We study the effects of airport ownership (private vs. government) on welfare in the presence of airport complementarity, where each airport is located in a different country. Considering Cournot competition in the airline market, the unique Nash equilibrium is such that the two countries privatize their airports, even though both countries are better off, from a welfare perspective, with public (government-owned) airports. Considering a differentiated Bertrand competition in the airline market, the same result prevails if the cross price elasticities are sufficiently high, otherwise the symmetric government-ownership of airports may also be a Nash equilibrium.