Patients can express this reservation utility by where they rank their own donor in their preferences. 4. Depending on how this priority is given, patients may need to be aware of the current population of the queue to evaluate the desirability of the w option. Liran Einav and Muriel Niederle have shared with us interesting ideas about how to further model the desirability of the w option dynamically, taking into account that others may enter the queue with high priority, but we will not pursue this here. KIDNEY EXCHANGE 467 III.B. The Exchange Mechanism For the mechanism defined below, we assume that when one among multiple w-chains must be selected, a fixed chain selection rule is invoked. We will consider a number of such rules, and their implications for incentives, efficiency, and equity. Throughout the procedure kidneys are assigned to patients through a series of exchanges. Some patients and their assigned kidneys will be immediately removed from the procedure, while others will remain with their assignments but they will assume a passive role. So at any point in the procedure, some agents may no longer be participants, some participants will be active, and the others passive. For a given kidney exchange problem, the TTCC mechanism determines the exchanges as follows. 1. Initially, all kidneys are available, and all agents are active. At each stage of the procedure each remaining active patient ti points to his most preferred remaining unassigned kidney or to the wait-list option w, whichever is more preferred, each remaining passive patient continues to point to his assignment, and each remaining kidney ki points to its paired recipient ti. 2. By Lemma 1, there is either a cycle, or a w-chain, or both. (a) Proceed to Step 3 if there are no cycles. Otherwise, locate each cycle, and carry out the corresponding exchange (i.e., each patient in the cycle is assigned the kidney he is pointing to). Remove all patients in a cycle together with their assignments. (b) Each remaining patient points to his top choice among remaining kidneys, and each kidney points to its paired recipient. Locate all cycles, carry out the corresponding exchanges, and remove them. Repeat until no cycle exists. 3. If there are no pairs left, we are done. Otherwise, by Lemma 1 each remaining pair is the tail of a w-chain. Select only one of the chains with the chain selection rule. The assignment is final for the patients in the selected w-chain. The chain selection rule also determines whether the selected w-chain is removed and the associated exchanges are all immediately assigned (including the kidney at the tail, which is designated to go to a patient on the cadaver queue), or if the selected w-chain is kept in the 468 QUARTERLY JOURNAL OF ECONOMICS procedure although each patient in it is passive henceforth.5 4. After a w-chain is selected, new cycles may form. Repeat Steps 2 and 3 with the remaining active patients and unassigned kidneys until no patient is left. At the end of the procedure, each patient with a living donor is assigned a kidney (or a high priority place on the waiting list). However, that does not necessarily mean each of these patients receives a transplant. In particular, a minimal cycle (ki,ti) consisting of a single patient-donor pair may be a pair that was not offered a sufficiently desirable kidney in the current exchange, and chooses to wait in the hope of exchanging for a high quality living donor kidney the next time the exchange is run, after new donors have entered the system. Suppose that patients are ordered in a priority-list based on their indices starting with the patient with the smallest index. We use the following chain selection rule: choose the longest w-chain. In case the longest w-chain is not unique, choose the w-chain with the highest priority patient; if the highest priority patient is part of more than one, choose the w-chain with the second highest priority patient, and so on. Keep the selected w-chains until the termination. The execution of the TTCC mechanism is given in Figures I–V. 5. The relevance of the last point is the following: whenever a w-chain (k 1,t 1,..., k m,t m) is selected, even though the assignments of all patients in the w-chain are finalized, the kidney k 1 at the tail of the w-chain can be utilized in two possible ways. It can immediately be offered to the waiting list (in which case the w-chain is removed), or it may be made available to the remaining patients as the process continues, and hence the selected w-chain may possibly grow later on, although the patients already in it are not k4 w k1 k3 k10 . It is worth emphasizing that the chain selection policy does not affect a patient who is at the head of a chain. Since he points to the wait-list option, he will eventually be selected regardless of the chain selection rule. However, whether his intended donor’s kidney is offered to the cadaveric waiting list or another patient with a living donor depends on the rule. Depending on policy priorities, one may consider adopting a number of alternative chain selection rules. For example: a. Choose minimal w-chains, and remove them. b. (c.) Choose the longest w-chain, and remove it (keep it). If the longest w-chain is not unique, then use a tiebreaker to choose among them. FIGURE I Example 1, Round 1 There is a single cycle