N-player cournot and price-quantity function mixed competition

We study the value of network providers committing to offering a quantity of bandwidth to a market versus having the amount of bandwidth offered be conditional on the prices that the market settles upon. For instance a cable television ISP has the option to shift capacity from Internet service to television channels if the market price for Internet service is low, and thus such a provider can avoid committing to a fixed capacity being devoted to Internet service. To study the issue, we consider a two-stage game. In the first stage, competing network providers either commit to set a quantity of bandwidth to offer to the market, or choose to offer bandwidth to the market according to a function relating price to quantity. If they choose the later option, the network provider initially chooses a slope for their function. In the second stage, the quantity players choose the quantity to offer, where as the function players choose the offset term of their function. We show that a unique Nash equilibrium exists for the second stage play, and that it is the only subgame-perfect equilibrium for each provider to choose a quantity commitment in the first stage. We also show that a quantity commitment is not always a subgame-perfect equilibrium when demand uncertainty is introduced to the model.