Undercut-proof Subgame Perfect Equilibrium of a Pricing Game

Two firms selling a homogenous product to two types of buyers are involved in a sequential pricing game with zero costs. The pricing strategy available involves a fixed price and a royalty. It is shown that there exists a unique subgame perfect equilibrium with positive profits to both firms if and only if buyers differ significantly in their willingness to pay. In particular, the incumbent sets a positive royalty and sells to the low demand buyer while the entrant only charges a fixed price and sells to the high demand buyer, resulting in an undercut-proof subgame perfect equilibrium.