We consider a game-theoretical variant of the Steiner forest problem, in which each of k users i strives to connect his terminal pair (s_i, t_i) of vertices in an undirected, edge-weighted graph G. In [1] a natural primal-dual algorithm was shown which achieved a 2-approximate budget balanced cross-monotonic cost sharing method for this game. We derive a new linear programming relaxation from the techniques of [1] which allows for a simpler proof of the budget balancedness of the algorithm from [1]. Furthermore we show that this new relaxation is strictly stronger than the well-known undirected cut relaxation for the Steiner forest problem. We conclude the paper with a negative result, arguing that no cross-monotonic cost sharing method can achieve a budget balance factor of less than 2 for the Steiner tree and Steiner forest games. This shows that the results of [1,2] are essentially tight.
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