A More Reliable Greedy Heuristic for Maximum Matchings in Sparse Random Graphs

We propose a new greedy algorithm for the maximum cardinality matching problem. We give experimental evidence that this algorithm is likely to find a maximum matching in random graphs with constant expected degree c>0, independent of the value of c. This is contrary to the behavior of commonly used greedy matching heuristics which are known to have some range of c where they probably fail to compute a maximum matching

Coalitions and Cliques in the School Choice Problem

The school choice mechanism design problem focuses on assignment mechanisms matching students to public schools in a given school district. The well-known Gale Shapley Student Optimal Stable Matching Mechanism (SOSM) is the most efficient stable mechanism proposed so far as a solution to this problem. However its inefficiency is well-documented, and recently the Efficiency Adjusted Deferred Acceptance Mechanism (EADAM) was proposed as a remedy for this weakness. In this note we describe two related adjustments to SOSM with the intention to address the same inefficiency issue. In one we create possibly artificial coalitions among students where some students modify their preference profiles in order to improve the outcome for some other students. Our second approach involves trading cliques among students where those involved improve their assignments by waiving some of their priorities. The coalition method yields the EADAM outcome among other Pareto dominations of the SOSM outcome, while the clique method yields all possible Pareto optimal Pareto dominations of SOSM. The clique method furthermore incorporates a natural solution to the problem of breaking possible ties within preference and priority profiles. We discuss the practical implications and limitations of our approach in the final section of the article

Strategic schools under the Boston mechanism revisited

We show that Ergin & Sönmez's (2006) results which show that for schools it is a dominant strategy to truthfully rank the students under the Boston mechanism, and that the Nash equilibrium outcomes in undominated strategies of the induced game are stable, rely crucially on two assumptions. First, (a) that schools need to be restricted to find all students acceptable, and (b) that students cannot observe the priorities set by the schools before submitting their preferences. We show that relaxing either assumption eliminates the strategy dominance, and that Nash equilibrium outcomes in undominated strategies for the simultaneous induced game in case (a) and subgame perfect Nash equilibria in case (b) may contain unstable matchings. We also show that when able to manipulate capacities, schools may only have an incentive to do so if students submit their preferences after observing the reported capacities

Pairwise Kidney Exchange

In connection with an earlier paper on the exchange of live donor kidneys (Roth, S”nmez, and šnver 2004) the authors entered into discussions with New England transplant surgeons and their colleagues in the transplant community, aimed at implementing a Kidney Exchange program. In the course of those discussions it became clear that a likely first step will be to implement pairwise exchanges, between just two patient-donor pairs, as these are logistically simpler than exchanges involving more than two pairs. Furthermore, the experience of these surgeons suggests to them that patient and surgeon preferences over kidneys should be 0-1, i.e. that patients and surgeons should be indifferent among kidneys from healthy donors whose kidneys are compatible with the patient. This is because, in the United States, transplants of compatible live kidneys have about equal graft survival `robabilities, regardless of the closeness of tissue types between patient and dOnor (unless there is a rare perfect match). In the present paper we show that, although thd pairwise constraint eliminates some potential exchanges, there is a wide class of constrained-efficient mechanisms 4hat are strategy-proof when patient-donor pairs and surgeons have 0-1 preferences. This class of meahanisms includes deterministic mechanisms that would accomodate the kinds of priority setting that organ banks currently use for the allocation of cadaver organs, as well as stochastic mechanisms that allow considerations of distributive justice to be addressed.

Pairwise Kidney Exchange

The theoretical literature on exchange of indivisible goods finds natural application in organizing the exchange of live donor kidneys for transplant. However, in kidney exchange, there are constraints on the size of feasible exchanges. Initially, kidney exchanges are likely to be pairwise exchanges, between just two patient-donor pairs, as these are logistically simpler than larger exchanges. Furthermore, the experience of many American surgeons suggests to them that preferences over kidneys are approximately 0-1, i.e. that patients and surgeons should be largely indifferent among healthy donors whose kidneys are compatible with the patient. This is because, in the United States, transplants of compatible live kidneys have about equal graft survival probabilities, regardless of the closeness of tissue types between patient and donor. We show that, although the pairwise constraint eliminates some potential exchanges, there is a wide class of constrained-efficient mechanisms that are strategy-proof when patient-donor pairs and surgeons have 0-1 preferences. This class of mechanisms includes deterministic mechanisms that would accomodate the kinds of priority setting that organ banks currently use to allocate cadaver organs, as well as stochastic mechanisms that allow distributive justice issues to be

Defining Equitable Geographic Districts in Road Networks via Stable Matching

We introduce a novel method for defining geographic districts in road networks using stable matching. In this approach, each geographic district is defined in terms of a center, which identifies a location of interest, such as a post office or polling place, and all other network vertices must be labeled with the center to which they are associated. We focus on defining geographic districts that are equitable, in that every district has the same number of vertices and the assignment is stable in terms of geographic distance. That is, there is no unassigned vertex-center pair such that both would prefer each other over their current assignments. We solve this problem using a version of the classic stable matching problem, called symmetric stable matching, in which the preferences of the elements in both sets obey a certain symmetry. In our case, we study a graph-based version of stable matching in which nodes are stably matched to a subset of nodes denoted as centers, prioritized by their shortest-path distances, so that each center is apportioned a certain number of nodes. We show that, for a planar graph or road network with $n$ nodes and $k$ centers, the problem can be solved in $O(n\sqrt{n}\log n)$ time, which improves upon the $O(nk)$ runtime of using the classic Gale-Shapley stable matching algorithm when $k$ is large. Finally, we provide experimental results on road networks for these algorithms and a heuristic algorithm that performs better than the Gale-Shapley algorithm for any range of values of $k$.Comment: 9 pages, 4 figures, to appear in 25th ACM SIGSPATIAL International Conference on Advances in Geographic Information Systems (ACM SIGSPATIAL 2017) November 7-10, 2017, Redondo Beach, California, US

Fictitious students creation incentives in school choice problems

We address the question of whether schools can manipulate the student-optimal stable mechanism by creating fictitious students in school choice problems. To this end, we introduce two different manipulation concepts, where one of them is stronger. We first demonstrate that the student-optimal stable mechanism is not even weakly fictitious student-proof under general priority structures. Then, we investigate the same question under acyclic priority structures. We prove that, while the student-optimal stable mechanism is not strongly fictitious student-proof even under the acyclicity condition, weak fictitious student-proofness is achieved under acyclicity. This paper, hence, shows a way to avoid the welfare detrimental fictitious students creation (in the weak sense) in terms of priority structures

Minimal Envy and Popular Matchings

We study ex-post fairness in the object allocation problem where objects are valuable and commonly owned. A matching is fair from individual perspective if it has only inevitable envy towards agents who received most preferred objects -- minimal envy matching. A matching is fair from social perspective if it is supported by majority against any other matching -- popular matching. Surprisingly, the two perspectives give the same outcome: when a popular matching exists it is equivalent to a minimal envy matching. We show the equivalence between global and local popularity: a matching is popular if and only if there does not exist a group of size up to 3 agents that decides to exchange their objects by majority, keeping the remaining matching fixed. We algorithmically show that an arbitrary matching is path-connected to a popular matching where along the path groups of up to 3 agents exchange their objects by majority. A market where random groups exchange objects by majority converges to a popular matching given such matching exists. When popular matching might not exist we define most popular matching as a matching that is popular among the largest subset of agents. We show that each minimal envy matching is a most popular matching and propose a polynomial-time algorithm to find them