Airlines often cooperate with partners within strategic alliances to offer their customers itineraries beyond their own networks. However, despite the cooperation, the alliance members often remain competitors on other routes and compete for customers. This thesis takes into account the competition between two alliance partners and models it in a linear program. An algorithm is developed to compute optimal capacity allocations in pure Nash equilibria based on the model. For cases, in which a Nash equilibrium does not exist or cannot be found within a pre-defined time, a heuristic approach is described to compute an approximate Nash equilibrium. Computational studies show the applicability of the approaches in real-sized airline networks. Finally, suggestions for future research are made.