Algorithms for Combinatorial Coalition Formation and Payoo Division in an Electronic Marketplace 1

In an electronic marketplace coalition formation allows buyers to enjoy a price discount for each item while combinatorial auction enables buyers to place bids for a bundle of items that are complementary. Coalition formation and combinatorial auction both help to improve the e ciency of a market and have received much attention from economists and computer scientists. But neither in laboratories nor in practice has there been literature on the situations where both coalition formation and combinatorial auctions exist. In this paper we consider an e-market where each buyer places a bid on a combination of items with a reservation cost, and sellers o er price discounts for each item based on volumes. We call coalition formation under this condition a Combinatorial Coalition Formation (CCF) problem since coalition formation is motivated by price discounts on single items while multiple items are complementary for buyers. By arti cially dividing the reservation cost of each buyer appropriately among the items we can construct optimal coalitions with respect to each item. We then try to make these coalitions satisfy the complementarity of the items, and thus induce the optimal solution. Based on this idea we present polynomial-time algorithms to nd a semi-optimal solution of CCF and a payo division scheme that is in the core of the coalition when linear price functions are applied, and in the pseudo-core when general price functions are applied. Simulation results show that the algorithms obtain solutions in a satisfactory ratio to the optimal value.