Tâtonnement Mechanisms for Combinatorial Exchanges

Combinatorial exchanges are double sided marketplaces with multiple sellers and multiple buyers trading with the help of combinatorial bids. The allocation and other associated problems in such exchanges are known to be among the hardest solve among all economic mechanisms. In this paper, we develop computationally efficient iterative mechanisms for clearing combinatorial exchanges. The mechanisms we propose are called tâtonnement mechanisms. These mechanisms allow prices in successive iterations to change non-monotonically and are therefore more general than iterative auctions. Interestingly, we use a Lagrangian relaxation of the allocation problem to develop the iterative tâtonnement mechanisms. We show that the Lagrangian multipliers can be interpreted as the prices of the items set by the exchange in each iteration. We formulate a relaxed linear programming problem in each iteration which gives integer solutions and therefore reduces the computational complexity. Through extensive numerical experiments, we show that our algorithm produces high quality solutions and is computationally efficient.