In recent years, certain formalizations of combinatorial negotiation settings, most notably combinatorial auctions, have become an important research topic in the AI community. A pervasive assumption has been that of no externalities: the agents deciding on a variable (such as whether a trade takes place between them) are the only ones affected by how this variable is set. To date, there has been no widely studied formalization of combinatorial negotiation settings with externalities. In this paper, we introduce such a formalization. We show that in a number of key special cases, it is NP-complete to find a feasible nontrivial solution (and therefore the maximum social welfare is completely inapproximable). However, for one important special case, we give an algorithm which converges to the solution with the maximal concession by each agent (in a linear number of rounds for utility functions that decompose into piecewise constant functions). Maximizing social welfare, however, remains NP-complete even in this setting. We also demonstrate a special case which can be solved in polynomial time by linear programming.
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