Approximability of Economic Equilibrium in Housing Markets with - PowerPoint PPT Presentation

Approximability of Economic Equilibrium in Housing Markets with Duplicate Houses Katarína Cechlárová, PF UPJŠ Košice and Eva Jelínková, MFF UK Praha Economic equilibrium in housing markets K. Cechlárová & E. Jelínková

Basic notions Definition A housing market is a quadruple M = ( A, H, ω, P ) where A is a set of n agents, H is a set of m house types ω : A → H is the endowment function preference profile P is an n -tuple of agents’ preferences, i.e. linearly ordered lists P ( a ) of acceptable house types Example. A = { a 1 , a 2 , . . . , a 7 } ; H = { h 1 , h 2 , h 3 , h 4 } ω ( a 1 ) = h 1 ; P ( a 1 ) : h 4 , h 3 , h 2 , h 1 ω ( a 2 ) = h 4 ; P ( a 2 ) : ( h 1 , h 3 ) , h 4 ω ( a 3 ) = h 1 ; P ( a 3 ) : h 2 , h 4 , h 1 ω ( a 4 ) = h 2 ; P ( a 4 ) : ( h 1 , h 3 ) , h 4 , h 2 ω ( a 5 ) = h 2 ; P ( a 5 ) : h 4 , h 1 , h 2 ω ( a 6 ) = h 3 ; P ( a 6 ) : h 4 , h 3 ω ( a 7 ) = h 4 ; P ( a 7 ) : h 3 , h 1 , h 4 Economic equilibrium in housing markets K. Cechlárová & E. Jelínková

Basic notions Definition A housing market is a quadruple M = ( A, H, ω, P ) where A is a set of n agents, H is a set of m house types ω : A → H is the endowment function preference profile P is an n -tuple of agents’ preferences, i.e. linearly ordered lists P ( a ) of acceptable house types Example. A = { a 1 , a 2 , . . . , a 7 } H = { h 1 , h 2 , h 3 , h 4 } ω ( a 1 ) = h 1 ; P ( a 1 ) : h 4 , h 3 , h 2 , h 1 acceptable houses ω ( a 2 ) = h 4 ; P ( a 2 ) : ( h 1 , h 3 ) , h 4 ties ω ( a 3 ) = h 1 ; P ( a 3 ) : h 2 , h 4 , h 1 strict preferences ω ( a 4 ) = h 2 ; P ( a 4 ) : ( h 1 , h 3 ) , h 4 , h 2 trichotomous preferences ω ( a 5 ) = h 2 ; P ( a 5 ) : h 4 , h 1 , h 2 ω ( a 6 ) = h 3 ; P ( a 6 ) : h 4 , h 3 ω ( a 7 ) = h 4 ; P ( a 7 ) : h 3 , h 1 , h 4 Economic equilibrium in housing markets K. Cechlárová & E. Jelínková

Basic notions Definition A housing market is a quadruple M = ( A, H, ω, P ) where A is a set of n agents, H is a set of m house types ω : A → H is the endowment function preference profile P is an n -tuple of agents’ preferences, i.e. linearly ordered lists P ( a ) of acceptable house types Example. A = { a 1 , a 2 , . . . , a 7 } H = { h 1 , h 2 , h 3 , h 4 } ω ( a 1 ) = h 1 ; P ( a 1 ) : ( h 4 , h 3 , h 2 ) , h 1 trichotomous preferences ω ( a 2 ) = h 4 ; P ( a 2 ) : ( h 1 , h 3 ) , h 4 each agent has: ω ( a 3 ) = h 1 ; P ( a 3 ) : ( h 2 , h 4 ) , h 1 1. better house types ω ( a 4 ) = h 2 ; P ( a 4 ) : ( h 1 , h 3 , h 4 ) , h 2 2. type of his own house ω ( a 5 ) = h 2 ; P ( a 5 ) : ( h 4 , h 1 ) , h 2 3. unacceptable houses ω ( a 6 ) = h 3 ; P ( a 6 ) : h 4 , h 3 ω ( a 7 ) = h 4 ; P ( a 7 ) : ( h 3 , h 1 ) , h 4 Economic equilibrium in housing markets K. Cechlárová & E. Jelínková

Further notation Definition A function x : A → H is an allocation if there exists a bijection π on A such that x ( a ) = ω ( π ( a )) for each a ∈ A . Each allocation consists of trading cycles ω ( a 1 ) = h 1 ; P ( a 1 ) : h 4 , h 3 , h 2 , h 1 take trading cycles ω ( a 2 ) = h 4 ; P ( a 2 ) : ( h 1 , h 3 ) , h 4 ( a 1 , a 7 , a 6 , a 2 )( a 3 , a 4 , a 5 ) ω ( a 3 ) = h 1 ; P ( a 3 ) : h 2 , h 4 , h 1 this means ω ( a 4 ) = h 2 ; P ( a 4 ) : ( h 1 , h 3 ) , h 4 , h 2 x ( a 1 ) = h 4 ; ω ( a 5 ) = h 2 ; P ( a 5 ) : h 4 , h 1 , h 2 x ( a 7 ) = h 3 ; ω ( a 6 ) = h 3 ; P ( a 6 ) : h 4 , h 3 x ( a 6 ) = h 4 ; ω ( a 7 ) = h 4 ; P ( a 7 ) : h 3 , h 1 , h 4 x ( a 2 ) = h 1 etc. Economic equilibrium in housing markets K. Cechlárová & E. Jelínková

Economic equilibrium Definition A pair ( p, x ) , where p : H → R is a price function and x is an allocation on A is an economic equilibrium for market M if for each a ∈ A , house x ( a ) is of type that is among the most preferred house types in his budget set, i.e. S = B a ( p ) = { h ∈ H ; p ( h ) ≤ p ( ω ( a )) } . Lema If ( p, x ) is an economic equilibrium for market M then p ( x ( a )) = p ( ω ( a )) for each a ∈ A . Economic equilibrium in housing markets K. Cechlárová & E. Jelínková

Example: equilibrium ω ( a 1 ) = h 1 ; P ( a 1 ) : h 4 , h 3 , h 2 , h 1 ω ( a 2 ) = h 4 ; P ( a 2 ) : ( h 1 , h 3 ) , h 4 ω ( a 3 ) = h 1 ; P ( a 3 ) : h 2 , h 4 , h 1 ω ( a 4 ) = h 2 ; P ( a 4 ) : ( h 1 , h 3 ) , h 4 , h 2 ω ( a 5 ) = h 2 ; P ( a 5 ) : h 4 , h 1 , h 2 ω ( a 6 ) = h 3 ; P ( a 6 ) : h 4 , h 3 ω ( a 7 ) = h 4 ; P ( a 7 ) : h 3 , h 1 , h 4 Take p ( h j ) = p for all j and ( a 1 , a 7 , a 6 , a 2 )( a 3 , a 4 , a 5 ) Not equilibrium, since x ( a 5 ) = h 1 and this is the second-choice house Economic equilibrium in housing markets K. Cechlárová & E. Jelínková

Example: equilibrium ω ( a 1 ) = h 1 ; P ( a 1 ) : h 4 , h 3 , h 2 , h 1 ω ( a 2 ) = h 4 ; P ( a 2 ) : ( h 1 , h 3 ) , h 4 ω ( a 3 ) = h 1 ; P ( a 3 ) : h 2 , h 4 , h 1 ω ( a 4 ) = h 2 ; P ( a 4 ) : ( h 1 , h 3 ) , h 4 , h 2 ω ( a 5 ) = h 2 ; P ( a 5 ) : h 4 , h 1 , h 2 ω ( a 6 ) = h 3 ; P ( a 6 ) : h 4 , h 3 ω ( a 7 ) = h 4 ; P ( a 7 ) : h 3 , h 1 , h 4 Take p ( h j ) = p for all j and ( a 1 , a 7 , a 6 , a 2 )( a 3 , a 4 , a 5 ) Not equilibrium, since x ( a 5 ) = h 1 and this is the second-choice house Observation: In this example there is no equilibrium with equal prices, as demand for houses of type h 4 is 3, while the supply is only 2. Economic equilibrium in housing markets K. Cechlárová & E. Jelínková

Brief history Walras 1874: notion of equilibrium Arrow, Debreu 1954: notion of exchange economy equilibrium exist if commodites are infinitely divisible Deng, Papadimitriou, Safra 2002: if commodities are indivisible, decision about the equilibrium existence is NPC Shapley, Scarf 1974: housing market m = n ; each house different Gale 1974: proof of equilibrium existence by TTC algorithm Economic equilibrium in housing markets K. Cechlárová & E. Jelínková

Brief history Walras 1874: notion of equilibrium Arrow, Debreu 1954: notion of exchange economy equilibrium exist if commodites are infinitely divisible Deng, Papadimitriou, Safra 2002: if commodities are indivisible, decision about the equilibrium existence is NPC Shapley, Scarf 1974: housing market m = n ; each house different Gale 1974: proof of equilibrium existence by TTC algorithm Economic equilibrium in housing markets K. Cechlárová & E. Jelínková

Brief history Walras 1874: notion of equilibrium Arrow, Debreu 1954: notion of exchange economy equilibrium exist if commodites are infinitely divisible Deng, Papadimitriou, Safra 2002: if commodities are indivisible, decision about the equilibrium existence is NPC Shapley, Scarf 1974: housing market m = n ; each house different Gale 1974: proof of equilibrium existence by TTC algorithm Economic equilibrium in housing markets K. Cechlárová & E. Jelínková

Brief history Walras 1874: notion of equilibrium Arrow, Debreu 1954: notion of exchange economy equilibrium exist if commodites are infinitely divisible Deng, Papadimitriou, Safra 2002: if commodities are indivisible, decision about the equilibrium existence is NPC Shapley, Scarf 1974: housing market m = n ; each house different Gale 1974: proof of equilibrium existence by TTC algorithm Economic equilibrium in housing markets K. Cechlárová & E. Jelínková

Brief history Walras 1874: notion of equilibrium Arrow, Debreu 1954: notion of exchange economy equilibrium exist if commodites are infinitely divisible Deng, Papadimitriou, Safra 2002: if commodities are indivisible, decision about the equilibrium existence is NPC Shapley, Scarf 1974: housing market m = n ; each house different Gale 1974: proof of equilibrium existence by TTC algorithm Economic equilibrium in housing markets K. Cechlárová & E. Jelínková

Brief history Walras 1874: notion of equilibrium Arrow, Debreu 1954: notion of exchange economy equilibrium exist if commodites are infinitely divisible Deng, Papadimitriou, Safra 2002: if commodities are indivisible, decision about the equilibrium existence is NPC Shapley, Scarf 1974: housing market m = n ; each house different Gale 1974: proof of equilibrium existence by TTC algorithm Economic equilibrium in housing markets K. Cechlárová & E. Jelínková

Brief history Walras 1874: notion of equilibrium Arrow, Debreu 1954: notion of exchange economy equilibrium exist if commodites are infinitely divisible Deng, Papadimitriou, Safra 2002: if commodities are indivisible, decision about the equilibrium existence is NPC Shapley, Scarf 1974: housing market m = n ; each house different Gale 1974: proof of equilibrium existence by TTC algorithm Economic equilibrium in housing markets K. Cechlárová & E. Jelínková

Brief history Walras 1874: notion of equilibrium Arrow, Debreu 1954: notion of exchange economy equilibrium exist if commodites are infinitely divisible Deng, Papadimitriou, Safra 2002: if commodities are indivisible, decision about the equilibrium existence is NPC Shapley, Scarf 1974: housing market m = n ; each house different Gale 1974: proof of equilibrium existence by TTC algorithm Economic equilibrium in housing markets K. Cechlárová & E. Jelínková