Antisaccades are widely used in the study of voluntary behavioural control: a subject told to look in the opposite direction to a stimulus must suppress the automatic response of looking towards it, leading to delays and errors that are commonly believed to be generated by competing decision processes. However, currently we lack a precise model of the details of antisaccade behaviour, or indeed detailed quantitative data in the form of full reaction time distributions by which any such model could be evaluated. We measured subjects' antisaccade latency distributions and error rates, and found that we could account precisely for both distributions and errors with a model having three competing LATER processes racing to threshold. In an even more stringent test, we manipulated subjects' expectation of the stimulus, leading to large changes in behaviour that were nevertheless still accurately predicted. The antisaccade task is widely used in the laboratory and clinic because of the relative complexity and vulnerability of the underlying decision mechanisms: our model, grounded in detailed quantitative data, is a robust way of conceptualizing these processes.