We consider a game-theoretical variant of the Steiner forest problem in which each player j, out of a set of k players, strives to connect his terminal pair (s j , t j ) of vertices in an undirected, edge-weighted graph G. In this paper we show that a natural adaptation of the primaldual Steiner forest algorithm of Agrawal, Klein, and Ravi [SIAM J. Comput., 24 (1995), pp. 445-456] yields a 2-budget balanced and cross-monotonic cost sharing method for this game. We also present a negative

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... arguing that no cross-monotonic cost sharing method can achieve a budget balance factor of less than 2 for the Steiner tree game. This shows that our result is tight. Our algorithm gives rise to a new linear programming relaxation for the Steiner forest problem which we term the lifted-cut relaxation. We show that this new relaxation is stronger than the standard undirected cut relaxation for the Steiner forest problem.