Exchange Market Mechanisms without Money

We introduce and study the following exchange market mechanism problem without money: consider a set of agents who wish to obtain a set of items, and have a set of items to offer to others. An exchange market mechanism specifies for each agent a subset of items to give away, and another subset of items that he would receive in exchange. Each agent would like to maximize the number of items he receives from his wish list, but will have a large dis-utility if he gives away more items than what he receives. Our goal is to design truthful mechanisms that maximize the total number of items being exchanged. This problem is a generalization of the kidney exchange problem, and is motivated by several barter exchange websites on the Internet. We show that an exchange can be viewed as a collection of simple cycles in a directed bipartite graph with agents and items as the two sides. Any cycle represents a trade in which each participating agent gives and receives exactly one item. We study two variants of the exchange market problem: the lengthconstrained variant where the number of items exchanged in each cycle should be at most a given constant k ≥ 2, and the unconstrained variant where cycles of any length are allowed. For the length-constrained variant, we show that no truthful deterministic or randomized mechanism can achieve an approximation factor better than 3k+1 3k+2 and 3k+1.89 3k+2 respectively. We present a 1 8 -approximate truthful mechanism for the problem with k = 2. For the unconstrained version, we present a polynomial-time algorithm solving the optimal exchange market.