Distributed estimation of a parameter vector in a network of sensor nodes with ambiguous measurements is considered. The ambiguities are modelled by following a set-theoretic approach, that leads to each sensor employing a non-convex constraint set on the parameter vector. Consensus can be used to reach an estimate consistent with the measurements of all nodes, assuming that such an estimate exists, but unfortunately, such an approach leads to a non-convex problem. Using proper assumptions, the considered problem is decomposed into two sub-problems, where one is well studied in literature and the other is modelled as a non-cooperative game. An exact potential function is derived for this game, and an algorithm for its solution is given. Numerical results, consistent with the theoretical findings, demonstrate the efficacy of the proposed approach.