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"... When donating money to a (say, charitable) cause, it is possible to use the contemplated donation as a bargaining chip to induce other parties interested in the charity to donate more. Such negotiation is usually done in terms of matching offers, where one party promises to pay a certain amount if o ..."
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When donating money to a (say, charitable) cause, it is possible to use the contemplated donation as a bargaining chip to induce other parties interested in the charity to donate more. Such negotiation is usually done in terms of matching offers, where one party promises to pay a certain amount if others pay a certain amount. However, in their current form, matching offers allow for only limited negotiation. For one, it is not immediately clear how multiple parties can make matching offers at the same time without creating circular dependencies. Also, it is not immediately clear how to make a donation conditional on other donations to multiple charities when the donor has different levels of appreciation for the different charities. In both these cases, the limited expressiveness of matching offers causes economic loss: it may happen that an arrangement that all parties (donors as well as charities) would have preferred cannot be expressed in terms of matching offers and will therefore not occur. In this paper, we introduce a bidding language for expressing very general types of matching offers over multiple charities. We formulate the corresponding clearing problem (deciding how much each bidder pays, and how much each charity receives), and show that it cannot be approximated to any ratio in polynomial time unless P=NP, even in very restricted settings. We give a mixed integer program formulation of the clearing problem, and show that for concave bids, the program reduces to a linear program. We then show that the clearing problem for a subclass of concave bids is at least as hard as the decision variant of linear programming. We also consider the case where each charity has a target amount, and bidders’ willingness-to-pay functions are concave. Here, we show that the optimal surplus can be approximated to a ratio m, the number of charities, in polynomial ∗A short, early conference version (EC-04) of this paper was based on work supported by the National