits by imposing an equally-spaced grid of times at which the players can adjust their actions. We study the set of payoffs which can be obtained in strongly symmetric subgame perfect equilibria of the resulting stochastic game and analyze its limit as the grid size goes to zero. After computing an upper bound on the extent of experimentation that can be sustained in the limit, we construct equilibria that get arbitrarily close to this bound as the grid size vanishes. These equilibria involve two-state automata with a normal and a punishment state; a public randomization device governs the transitions from the latter to the former. We find that efficient behavior is sustainable in the limit if and only if news are “small”; for “big” news, the high level of optimism after a news event makes the wedge between the best and worst continuation payoffs too small to deter deviations from the efficient path of play. This result extends to subgame perfect equilibria that are not strongly symmetric. In the continuous-time limit, therefore, the equilibria that we construct realize the maximal efficiency gain relative to the Markov perfect equilibria that have been the focus of the strategic-experimentation literature so far.