Making a decision and reporting your confidence in the accuracy of that decision are thought to reflect a similar mechanism: the accumulation of evidence. Previous research has shown that choices and reaction times are well accounted for by a computational model assuming noisy accumulation of evidence until crossing a decision boundary (e.g., the drift diffusion model). Decision confidence can be derived from the amount of evidence following post-decision evidence accumulation. Currently, the stopping rule for post-decision evidence accumulation is underspecified. Inspired by recent neurophysiological evidence, we introduce additional confidence boundaries that determine the termination of post-decision evidence accumulation. If this conjecture is correct, it implies that confidence judgments should be subject to the same strategic considerations as the choice itself, i.e. a tradeoff between speed and accuracy. To test this prediction, we instructed participants to make fast or accurate decisions, and to give fast or carefully considered confidence judgments. Results show that our evidence accumulation model with additional confidence boundaries successfully captured the speed-accuracy tradeoffs seen in both decisions and confidence judgments. Most importantly, instructing participants to make fast versus accurate decisions influenced the decision boundaries, whereas instructing participants to make fast versus careful confidence judgments influenced the confidence boundaries. Our data show that the stopping rule for confidence judgments can be well understood within the context of evidence accumulation models, and that the computation of decision confidence is under strategic control.