[The existence of perfect equilibrium is demonstrated for a class of games with compact space of histories and continuous payoffs, and in which the set of actions feasible at any given period is a lower hemicontinuous correspondence of the previous history of the game. The proof is by construction. A set of histories is constructed, each of which is the equilibrium path of some perfect equilibrium point of the game. Also, any equilibrium path is a member of this set. The construction therefore provides a characterization of perfect equilibrium.]

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.