preferences, that means he and his donor do not wish to participate in an exchange. If patient ti ranks ki on top of w, that means he and his donor do not consider exchanging kidney ki with a priority in the cadaveric kidney waiting list. We can now formalize a (static) kidney exchange problem consisting of a set of donor-recipient pairs {(k1,t1), . . . , (kn,tn)}, a set of compatible kidneys Ki K {k1,...,kn} for each patient ti, and a strict preference relation Pi over Ki {ki,w} for each patient ti. The outcome of a kidney exchange problem is a matching of kidneys/wait-list option to patients such that each patient ti is either assigned a kidney in Ki {ki} or the wait-list option w, and no kidney can be assigned to more than one patient although the wait-list option w can be assigned to several patients. A kidney exchange mechanism selects a matching for each kidney exchange problem. We are almost ready to introduce the Top Trading Cycles and Chains (TTCC) mechanism, a generalization of the TTC mechanism, for kidney exchange. First, we give a few definitions and observations to facilitate the description of the mechanism. KIDNEY EXCHANGE 465 III.A. Cycles and w-Chains The mechanism relies on an algorithm consisting of several rounds. In each round each patient ti points either toward a kidney in Ki {ki} or toward w, and each kidney ki points to its paired recipient ti. A cycle is an ordered list of kidneys and patients (k 1,t 1,k 2,t 2,..., k m,t m) such that kidney k 1 points to patient t 1, patient t 1 points to kidney k 2,..., kidney k m points to patient t m, and patient t m points to kidney k 1. Cycles larger than a single pair are associated with direct exchanges, very much like the paired-kidney-exchange programs, but may involve more than two pairs, so that patient t 1 is assigned kidney k 2, patient t 2 is assigned kidney k 3,..., patient t m is assigned kidney k 1. Note that each kidney or patient can be part of at most one cycle and thus no two cycles intersect. A w-chain is an ordered list of kidneys and patients (k 1,t 1,k 2,t 2,..., k m,t m) such that kidney k 1 points to patient t 1, patient t 1 points to kidney k 2,..., kidney k m points to patient t m, and patient t m points to w. We refer to the pair (k m,t m) whose patient receives a cadaver kidney in a w-chain as the head and the pair (k 1,t 1) whose donor donates to someone on the cadaver queue as the tail of the w-chain. W-chains are associated with indirect exchanges but unlike in a cycle, a kidney or a patient can be part of several w-chains. One practical possibility is choosing among w-chains with a well-defined chain selection rule, very much like the rules that establish priorities on the cadaveric waiting list. The current pilot indirect exchange programs in the United States choose the minimal w-chains, consisting of a single donor-recipient pair, but this may not be efficient. Selection of longer w-chains will benefit other patients as well, and therefore the choice of a chain selection rule has efficiency implications (see Theorem 1). Chain selection rules may also be used for specific policy objectives such as increasing the inflow of type O living donor kidneys to the cadaveric waiting list. Whenever w-chain (k 1,t 1,..., k m,t m) is selected in the algorithm, patient t 1 is assigned kidney k 2, patient t 2 is assigned kidney k 3,..., patient t m1 is assigned kidney k m, patient t m receives high priority for the next compatible kidney in the cadaveric waiting list, and kidney k 1 is offered either to the cadaveric waiting list or to another patient with a paired donor. 466 QUARTERLY JOURNAL OF ECONOMICS LEMMA 1. Consider a graph in which both the patient and the kidney of each pair are distinct nodes as is the wait-list option w. Suppose that each patient points either toward a kidney or w, and each kidney points to its paired recipient. Then either there exists a cycle, or each pair is the tail of some w-chain. We can now introduce the TTCC mechanism. Because the exchange mechanism interacts with many parts of the kidney transplant environment, it will clarify the discussion to start by indicating which parts of the environment we take as fixed for our present purpose. First, we take the operation of the cadaver queue as fixed. The cadaver queue can be thought of as a stochastic arrival process of cadavers and patients, interacting with a scoring rule that determines which patients are offered which cadaver kidneys. We also take as fixed how patients whose donors donate a kidney to someone on the queue are given high priority on the queue, e.g., by being given points in the scoring rule.4 We also take as given the size of the live kidney exchange; i.e., the set of patient-donor pairs is taken to be fixed. In practice, the set of patient-donor pairs will grow as the geographic area served by the kidney exchange is increased, or as the time between exchanges is increased. A larger pool of possible exchanges will increase the potential efficiency gains that can be realized by exchange, but will also increase the size of the trading cycles/ w-chains that might be needed to achieve these efficiencies. We will keep track of both of these when we report simulations. Both the operation of the cadaver queue, and the frequency and scope of the kidney exchange will influence patients’ “reservation utility,” i.e., how they compare various opportunities for direct or indirect exchange to the option of not making any exchange now, but waiting for a future opportunity.