Governments often subsidize poorer groups in society to ensure their access to new drugs. We analyse the optimal income‐based price subsidies in a strategic environment. We show that universal access is less likely to arise when price arbitrage prevents international price discrimination. When this is not the case, under some income ranges, bilateral universal coverage can be supported by equilibrium subsidies together with bilateral partial provision. In such a case, international health policy coordination becomes relevant. We also show that asymmetric universal access to medicines across countries can arise, even when countries are ex ante symmetric, when international price discrimination is possible and governments cannot design subsidies proportional to either income or quality.