Empirical findings suggest that the mammalian brain has two decision-making systems that act at different speeds. We represent the faster system using standard signal detection theory. We represent the slower (but more accurate) cortical system as the integration of sensory evidence over time until a certain level of confidence is reached. We then consider how two such systems should be combined optimally for a range of information linkage mechanisms. We conclude with some performance predictions that will hold if our representation is realistic.