This paper investigates perfect equilibrium in a model of a race in which two players are competing for an indivisible prize. The winner is the first player to reach the finishing line. It is shown that the behaviour of the winner of the race is often exactly as if he were the only player: the rival makes no difference. Even if competition does affect the winner's behaviour, it does so only in the first stage of the race and not thereafter. It is shown how several factors combine to determine which player will win: relative valuations of the prize, discount rates, efficiency at making progress and initial distances from the finishing line. Insofar as the model applies to patent races, it suggests that the potential competition faced by one firm in a patent race (e.g. an incumbent monopolist) may be of little or no consequence.