This paper develops a coherent theory of international co-operation 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 co-operation by means of a self-enforcing agreement. Moreover, I find that this number is not fixed but depends on the nature of the co-operation 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 co-operation--that is, co-operation involving 200 or so countries--by a self-enforcing treaty when the gains to co-operation are "small." Finally, I find that the ability of the international system to sustain co-operation does not hinge on whether the compliance norm of customary international law has been internalised by states or whether compliance must instead be enforced by the use of treaty-based sanctions. The constraint on international co-operation is free-rider deterrence, not compliance enforcement.