Short trading cycles: Paired kidney exchange with strict ordinal preferences
Introduction
The creation and design of living-donor kidney-exchange programs, which allow patient–donor pairs to exchange kidneys among themselves without the use of monetary transfers, have received significant attention recently. See Sönmez and Ünver (2013) for a recent survey.
The guidelines and legal frameworks governing kidney-exchange programs around the world enshrine the balance of the dual goals of efficiency and equity as the main desideratum. The National Organ Transplantation Act stipulates the equitable allocation of organs in the US, while the Council of Europe’s Convention on Human Rights and Biomedicine requires “equitable access to transplantation services” (Council of Europe, 2002). The United Network for Organ Sharing in the US has as its main goal the balancing of utility and justice (Wallis et al., 2011). Exchanges in Canada, Australia, and New Zealand place similar emphasis on equity (Malik and Cole, 2014; AKX, 2015; NRTS, 2017).
For legal and logistical reasons described in Section 2, the maximum number of pairs participating in each exchange cycle is usually limited in practice. Without randomization, such constraints prevent the equitable treatment of patients. Section 3 presents the model of constrained object exchange without monetary transfers with strict ordinal preferences. Within that context, Section 4 shows that deterministic mechanisms indeed fail to meet the desiderata in the presence of cycle-length constraints. Namely, anonymity and constrained Pareto optimality are incompatible with one another. Anonymity is the basic equity criterion considered by this paper and requires that mechanism participants’ identities are irrelevant for determining their expected allocations.1
Turning to the main results of the paper, Section 5 proposes a random mechanism in which each trade involves no more than two pairs, called the 2-cycle probabilistic serial (2CPS) mechanism. It is based on the Probabilistic Serial (PS) mechanism (Bogomolnaia and Moulin, 2001).2 The 2CPS mechanism is individually rational, incentivizing the participation of compatible patient–donor pairs. It is ordinally efficient,3 i.e. Pareto optimal with respect to first-order stochastic dominance, and, balancing the needs for efficiency and equity, also anonymous. The 2-cycle simultaneous eating (2CSE) mechanisms, a more general class of mechanisms, retains the first two properties and can be defined so that agents are categorized in priority classes and all within a given priority class receive equitable treatment.4 In the other main result of the paper, Section 6 shows that no mechanism satisfies a given cycle-length constraint while being individually rational, efficient, and weakly strategyproof.5
Following Nicoló and Rodríguez-Álvarez (2012) and subsequent work, I depart from the seminal kidney-exchange papers’ assumption of dichotomous preferences, which views all compatible kidneys as perfect substitutes. As discussed in Section 2, a variety of factors beyond blood- and tissue-type compatibility affect the survival rates of kidney grafts. Thus, reducing the problem to dichotomous compatibility-based preferences ignores welfare-relevant information. I assume strict preferences as in Nicoló and Rodríguez-Álvarez (2012).
More importantly, an individually rational mechanism taking non-dichotomous preferences into account incentivizes the participation of any patient, who is compatible with her related donor, by guaranteeing her a kidney with expected graft-survival rate at least as high as her donor’s. In contrast, if collapsing to dichotomous compatibility-based preferences, such a patient may receive a kidney with worse expected outcome than her donor’s. The participation of such compatible pairs would increase the transplantation rates for incompatible pairs.6 The literature on the participation of compatible pairs is discussed in Section 1.1.
The 2CSE and 2CPS mechanisms are based on a simultaneous-eating algorithm. The algorithm treats all kidneys as if they are infinitely divisible objects and all agents as if they are claiming increasing object shares in continuous time starting with their most preferred object. The algorithm ends when all objects have been completely claimed or, equivalently, when all patients have one unit of probability shares. The share that patient has claimed from object is treated as the probability with which patient receives kidney . This induces a symmetric probability-share matrix , where denotes the probability that patient–donor pairs and trade with each other or, if , the probability that pair does not participate in an exchange with another pair. Special care needs to be taken to guarantee that the resulting matrix represents a lottery over deterministic matchings satisfying the cycle-length constraint. The usual approach, based on the Birkhoff–von Neumann theorem, is not viable here. Instead, I use Edmonds’ characterization of the matching polytope (Edmonds, 1965), which implies sufficient and necessary conditions for a bistochastic matrix to be the representation of a lottery over two-way exchanges.
The first economic study of kidney exchange (Roth et al., 2004) considered a setting with strict ordinal preferences but without cycle constraints. Subsequent work, starting with Roth et al. (2005a), has accounted for the cycle constraints but has tended to assume dichotomous preferences. My assumptions primarily follow Nicoló and Rodríguez-Álvarez, 2012, Nicoló and Rodríguez-Álvarez, 2017. Nicoló and Rodríguez-Álvarez (2012) study a similar model, in which a publicly observed survival-probability matrix induces patients’ strict ordinal preferences over the kidneys. The only private information agents have is how they rank their outside option. They present an impossibility result (see Proposition 2), later generalized in Nicoló and Rodríguez-Álvarez (2013), and characterize when truth-telling is the unique protective strategy for the agents. Nicoló and Rodríguez-Álvarez (2017) propose a deterministic solution for the kidney-exchange problem under non-dichotomous and not necessarily strict preferences with the added assumption that all patients rank all kidneys in the same way, barring incompatibilities. Both papers also allow for the participation of compatible pairs.
Roth et al., 2004, Roth et al., 2005b were early voices advocating for the inclusion of compatible patient–donor pairs to kidney-exchange pools. Under dichotomous preferences and compatible-pair participation, Sönmez and Ünver (2014) propose a deterministic mechanism that finds a maximal-cardinality matching that minimizes the number of out-of-pair transplantations for compatible pairs. Sönmez et al. (2019) study how prioritizing patients with compatible donors in case of future graft failure would incentivize their participation. Andersson and Kratz (2019) study the positive market-thickness effects that the inclusion of “half-compatible” patient–donor pairs (i.e. blood-group incompatible pairs that could nevertheless undergo a compatible transplantation with immunosuppressant treatment) would have. They assume three-tiered preferences corresponding to the degree of compatibility. For another paper studying how randomization can improve equity in kidney exchanges under dichotomous preferences, see Yılmaz (2011).
The mechanism I propose here is based on the Probabilistic Serial mechanism, initially defined by Bogomolnaia and Moulin (2001) in the simple object-assignment setting. Since their seminal contribution, the PS mechanism has been adapted for ordinal preferences allowing indifferences (Katta and Sethuraman, 2006), for multi-unit demand (Kojima, 2009), and for property rights necessitating individual rationality (Yılmaz, 2009, Yılmaz, 2010) among many others. Bogomolnaia and Heo (2012) and Hashimoto et al. (2014) offer characterizations of the PS mechanism; see Kesten et al. (2017) and Chang and Chun (2017) for some recent related results.
Section snippets
Background on kidney exchange
Kidney transplantation is generally the only long-term treatment for end-stage renal disease. Most living-donor transplantations occur when a patient in need of a transplant finds a compatible donor willing to donate one of her kidneys. Even if incompatible, they can still effect a transplantation if there is another patient–donor pair, such that the donor of each pair is compatible with the patient of the other. Then the patient in the first pair receives a kidney from the donor in the second
Model
Let be a set of patient–donor pairs.9 Each patient has a strict preference order over .10 11
Deterministic mechanisms with cycle constraints
This section shows that the main desiderata (individual rationality, efficiency, anonymity, and strategyproofness) are generally incompatible for deterministic mechanisms in the presence of cycle constraints. First, there does not exist a deterministic mechanism that is fair (in the sense of anonymity) and efficient (in the sense of -constrained efficiency).
Proposition 1 For any , there does not exist an anonymous -constrained efficient deterministic mechanism for all .
Therefore, since efficiency
The 2-cycle probabilistic serial mechanism
As noted above, any random matching can be represented as a bistochastic matrix. The Birkhoff–von Neumann theorem supplies the converse: any bistochastic matrix represents one or more elements of . This section starts by presenting an analogous result regarding the matrix representation of elements of , the set of random matchings satisfying the -cycle constraint.15
Incentives and impossibility results
This section demonstrates that, while the 2CPS mechanism is not weakly strategyproof, there is no mechanism in the cycle-constrained environment that satisfies weak strategyproofness while retaining individual rationality and efficiency.
Proposition 5 The 2CPS mechanism is not weakly strategyproof for.
The 2CPS’ properties are, however, the best possible as there does not exist a mechanism that is individually rational, -constrained ex-post efficient, and weakly strategyproof. This fact is demonstrated by
Conclusion
This paper proposes a random mechanism suitable for the setting of two-way kidney exchange with strict ordinal preferences. The mechanism is individually rational so it provides sufficient incentives for compatible patient–donor pairs to enter the kidney-exchange program leading to a substantially thicker matching market. The mechanism is also ordinally efficient and anonymous, respecting the dual objectives of efficiency and equity. This section concludes by presenting simulation results on
Acknowledgments
This paper was previously circulated as the first half of a longer manuscript with the same title. I am particularly grateful to Haluk Ergin, and the two anonymous reviewers, as well as to David Ahn, Satoshi Fukuda, Yuichiro Kamada, Fuhito Kojima, Maciej Kotowski, C. Matthew Leister, Simon Loertscher, Antonio Miralles, Michèle Müller-Itten, Chris Shannon, Yong Song, and Utku Ünver. I am grateful to Laura Waring for excellent research assistance. The work was partly supported by National Science
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2022, Journal of Economic Theory