These are two papers by our new Nobel Laureate, Alvin Roth, one from 2004 and the other from 2012 - the field has evolved significantly:
[In 2004]: Most transplanted kidneys are from cadavers, but there are also many transplants from live donors. Recently, there have started to be kidney exchanges involving two donor-patient pairs such that each donor cannot give a kidney to the intended recipient because of immunological incompatibility, but each patient can receive a kidney from the other donor. Exchanges are also made in which a donor-patient pair makes a donation to someone waiting for a cadaver kidney, in return for the patient in the pair receiving high priority for a compatible cadaver kidney when one becomes available. There are stringent legal/ethical constraints on how exchanges can be conducted. We explore how larger scale exchanges of these kinds can be arranged efficiently and incentive compatibly, within existing constraints. The problem resembles some of the “housing” problems studied in the mechanism design literature for indivisible goods, with the novel feature that while live donor kidneys can be assigned simultaneously, cadaver kidneys cannot. In addition to studying the theoretical properties of the proposed kidney ex- change, we present simulation results suggesting that the welfare gains from larger scale exchange would be substantial, both in increased number of feasible live donation transplants, and in improved match quality of transplanted kidneys (Roth, Sonmez, and Unver, 2004).
[In 2012]: Consider for example the market for residential real estate in the United States. It is not uncommon for purchase and sales agreements to have contingency clauses saying that the purchase of a house is contingent on the buyer’s successful sale of his own home in time to close the deal. Thus a given sale may go through only if it is part of a chain of sales that may begin with a non-contingent buyer (e.g. a buyer who does not have a home that he must sell before he can buy another). Such a non-contingent buyer plays much the same role as a non-directed donor here. Similarly, bridge loans that allow a purchase to go through before the buyer has sold his own home allow for such chains to be non-simultaneous.
In conclusion, an obstacle to bilateral barter is the need to find double coincidences of wants. Developing the capability to arrange trades in longer cycles and chains helps overcome this barrier, in markets like kidney exchange in which money and prices are not available to clear the market, and very likely in other markets as well, in which credit and cash constraints prevent prices from completely clearing the market. In the case of kidney exchange, long non-simultaneous chains of the sort proposed in Roth et al. (2006) are proving increasingly important, and the present paper offers some new insight into why this is the case (Roth et al, 2012).
James Buchanan said that economists should study exchange, Roth has done this in extremely important areas such as kidney transplants, even when prices are not available. Through sophisticated algorithms he has gone beyond studying exchange to the creation of exchange, so to speak.
Some might think that markets in areas like organs are repugnant. Roth has written about this as well, see his article "Repugnance as a Constraint on Markets."