Supermodular game for energy efficient TOA-based positioning

In this paper we address the problem of minimizing the energy cost of positioning a node in a wireless sensor network, using time of arrival measurements. A sensor needs to receive at least three distance measurements to known anchors in order to position itself. The accuracy of its position estimation depends on the signal to noise ratio of the beacons from the anchor nodes, whose power levels are to be selected according to a twofold criterion: minimum power level and desired positioning quality for users, determined by the error covariance metric. We derive a solution based on modeling the positioning problem as a non-cooperative game. We show that the resulting game is Supermodular and that it possesses a unique Nash Equilibrium, which can be quickly reached with best response dynamics. Finally, in the numerical results we find the price of anarchy of our game.