Dari-Mattiacci, G. and Parisi, F. (2006). Substituting complements Journal of Competition Law and Economics, 2(3):333--347.

The presence of multiple sellers in the provision of (nonsubstitutable) complementary goods leads to outcomes that are worse than those generated by a monopoly (with a vertically integrated production of complements), a problem known in the economic literature as complementary oligopoly and recently popularized in the legal literature as the tragedy of the anticommons. We ask the following question: how many substitutes for each complement are necessary to render the presence of multiple sellers preferable to a monopoly? Highlighting the asymmetries between Cournot (quantity) and Bertrand (price) competition and their dual models, we show that the results crucially depend on whether firms compete by controlling price or quantity. Two substitutes per component are sufficient when firms choose price. However, when firms choose quantity, the availability of substitutes, regardless of their number, is ineffective. Considering more complex cases of multi-complementarity, we ask the related question of how many complements need to be substitutable and offer comments on equilibrium prices and quantities under different scenarios.