Demand-Insensitive price relationships in competitive private commons

We introduce an economic model for private commons that consists of network providers serving a fixed primary demand and making strategic pricing decisions to improve their revenues by providing service to a secondary demand. For general forms of secondary demand, we establish the existence and uniqueness of two critical prices for each provider: the break-even price and the market sharing price. The prior determines service profitability while the latter determines a provider's willingness to share the market. We further show that the market sharing price is always greater than the break-even price, leading to a price interval in which a provider is both profitable and willing to share the market. Making use of these results, we shed insight into the nature of market outcomes (Nash equilibria) when two providers compete to attract secondary demand: (i) if the market sharing intervals of the two providers overlap, then the providers end up sharing the market; (ii) else, the provider with the lower break-even price captures the entire market as the result of a price war.