We analyse a stylized model of the world grain market characterized by a small oligopoly of traders with market power on both the supply and demand side. Crops are stochastic and exporting countries can impose export tariffs to protect domestic food prices. Our first results is that export tariffs are strategic complements and that for poor harvests equilibrium tariffs can explode (shedding some light on recent volatility in world food prices). We also show that the strategic interplay between governments of export countries and traders can give rise to a number of peculiar comparative statics. For example, it can be in the interest of traders to have poor harvests in one of the countries. Finally, we demonstrate that traders as well as consumers in import countries can benefit from cooperation between grain exporting countries.