This paper develops a coherent theory of international cooperation relying on the twin assumptions of individual and collective rationality. Using a linear version of the N-player prisoners' dilemma game, I provide a formal proof of Olson's conjecture that only a "small" number of countries can sustain full cooperation by means of a self-enforcing agreement. Moreover, I find that this number is not fixed but depends on the nature of the cooperation problem; for some problems, three countries will be "too many," while for others even 200 countries will be a "small" number. In addition, I find that the international system is only able to sustain global cooperation--that is, cooperation involving 200 or so countries--by a self-enforcing treaty when the gains to cooperation are "small." Finally, I find that the ability of the international system to sustain cooperation does not hinge on whether the compliance norm of customary international law has been internalized by states or whether compliance must instead be enforced by the use of treaty-based sanctions. The constraint on international cooperation is free-rider deterrence, not compliance enforcement.