Strategy-Proof and Efficient Kidney Exchange Using a Credit Mechanism

We present a credit-based matching mechanism for dynamic barter markets—and kidney exchange in particular—that is both strategy proof and efficient, that is, it guarantees truthful disclosure of donor-patient pairs from the transplant centers and results in the maximum global matching. Furthermore, the mechanism is individually rational in the sense that, in the long run, it guarantees each transplant center more matches than the center could have achieved alone. The mechanism does not require assumptions about the underlying distribution of compatibility graphs—a nuance that has previously produced conflicting results in other aspects of theoretical kidney exchange. Our results apply not only to matching via 2-cycles: the matchings can also include cycles of any length and altruist-initiated chains, which is important at least in kidney exchanges. The mechanism can also be adjusted to guarantee immediate individual rationality at the expense of economic efficiency, while preserving strategy proofness via the credits. This circumvents a well-known impossibility result in static kidney exchange concerning the existence of an individually rational, strategy-proof, and maximal mechanism. We show empirically that the mechanism results in significant gains on data from a national kidney exchange that includes 59% of all US transplant centers. Introduction In the United States alone, over 3.8 million people—roughly 1.6% of the population—have kidney disease.1 For many, kidney disease will progress to outright kidney failure—and with it the need for a kidney transplant. Transplant organs can be sourced from cadavers or willing living donors. However, there is a severe supply and demand mismatch with donor organs; in 2013, 36,395 people were added to the US national kidney waiting list, while only 16,462 left due to receiving a living or deceased donor kidney.2 Furthermore, roughly half of the over 100,000 candidates on the US list have been waiting for a kidney for more than two years. Kidney exchange aims to reduce the transplant organ supply-demand imbalance by making it easier to match willCopyright c 2015, Association for the Advancement of Artificial Intelligence (www.aaai.org). All rights reserved. The 2012 US National Health Interview Survey (Blackwell, Lucas, and Clarke 2014) counts patients with “weak or failing kidneys” (see their Tables 7 and 8 for methodology details). 2 http://optn.transplant.hrsa.gov ing donors to needy patients. A person in need of a kidney may have one or more healthy donors with two healthy kidneys who are willing but unable to donate a kidney to that person, typically due to a medical incompatibility like a blood or tissue type mismatch. These incompatible donorpatient pairs can exchange donors with other pairs, through cycles or chains of exchanges, in such a way that each involved pair’s donor gives to a (medically compatible) patient of another pair. Recommending a “good” set of swaps is thus the basic kidney exchange problem. In recent years, numerous multi-transplant center kidney exchanges have been fielded around the world, where donorpatient pairs from different transplant centers can switch donors. Multi-center kidney exchange programs are run through clearinghouses that recommend a matching from the full set of donor-patient pairs. Transplant centers have different incentives than patients and donors. Performing an organ transplant surgery is typically very profitable to a center. Thus, centers may have incentive to only reveal some subset of their donor-patient pairs to the clearinghouse, and match other pairs internally at their center.3 Such strategic behavior is rampant: today most transplant centers not only hide their easy-to-match pairs, but all their internally matchable pairs (Stewart et al. 2013). It has also been shown to reduce the overall efficiency of kidney exchange in theory (Ashlagi and Roth 2014). While most prior work on mechanism design for kidney exchange has focused on static models, in this paper we consider a more realistic multi-period dynamic model. For this dynamic model, we study how to construct a mechanism that makes it a dominant strategy for each center to reveal its pairs truthfully, and hence maximizes the number of pairs in the central pool from which the clearinghouse will construct a global matching. Our contributions in this work are the following. First, we design a credit-based strategy-proof matching mechanism for the dynamic model that considers the incentives of the different transplant centers. The mechanism is efficient and guarantees long-term individual rationality (IR). Our results apply not only to matching via 2-cycles—a restriction present in some kidney exchange papers: our matchings can also include cycles of any length and altruist-initiated chains, which is important at least in Internal matches are also logistically easier to handle. kidney exchanges. Second, we show that the mechanism can be adjusted to guarantee immediate individual rationality at the expense of economic efficiency, while preserving strategy proofness via the credits. Our experiments show that the efficiency loss is very small in practice. This variant of the mechanism circumvents—via the use of the credits—a well-known impossibility result, proved for static kidney exchange, that no IR mechanism is both maximal and strategy proof. Our mechanism does not require assumptions about the underlying distribution of compatibility graphs—a nuance that has previously produced conflicting results in other aspects of theoretical kidney exchange. Finally, experiments with the mechanism, both on real data from a large fielded kidney exchange in the US as well as from a data generator, show that the number of resulting matches is substantially greater than without the mechanism. Related Work The idea of kidney exchange was introduced by Rapaport (1986), and the first organized kidney exchange started in 2003 (Roth, Sönmez, and Ünver 2004). The topic has attracted researchers from non-medical fields such as economics (Roth, Sönmez, and Ünver 2005; 2007b; Akbarpour, Li, and Gharan 2014) and computer science (Abraham, Blum, and Sandholm 2007; Biró, Manlove, and Rizzi 2009; Dickerson, Procaccia, and Sandholm 2012a; 2013; 2014; Dickerson and Sandholm 2014; Anshelevich et al. 2013; Liu, Tang, and Fang 2014; Li et al. 2014). The National Organ Transplant Act of 1984 makes it illegal to buy or sell a kidney in the US, thus making donation the only viable option for kidney transplantation (Roth 2007; Leider and Roth 2010). Similar legislation exists throughout most of the world. The initial proposal for a large-scale kidney exchange that was made by Roth, Sönmez, and Ünver (2004) included the ability to use both cycles (where each donor donates a kidney to the next patient, with the final donor donating her kidney to the first patient) and chains (that are like cycles except that the cycle does not close).4 In the bulk of the work on mechanism design for kidney exchange (including our paper), the agents that need to be incentivized are the transplant centers (not the donorpatient pairs or altruistic donors) (Ashlagi and Roth 2014; Toulis and Parkes 2011; Sönmez and Ünver 2013; Ashlagi et al. 2013). A center can decide to reveal none, some, or all of its pairs and altruists to the clearinghouse. Roth, Sönmez, and Ünver (2007a) proved that no individually rational (IR) and maximal mechanism can also be strategy proof (in a static model, where patients and donors do not arrive and leave over time). According to their definition, a matching is maximal if there exists no larger matching that fully encompasses the former. One of the main contributions of our paper is the circumvention of this impossibility result via a credit mechanism in a dynamic model, and real kidney exchanges are dynamic. Ashlagi and Roth (2014) continTheir proposal was to have chains start with a pair that has received a kidney from the deceased-donor waiting list. In practice, chains start with an altruistic donor that does not expect anything in return. ued their work in the static model, showing that the efficiency loss due to lack of strategy proofness can be very high but that there exists an ✏-Bayesian incentive compatible IR mechanism that tends to become nearly efficient in the large under a dense random graph model. Ashlagi et al. (2013) presented a randomized strategyproof IR mechanism for a static setting that delivers at least 50% of efficiency. Their model, though, is for 2-cycles only, while ours supports cycles and chains of any length. Model We start from the standard kidney exchange model with a set of n transplant centers, T = {⌧1, . . . , ⌧n}, and a central clearinghouse. The process is modeled as a game that is divided into time periods: at each period, each center ⌧