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Unformatted text preview: Matching Markets: Lecture 7 Georgios Katsenos Department of Economics, University of Hannover December 03, 2009 Housing Market Housing Market: Agents enter the market already owning one house. They can improve upon their property via exchange. It defers from the house allocation problem in assuming the presence of endowments. It is essentially a pureexchange economy with discrete resources (1 unit of each commodity). Problem formulated by Shapley and Scarf (1974). The algorithm presented is attributed to David Gale. Housing Market The Housing Market: Connection to the house allocation problem: Assign the houses randomly, by lottery. Allow the agents to exchange houses. How does the expected solution relate to the outcome of the serial dictatorship with randomly assigned priorities? Important application: Exchange of live kidney donors. Formal Model Model: The setting is described by { ( a i ,h i ,P i ) } n i =1 , where A = { a 1 ,...,a n } is the set of agents. H = { h 1 ,...,h n } is the set of houses. The house h i is initially owned by the agent a i . P i is agent a i s preference ordering of the houses. We assume that the preferences are strict. Matchings Matching: A matching in { ( a i ,h i ,P i ) } n i =1 is a 11 function : A H ; that is, if a 6 = a , then ( a ) 6 = ( a ). Pareto Efficient Matching: Defined exactly as in the housing allocation market: There is no matching in which all agents are weakly betteroff and at least one agent is strictly betteroff. Individually Rational Matching: A matching such that ( a ) a h a , for all a A . No agent gets a worse house than the one he had initially. Mechanisms Mechanism: A mechanism associates each market { ( a i ,h i ,P i ) } n i =1 with a matching via a revelation game. Pareto Efficient Mechanism: A mechanism that produces Pareto efficient matchings, for any preferences reported. Individually Rational Mechanism: A mechanism that produces individually rational matchings, for any preferences reported. Strategyproof Mechanism: A mechanism in which truthtelling, Q i = P i , is a dominant strategy for all agents. Stability: The Core The Core: Our concept of stability requires that no group of agents will be betteroff blocking a matching by exchanging only within the group. Definition: A matching is in the core of the market { ( a i ,h i ,P i ) } n i =1 if there is no coalition of agents B A and a matching such that For all a B , we have ( a ) { h i } a i B ; ( a ) a ( a ), for all a B ; ( a ) a ( a ), for at least one a B . Stability: The Core Remarks: In manytoone, two sided matching, stability is equivalent to the core....
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 Spring '08
 Serra
 Economics

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