Uncertain Price Competition in a Duopoly: Impact of Heterogeneous Availability of the Commodity Under Sale

We study price competition in a duopoly with an arbitrary number of buyers. Each seller can offer multiple units of a commodity depending on the availability of the commodity which is random and may be different for different sellers. Each seller seeks to select a price that will be attractive to the buyers and also fetch adequate profits. The selection will in general depend on the number of units available with the seller and also that of its competitor - the seller may only know the statistics of the latter. The setting captures a microgrid network or a Cognitive Radio Network (CRN) in which excess power units and unused spectrum bands constitute the respective commodities of sale. We analyze this price competition as a game. We prove the existence and uniqueness of Nash Equilibrium (NE) in the class of symmetric equilibria, and explicitly compute the price distribution in the NE.