This paper examines the foreign direct investment (FDI) versus exports decision of foreign oligopolistic firms under cost heterogeneity. An additional motivation for firms to invest abroad is the technological sourcing via spillovers, which flow from the host more efficient firm to foreign less advantaged firms. For intermediate values of the set‐up costs associated with FDI entry, it is shown that foreign firms choose opposite entry strategies. An equilibrium where the less efficient foreign firm exports whereas the more efficient invests is more likely to happen when foreign firms become more heterogeneous, the larger the trade costs and not too big oligopolistic profitability. Interestingly, the opposite may also be an equilibrium thus finding that the more efficient firm does not choose to invest, a result that emphasizes the relevance of the strategic setting under consideration. The latter result identifies a market failure since welfare in the host market is higher when both firms undertake FDI; a finding that calls attention to how appropriate are host government policies towards internationalization strategies.