Pricing of partially compatible products

In this paper, we examine a duopolistic market where the two firms compete to sell a system of components. Components are digital (firms haveunlimited supply at no marginal cost), and customers are homogeneous in their component preferences. Each customer will assemble a utility maximizing system by purchasing each necessary component from one of the two firms. While components from the same firm are always compatible, pairwise compatibility of components from rival firms may vary; in addition to utility due to the quality of the system purchased, customers have negative utility for purchasing incompatible parts. We investigate algorithms and hardness results for profit-maximizing decisions of the firms with regards to their price-setting, component value-enhancing and compatibility-enabling strategies. The users' behavior can be modeled as a minimum cut computation, and the company's strategies require addressing novel and interesting questions about graph cuts and flows. We develop a polynomial-time algorithm for finding profit-maximizing prices if the qualities and compatibilities are fixed. On the other hand, we show that finding profit-maximizing quality improvements is equivalent to the Maximum Size Bounded Capacity Cut problem, and thus NP-complete. Finally, for the problem of improving compatibilities to maximize the price, we give polynomial approximation hardness results even in very restricted cases, but show that if all components have uniform prices, and quality differences are small, then an approximation can be found in polynomial time.