We study a contest problem in which two players compete on a continuum of battlefields by spending resources subject to some constraints on their strategies. Following Myerson (1993), we assume that each player's resource allocation on each battlefield is an independent random draw from the same distribution. Within each battlefield, the player who allocates a higher level of resources wins, but both players incur costs for resource allocation. To analyze this problem, we introduce a systematic way to identify equilibrium allocation strategies and show that any equilibrium strategy of a player renders the rival indifferent in terms of the “constraint-adjusted payoff”. Using this, we provide a complete characterization of equilibrium allocation strategies. We also show that whereas a symmetric budget constraint may induce players to spend more resources than in the case with no constraint, asymmetric budgets that are proportional to players' values of winning would eliminate this possibility.