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Matching-7

# Matching-7 - Matching Markets Lecture 7 Georgios Katsenos...

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Matching Markets: Lecture 7 Georgios Katsenos Department of Economics, University of Hannover December 03, 2009

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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 pure-exchange 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.

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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 1-1 function μ : A H ; that is, if a 6 = a 0 , then μ ( a ) 6 = μ ( a 0 ). Pareto Efficient Matching: Defined exactly as in the housing allocation market: There is no matching in which all agents are weakly better-off and at least one agent is strictly better-off. Individually Rational Matching: A matching μ such that μ ( a ) a h a , for all a A . No agent gets a worse house than the one he had initially.

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Mechanisms Mechanism: A mechanism φ associates each market { ( a i , h i , P i ) } n i =1 with a matching μ 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. Strategy-proof Mechanism: A mechanism in which truth-telling, 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 better-off blocking a matching by exchanging only within the group. Definition: A matching μ 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 ν such that - For all a B , we have ν ( a ) ∈ { h i } a i B ; - ν ( a ) a μ ( a ), for all a B ; - ν ( a ) a μ ( a ), for at least one a B .

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Stability: The Core Remarks: - In many-to-one, two sided matching, stability is equivalent to the core. - Every matching in the core is Pareto efficient. Simply let B = A . - Every matching in the core is individually rational. Simply let B = { a } , for any a A .
Stability: The Core Remark: A Pareto efficient and individually rational matching may not be in the core. Consider the following market: P ( a 1 ) = h 3 , h 1 , h 2 P ( a 2 ) = h 3 , h 2 , h 1 P ( a 3 ) = h 1 , h 2 , h 3 The matching μ = h 1 h 3 h 2 Pareto efficient and individually rational.

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