Stochastic characterization of the two band two player spectrum sharing game

We consider two pairs of communicating users sharing two bands of spectrum under a sum band power constraint. Our earlier work proposed a natural spectrum sharing game for this problem and characterized the Nash equilibria as a function of the signal and interference distances, when the positions of the four nodes were assumed fixed. In this work, we derive i) the joint distribution of the interference distances conditioned on the transmitter separation distance, as well as ii) the unconditioned interference distance distribution when we place one transmitter at the origin and the second uniformly at random over a disk. This allows us to compute the distribution of the random Nash equilibria and random prices of anarchy and stability as a function of the random interference distances. We leverage the analysis to give an asymptotic expression for the coupling probability in a game where the transmitter positions form a (low density) Poisson process, which may be interpreted as the fraction of players essentially playing a two player game.