On Pareto optimal balanced exchanges

We investigate a market without money in which every agent offers indivisible goods in multiple copies, in exchange for goods of other agents. The exchange must be balanced in the sense that each agent should receive a quantity of good(s) equal to the one she transfers to others. We describe the market in graph-theoretic terms hence we use the notion of circulations to describe a balanced exchange of goods. Each agent has strict preferences over the agents from which she will receive goods and an upper bound on the quantity of each transaction, while a positive integer weight reflects the social importance of each unit exchanged. In this paper, we propose a simple variant of the Top Trading Cycles mechanism that finds a Pareto optimal circulation. We then offer necessary and sufficient conditions for a circulation to be Pareto optimal and, as a consequence, a easy recognition procedure. Last, we show that finding a maximum weight Pareto optimal circulation is NP-hard but becomes polynomial if weights are concordant with preferences.