Secondary spectrum oligopoly market over large locations

We investigate a secondary spectrum market where each primary owns a channel over large number of locations. Each primary sells its channel to the secondaries in exchange of a price. However, the secondaries can not transmit simultaneously at interfering locations. A primary must select a price and a set on non-interfering locations for its available channel where the availability of a channel for sale evolves randomly. The set of non-interfering locations turns out to be an independent set in the conflict graph representation of the region. The primary needs to find a strategy for each possible channel state vector. We consider node symmetric conflict graphs which arise frequently in practice when the number of locations is large (potentially, infinite). Since there is a symmetry in the interference relationship, we also consider a symmetric relationship among the joint probability distribution of the channel state vectors. We show that that a symmetric NE exists and explicitly compute it. In the symmetric NE a primary randomizes equally among the maximum independent sets at a given channel state vector. The symmetric NE exhibits several important structural differences compared to the symmetric NE strategy for small number of locations which we have obtained in our earlier works. The conflict graph representation depends on the channel state vector, thus, it is a random graph. We also empirically and theoretically investigate the expected component size in random conflict graphs which governs the computation of maximum independent sets. Our analysis shows that the mean component size is in general moderate, however, it can be high when the channel availability probability is very high. We show that with random sampling method, a primary can govern the mean component size. We numerically evaluate the ratio of the expected payoff attained by primaries in the game and the payoff attained by primaries when all the primaries collude.