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We use the framework of random matching games and develop a two-country model of the world economy, in which two national currencies compete and may be circulated as media of exchange. There are multiple equilibria, which differ in the areas of circulation of the two currencies. In one equilibrium, the two national currencies are circulated only locally. In another, one currency is circulated as an international currency. There is also an equilibrium in which both currencies are accepted internationally. We also find an equilibrium in which the two currencies are directly exchanged. We first characterize the existence conditions of these equilibria in terms of the relative country size and the degree of economic integration and then use an evolutionary approach to equilibrium selection to explain the evolution of the international currency as the two economies become more integrated. Some welfare implications are also discussed. For example, a country can improve its national welfare by letting its own currency circulate internationally, provided the domestic circulation is controlled for. When the total supply is fixed, however, a resulting currency shortage may reduce the national welfare.