A general model of Bertrand-Edgeworth duopoly

This paper studies a class of two-player all-pay contests with externalities that encompass a general version of duopoly price competition. This all-pay contest formulation puts little restriction on production technologies, demand, and demand rationing. There are two types of possible equilibria: In the first type of equilibrium, the lower bound to pricing is the same for each firm, and the probability of any pricing tie above this price is zero. Each firm's equilibrium expected profit is their monopoly profit at the lower bound price. In the second type of equilibrium, one firm prices at the lower bound of the other firm's average cost and other firm prices according to a non-degenerate mixed strategy. This type of equilibrium can only occur if production technologies are sufficiently different across firms. We derive necessary and sufficient conditions for the existence of pure strategy equilibrium and use these conditions to demonstrate the fragility of deterministic outcomes in pricing games.