  Sensor network localization (SNL) is a challenging problem due to its
inherent non-convexity and the effects of noise in inter-node ranging
measurements and anchor node position. We formulate a non-convex SNL problem as
a multi-player non-convex potential game and investigate the existence and
uniqueness of a Nash equilibrium (NE) in both the ideal setting without
measurement noise and the practical setting with measurement noise. We first
show that the NE exists and is unique in the noiseless case, and corresponds to
the precise network localization. Then, we study the SNL for the case with
errors affecting the anchor node position and the inter-node distance
measurements. Specifically, we establish that in case these errors are
sufficiently small, the NE exists and is unique. It is shown that the NE is an
approximate solution to the SNL problem, and that the position errors can be
quantified accordingly. Based on these findings, we apply the results to case
studies involving only inter-node distance measurement errors and only anchor
position information inaccuracies.
