Combinatorial Coalition Formation for multi-item group-buying with heterogeneous customers

A group-buying market may offer multiple items with non-additive values (i.e., items may be complementary or substitutable), to buyers who are often heterogeneous in their item valuations. In such a situation, the formation of buying groups should concentrate buyers for common items while taking into consideration buyers' heterogeneous preferences over item bundles. Also, it should permit non-uniform cost sharing among buyers in the same group, which benefits all buyers by drawing more group-buying participants. We introduce the concept of Combinatorial Coalition Formation (CCF), which allows buyers to announce reserve prices for combinations of items. These reserve prices, along with the sellers' price-quantity curves for each item, are used to determine the formation of buying groups for each item. Moreover, buyers in the same group may not necessarily all pay the same price. The objective of CCF is to maximize buyers' total surplus. Determining the optimal coalition configuration in CCF is NP-hard, and the stability of such a configuration relies on the cost sharing rule within each group. We thus propose a heuristic algorithm for CCF based on augmented greedy selections, along with a cost sharing rule satisfying certain stability properties. Simulation results show that our approximate algorithm generates fairly good solutions compared to the optimal results, and is greatly superior to a simpler distributed approach. Furthermore, our algorithm's performance is enhanced when items are complementary or strongly substitutable, especially in settings when the prices decrease either rapidly or slowly with the quantities. Evaluations of the sellers' revenue under CCF demonstrate that sellers should offer a more gradually decreasing price-quantity curve for complementary or strongly substitutable items, and a more abruptly decreasing curve for weakly substitutable items. In addition, sellers may benefit from greater sales generated by simpler price-quantity curves with fewer steps.