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Unformatted text preview: Housing Markets & Top Trading Cycles Tayfun S¨onmez 16th Jerusalem Summer School in Economic Theory “Matching, Auctions, and Market Design” House Allocation Problems: A Collective Ownership Economy A house allocation problem (Hylland & Zeckhauser, JPE 1979) is a triple h I, H, R i . I : set of agents H : set of houses P : list of preferences over houses For simplicity assume: 1.  H  =  I  , and 2. the preferences are strict. The Outcome: A Matching A (house) matching µ : I → H is a onetoone and onto function from I to H . With everyday language it is an assignment of houses to agents such that 1. every agent is assigned one house, and 2. no house is assigned to more than one agent. Housing Markets: A Basic Exchange Economy A housing market (Shapley & Scarf, JPE 1974) is a 4tuple h I, H, R, µ i . I : set of agents H : set of houses with  H  =  I  P : list of strict preferences over houses µ : initial endowment matching A matching η is in the core of the housing market ( I, H, R, µ ) if there is no coalition T ⊆ I and matching ν such that 1. ν ( i ) ∈ µ ( T ) for all i ∈ T , 2. ν ( i ) R i η ( i ) for all i ∈ T , 3. ν ( i ) P i η ( i ) f o r s o m e i ∈ T . Gale’s Top Trading Cycles Algorithm (Described in Shapley & Scarf, attributed to David Gale) Step 1 : Each agent “points to” the owner of his favorite house. Since there are finite number of agents, there is at least one cycle . Each agent in a cycle is assigned the house of the agent he points to and removed from the market with his assignment. If there is at least one remaining agent, proceed with the next step. Step t : Each remaining agent points to the owner of his favorite house among the remaining houses. Every agent in a cycle is assigned the house of the agent he points to and removed from the market with his assignment. If there is at least one remaining agent, proceed with the next step. Important Properties of the Core Theorem (Roth & Postlewaite, JME 1977): There is a unique matching in the core of each housing market. Moreover, this matching is the unique competitive allocation. A direct matching mechanism is a systematic procedure to select a matching for each problem. Theorem (Roth, Economics Letters 1982): Core (as a direct mechanism) is strategyproof . Theorem (Ma, IJGT 1994): Core is the only mechanism that is Pareto eﬃcient, individually rational, and strategy proof . Mechanisms for House Allocation An ordering f : { 1 , . . . , n } → A is a onetoone and onto function. Simple serial dictatorship induced by f : Agent who is ordered first (by the ordering f ) gets her top choice; agent ordered second gets his top choice among those remaining; and so on....
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 Spring '10
 SHALOM
 Linear Algebra, Algebra, Game Theory, D. E. Marsh, British H class submarine

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