Itai Ashlagi

Nonsimultaneous Chains and Dominos in Kidney‐ Paired Donation—Revisited

Since 2008, kidney exchange in America has grown in part from the incorporation of nondirected donors in transplant chains rather than simple exchanges. It is controversial whether these chains should be performed simultaneously ‘domino‐paired donation’, (DPD) or nonsimultaneously ‘nonsimultaneous extended altruistic donor, chains (NEAD). NEAD chains create ‘bridge donors’ whose incompatible recipients receive kidneys before the bridge donor donates, and so risk reneging by bridge donors, but offer the opportunity to create more transplants by overcoming logistical barriers inherent in simultaneous chains. Gentry et al. simulated whether DPD or NEAD chains would produce more transplants when chain segment length was limited to three transplants, and reported that DPD performed at least as well as NEAD chains. As this finding contrasts with the experience of several groups involved in kidney‐paired donation, we performed simulations that allowed for longer chain segments and used actual patient data from the Alliance for Paired Donation. When chain segments of 4–6 transplants are allowed in the simulations, NEAD chains produce more transplants than DPD. Our simulations showed not only more transplants as chain length increased, but also that NEAD chains produced more transplants for highly sensitized and blood type O recipients.

Monotonicity and Implementability

Consider an environment with a finite number of alternatives, and agents with private values and quasilinear utility functions. A domain of valuation functions for an agent is a monotonicity domain if every finite-valued monotone randomized allocation rule defined on it is implementable in dominant strategies. We fully characterize the set of all monotonicity domains. Copyright 2010 The Econometric Society.

Resource selection games with unknown number of players

In the context of pre-Bayesian games we analyze resource selection games with unknown number of players. We prove the existence and uniqueness of a symmetric safety-level equilibrium in such games and show that in a game with strictly increasing linear cost functions every player benefits from the common ignorance about the number of players. In order to perform the analysis we define safety-level equilibrium for pre-Bayesian games, and prove that it exists in a compact-continuous-concave setup; in particular it exists in a finite setup.

Free riding and participation in large scale, multi-hospital kidney exchange

As multi-hospital kidney exchange has grown, the set of players has grown from patients and surgeons to include hospitals. Hospitals can choose to enroll only their hard-to-match patient-donor pairs, while conducting easily-arranged exchanges internally. This behavior has already been observed. We show that as the population of hospitals and patients grows the cost of making it individually rational for hospitals to participate fully becomes low in almost every large exchange pool (although the worst-case cost is very high), while the cost of failing to guarantee individual rationality is high, in lost transplants. We identify a mechanism that gives hospitals incentives to reveal all patient-donor pairs. We observe that if such a mechanism were to be implemented and hospitals enrolled all their pairs, the resulting patient pools would allow efficient matchings that could be implemented with two and three way exchanges.

Stability in Large Matching Markets with Complementarities

Labor markets can often be viewed as many-to-one matching markets. It is well known that if complementarities are present in such markets, a stable matching may not exist. We study large random matching markets with couples. We introduce a new matching algorithm and show that if the number of couples grows slower than the size of the market, a stable matching will be found with high probability. If however, the number of couples grows at a linear rate, with constant probability not depending on the market size, no stable matching exists. Our results explain data from the market for psychology interns.

Finding long chains in kidney exchange using the traveling salesman problem

Significance There are currently more than 100,000 patients on the waiting list in the United States for a kidney transplant from a deceased donor. To address this shortage, kidney exchange programs allow patients with living incompatible donors to exchange donors through cycles and chains initiated by altruistic nondirected donors. To determine which exchanges will take place, kidney exchange programs use algorithms for maximizing the number of transplants under constraints about the size of feasible exchanges. This problem is NP-hard, and algorithms previously used were unable to optimize when chains could be long. We developed two algorithms that use integer programming to solve this problem, one of which is inspired by the traveling salesman, that together can find optimal solutions in practice. As of May 2014 there were more than 100,000 patients on the waiting list for a kidney transplant from a deceased donor. Although the preferred treatment is a kidney transplant, every year there are fewer donors than new patients, so the wait for a transplant continues to grow. To address this shortage, kidney paired donation (KPD) programs allow patients with living but biologically incompatible donors to exchange donors through cycles or chains initiated by altruistic (nondirected) donors, thereby increasing the supply of kidneys in the system. In many KPD programs a centralized algorithm determines which exchanges will take place to maximize the total number of transplants performed. This optimization problem has proven challenging both in theory, because it is NP-hard, and in practice, because the algorithms previously used were unable to optimally search over all long chains. We give two new algorithms that use integer programming to optimally solve this problem, one of which is inspired by the techniques used to solve the traveling salesman problem. These algorithms provide the tools needed to find optimal solutions in practice.

Mix and match: A strategyproof mechanism for multi-hospital kidney exchange

As kidney exchange programs are growing, manipulation by hospitals becomes more of an issue. Assuming that hospitals wish to maximize the number of their own patients who receive a kidney, they may have an incentive to withhold some of their incompatible donor–patient pairs and match them internally, thus harming social welfare. We study mechanisms for two-way exchanges that are strategyproof, i.e., make it a dominant strategy for hospitals to report all their incompatible pairs. We establish lower bounds on the welfare loss of strategyproof mechanisms, both deterministic and randomized, and propose a randomized mechanism that guarantees at least half of the maximum social welfare in the worst case. Simulations using realistic distributions for blood types and other parameters suggest that in practice our mechanism performs much closer to optimal.

An optimal lower bound for anonymous scheduling mechanisms

We consider the problem of designing truthful mechanisms to minimize the makespan on m unrelated machines. In their seminal paper, Nisan and Ronen [14] showed a lower bound of 2, and an upper bound of m, thus leaving a large gap. They conjectured that their upper bound is tight, but were unable to prove it. Despite many attempts that yield positive results for several special cases, the conjecture is far from being solved: the lower bound was only recently slightly increased to 2.61 [5,10], while the best upper bound remained unchanged. In this paper we show the optimal lower bound on truthful anonymous mechanisms: no such mechanism can guarantee an approximation ratio better than m. This is the first concrete evidence to the correctness of the Nisan-Ronen conjecture, especially given that the classic scheduling algorithms are anonymous, and all state-of-the-art mechanisms for special cases of the problem are anonymous as well.

Unbalanced random matching markets

We analyze large random matching markets with unequal numbers of men and women. Agents have complete preference lists that are uniformly random and independent, and we consider stable matchings under the realized preferences. We find that being on the short side of the market confers a large advantage. We characterize the mens average rank of their wives. For each agent, assign a rank of 1 to the agents most preferred partner, a rank of 2 to the next most preferred partner and so forth. If there are n men and n+1 women then, we show that with high probability, in any stable matching, the mens average rank of their wives is no more than 3 log n, whereas the womens average rank of their husbands is at least n(3 log n). If there are n men and (1+λ)n women for λ0 then, with high probability, in any stable matching the mens average rank of wives is O(1), whereas the womens average rank of husbands is λ (n). Moreover, we find that in each case, whp, the number of agents who have multiple stable partners is o(n). Thus our results imply a limited scope for manipulation in unbalanced random matching markets for mechanisms that implement a stable match. These results are in stark contrast with previously known results for random matching markets with an equal number of men and women. In such balanced random matching markets, the lattice of stable matches is large, with the two extreme points of the lattice, the men optimal stable match (MOSM) and the women optimal stable match (WOSM) possessing contrasting properties. The mens average rank of their wives is just log n under the MOSM, but as large as n/log n under the WOSM, and the opposite holds for the womens average rank of their husbands. Thus, the proposing side in the Gale-Shapley deferred acceptance algorithm is greatly advantaged in a balanced market, whereas we prove that in markets with even a slight imbalance, the MOSM and WOSM are almost identical. This reveals the balanced case to be a knife edge. Our proof uses an algorithm which calculates the WOSM from the MOSM through a sequence of proposals by men. A woman improves if, by divorcing her husband, she triggers a rejection chain that results in a proposal back to her from a more preferred man. The algorithm lends itself to a stochastic analysis, in which we show that most rejection chains are likely to end in a proposal to an unmatched woman. Simulations show that our results hold even for small markets.

Optimal Lower Bounds for Anonymous Scheduling Mechanisms

We consider the problem of designing truthful mechanisms on m unrelated machines, to minimize some optimization goal. Nisan and Ronen [Nisan, N., A. Ronen. 2001. Algorithmic mechanism design. Games Econom. Behav.35 166--196] consider the specific goal of makespan minimization, and show a lower bound of 2, and an upper bound of m. This large gap inspired many attempts that yielded positive results for several special cases, but very partial success for the general case: the lower bound was slightly increased to 2.61 by Christodoulou et al. [Christodoulou, G., E. Koutsoupias, A. Kovacs. 2010. Mechanism design for fractional scheduling on unrelated machines. ACM Trans. Algorithms (TALG)6(2) 1--18] and Koutsoupias and Vidali [Koutsoupias, E., A. Vidali. 2007. A lower bound of 1+phi for truthful scheduling mechanisms. Proc. 32nd Internat. Sympos. Math. Foundations Comput. Sci. (MFCS)], while the best upper bound remains unchanged. In this paper we show the optimal lower bound on truthful anonymous mechanisms: no such mechanism can guarantee an approximation ratio better than m. Moreover, our proof yields similar optimal bounds for two other optimization goals: the sum of completion times and the lp norm of the schedule.