8 research outputs found
On Houseswapping, the Strict Core, Segmentation, and Linear Programming
We consider the n-player houseswapping game of Shapley-Scarf (1974), with indiïŹerences in preferences allowed. It is well-known that the strict core of such a game may be empty, single-valued, or multivalued. We deïŹne a condition on such games called âsegmentabilityâ, which means that the set of players can be partitioned into a âtop trading segmentation.â It generalizes Galeâs well-known idea of the partition of players into âtop trading cyclesâ (which is used to ïŹnd the unique strict core allocation in the model with no indiïŹerence). We prove that a game has a nonempty strict core if and only if it is segmentable. We then use this result to devise an O(n 3 ) algorithm which takes as input any houseswapping game, and returns either a strict core allocation or a report that the strict core is empty. Finally, we are also able to construct a linear inequality system whose feasible regionâs extreme points precisely correspond to the allocations of the strict core. This last result parallels the results of Vande Vate (1989) and Rothblum (1991) for the marriage game of Gale and Shapley (1962)
A Solution for General Exchange Markets with Indivisible Goods when Indifferences are Allowed
It is well known that the core of an exchange market with indivisible goods is always non empty, although it may contain Pareto inefficient allocations. The strict core solves this shortcoming when indifferences are not allowed, but when agents' preferences are weak orders the strict core may be empty. On the other hand, when indifferences are allowed, the core or the strict core may fail to be stable sets, in the von Neumann and Morgenstern sense. We introduce a new solution concept that improves the behaviour of the strict core, in the sense that it solves the emptiness problem of the strict core when indifferences are allowed in the individuals' preferences and whenever the strict core is non-empty, our solution is included in it. We define our proposal, the MS-set, by using a stability property (m-stability) that the strict core fulfills. Finally, we provide a min-max interpretation for this new solution
Exchange of indivisible goods and indifferences: the Top Trading Absorbing Sets mechanisms
There is a wide range of economic problems involving the exchange of indivisible goods without monetary transfers, starting from the housing market model of the seminal paper of Shapley and Scarf [10] and including other problems like the kidney exchange or the school choice problems. For many of these models, the classical solution is the application of an algorithm/mechanism called Top Trading Cycles, attributed to David Gale, which satisïŹes good properties for the case of strict preferences. In this paper, we propose a family of mechanisms, called Top Trading Absorbing Sets mechanisms, that generalizes the Top Trading Cycles for the general case in which individuals can report indifferences, and preserves all its desirable properties.housing market, indifferences, top trading cycles, absorbing sets
Efficient Kidney Exchange: Coincidence of Wants in a Structured Market
Patients needing kidney transplants may have willing donors who cannot donate to them because of blood or tissue incompatibility. Incompatible patient-donor pairs can exchange donor kidneys with other such pairs. The situation facing such pairs resembles models of the âdouble coincidence of wants,â and relatively few exchanges have been consummated by decentralized means. As the population of available patient-donor pairs grows, the frequency with which exchanges can be arranged will depend in part on how exchanges are organized. We study the potential frequency of exchanges as a function of the number of patient-donor pairs, and the size of the largest feasible exchange. Developing infrastructure to identify and perform 3-way as well as 2-way exchanges will have a substantial effect on the number of transplants, and will help the most vulnerable patients. Larger than 3- way exchanges have much smaller impact. Larger populations of patient- donor pairs increase the percentage of patients of all kinds who can find exchanges.
Efficient Kidney Exchange: Coincidence of Wants in a Structured Market
Patients needing kidney transplants may have willing donors who cannot donate to them because of blood or tissue incompatibility. Incompatible patient-donor pairs can exchange donor kidneys with other such pairs. The situation facing such pairs resembles models of the "double coincidence of wants," and relatively few exchanges have been consummated by decentralized means. As the population of available patient-donor pairs grows, the frequency with which exchanges can be arranged will depend in part on how exchanges are organized. We study the potential frequency of exchanges as a function of the number of patient-donor pairs, and the size of the largest feasible exchange. Developing infrastructure to identify and perform 3-way as well as 2-way exchanges will have a substantial effect on the number of transplants, and will help the most vulnerable patients. Larger than 3-way exchanges have much smaller impact. Larger populations of patient-donor pairs increase the percentage of patients of all kinds who can find exchanges.
Designated school choice
Turkish government changed the high-school placement system for several concerns in 2018. The government as a designer designates and orders schools to each student in terms of location. Then students reveal their preference list over these designated schools. The government desires students to be assigned to as possible as the closest schools. However, studentsâ preference list is independent from the designation order. In this context, there is an incompatibility between the studentsâ preferences and the concern of the designer. The thesis will solve this sort of incompatibility. Two-Stage-Generalized-Priority-Mechanism proposed in the thesis finds the set of all possible designer-optimal matchings. At the second-stage, TSGPM yields the best designer-optimal matching in terms of the studentsâ preference list. At the last part of the thesis, strategic properties of the mechanism will be discusse
On Houseswapping, the Strict Core, Segmentation, and Linear Programming
We consider the n-player houseswapping game of Shapley-Scarf (1974), with indfferences in preferences allowed. It is well-known that the strict core of such a game may be empty, single-valued, or multi-valued. We define a condition on such games called "segmentability", which means that the set of players can be partitioned into a "top trading segmentation". It generalizes Gale's well-known idea of the partition of players into "top trading cycles" (which is used to find the unique strict core allocation in the model with no indifference). We prove that a game has a nonempty strict core if and only if it is segmentable. We then use this result to devise and O(n^3) algorithm which takes as input any houseswapping game, and returns either a strict core allocation or a report that the strict core is empty. Finally, we are also able to construct a linear inequality system whose feasible region's extreme points precisely correspond to the allocations of the strict core. This last result parallels the results of Vande Vate (1989) and Rothbum (1991) for the marriage game of Gale and Shapley (1962).Shapley-Scarf Economy, Strict Core, Linear Inequality System, Extreme Points