Uniqueness and purity in multi-agent matching problems Brendan Pass - PowerPoint PPT Presentation

Uniqueness and purity in multi-agent matching problems Brendan Pass (joint with Y.-H. Kim (UBC)) University of Alberta September 16, 2014 Brendan Pass (joint with Y.-H. Kim (UBC)) Uniqueness and purity in multi-agent matching problems

Multi-agent matching under transferable utility Probability measures µ i on compact X i ⊆ R n , i = 1 , 2 , ..., m . distributions of agent types . Brendan Pass (joint with Y.-H. Kim (UBC)) Uniqueness and purity in multi-agent matching problems

Multi-agent matching under transferable utility Probability measures µ i on compact X i ⊆ R n , i = 1 , 2 , ..., m . distributions of agent types . Surplus function s ( x 1 , x 2 , ..., x m ) Brendan Pass (joint with Y.-H. Kim (UBC)) Uniqueness and purity in multi-agent matching problems

Multi-agent matching under transferable utility Probability measures µ i on compact X i ⊆ R n , i = 1 , 2 , ..., m . distributions of agent types . Surplus function s ( x 1 , x 2 , ..., x m ) Matching measure: A probability measure γ on X 1 × X 2 × ... × X m whose marginals are the µ i . Γ( µ 1 , µ 2 , ..., µ m ) = set of all matchings. Brendan Pass (joint with Y.-H. Kim (UBC)) Uniqueness and purity in multi-agent matching problems

Multi-agent matching under transferable utility Probability measures µ i on compact X i ⊆ R n , i = 1 , 2 , ..., m . distributions of agent types . Surplus function s ( x 1 , x 2 , ..., x m ) Matching measure: A probability measure γ on X 1 × X 2 × ... × X m whose marginals are the µ i . Γ( µ 1 , µ 2 , ..., µ m ) = set of all matchings. A matching is stable if there exists functions u 1 ( x 1 ) , u 2 ( x 2 ) , ..., u m ( x m ) such that m � u i ( x i ) ≥ s ( x 1 , x 2 , ..., x m ) i =1 with equality γ almost everywhere (payoff functions). A division of the utility among matched agents. Brendan Pass (joint with Y.-H. Kim (UBC)) Uniqueness and purity in multi-agent matching problems

Variational formulation: multi-marginal optimal transport Shapley-Shubik (1972): A matching is stable if and only if it maximizes: � γ �→ s ( x 1 , x 2 , ..., x m ) d γ, X 1 × X 2 × ... × X m over Γ( µ 1 , µ 2 , ..., µ m ) A multi-marginal optimal transportation problem. Brendan Pass (joint with Y.-H. Kim (UBC)) Uniqueness and purity in multi-agent matching problems

Variational formulation: multi-marginal optimal transport Shapley-Shubik (1972): A matching is stable if and only if it maximizes: � γ �→ s ( x 1 , x 2 , ..., x m ) d γ, X 1 × X 2 × ... × X m over Γ( µ 1 , µ 2 , ..., µ m ) A multi-marginal optimal transportation problem. Existence of a stable matching is easy to show. What about uniqueness? Purity – is γ concentrated on a graph over x 1 ? Brendan Pass (joint with Y.-H. Kim (UBC)) Uniqueness and purity in multi-agent matching problems

Variational formulation: multi-marginal optimal transport Shapley-Shubik (1972): A matching is stable if and only if it maximizes: � γ �→ s ( x 1 , x 2 , ..., x m ) d γ, X 1 × X 2 × ... × X m over Γ( µ 1 , µ 2 , ..., µ m ) A multi-marginal optimal transportation problem. Existence of a stable matching is easy to show. What about uniqueness? Purity – is γ concentrated on a graph over x 1 ? When m = 2, the generalized Spence-Mirrlees, or twist condition yields uniqueness and purity: Injectivity of x 2 �→ D x 1 s ( x 1 , x 2 ) (Ex. s ( x 1 , x 2 ) = x 1 · x 2 .) Brendan Pass (joint with Y.-H. Kim (UBC)) Uniqueness and purity in multi-agent matching problems

Variational formulation: multi-marginal optimal transport Shapley-Shubik (1972): A matching is stable if and only if it maximizes: � γ �→ s ( x 1 , x 2 , ..., x m ) d γ, X 1 × X 2 × ... × X m over Γ( µ 1 , µ 2 , ..., µ m ) A multi-marginal optimal transportation problem. Existence of a stable matching is easy to show. What about uniqueness? Purity – is γ concentrated on a graph over x 1 ? When m = 2, the generalized Spence-Mirrlees, or twist condition yields uniqueness and purity: Injectivity of x 2 �→ D x 1 s ( x 1 , x 2 ) (Ex. s ( x 1 , x 2 ) = x 1 · x 2 .) Brenier ’87, Gangbo ’95, Caffarelli ’96, Gangbo-McCann ’96, Levin ’96: If µ 1 is absolutely continuous with respect to Lebesgue measure and s is twisted, the stable match γ is unique and pure. Brendan Pass (joint with Y.-H. Kim (UBC)) Uniqueness and purity in multi-agent matching problems

A condition for purity and uniqueness A set S ⊆ X 2 × X 3 ... × X m is an s -splitting set at a fixed x 1 ∈ X 1 if there exist functions u 2 ( x 2 ) , ..., u m ( x m ) such that � m i =2 u i ( x i ) ≥ s ( x 1 , ..., x m ) with equality on S . Brendan Pass (joint with Y.-H. Kim (UBC)) Uniqueness and purity in multi-agent matching problems

A condition for purity and uniqueness A set S ⊆ X 2 × X 3 ... × X m is an s -splitting set at a fixed x 1 ∈ X 1 if there exist functions u 2 ( x 2 ) , ..., u m ( x m ) such that � m i =2 u i ( x i ) ≥ s ( x 1 , ..., x m ) with equality on S . We say s is twisted on splitting sets if whenever S ⊆ X 2 × X 3 ... × X m is a splitting set at x 1 , ( x 2 , ..., x m ) �→ D x 1 s ( x 1 , x 2 , ..., x m ) is injective on S . Brendan Pass (joint with Y.-H. Kim (UBC)) Uniqueness and purity in multi-agent matching problems

A condition for purity and uniqueness A set S ⊆ X 2 × X 3 ... × X m is an s -splitting set at a fixed x 1 ∈ X 1 if there exist functions u 2 ( x 2 ) , ..., u m ( x m ) such that � m i =2 u i ( x i ) ≥ s ( x 1 , ..., x m ) with equality on S . We say s is twisted on splitting sets if whenever S ⊆ X 2 × X 3 ... × X m is a splitting set at x 1 , ( x 2 , ..., x m ) �→ D x 1 s ( x 1 , x 2 , ..., x m ) is injective on S . Kim-P (2013) : If µ 1 is absolutely continuous with respect to Lebesgue measure and s is twisted on splitting sets, the stable match γ is unique and pure. Brendan Pass (joint with Y.-H. Kim (UBC)) Uniqueness and purity in multi-agent matching problems

Example: One dimensional case m = 2: Recall classical (two marginal, one dimension) ∂ 2 s Spence-Mirrlees condition (supermodularity): ∂ x 1 ∂ x 2 > 0 – leads to positive assortative matching. ∂ 2 s ∂ x 1 ∂ x 2 < 0 (submodularity) is also twisted – leads to negative assortative matching. Brendan Pass (joint with Y.-H. Kim (UBC)) Uniqueness and purity in multi-agent matching problems

Example: One dimensional case m = 2: Recall classical (two marginal, one dimension) ∂ 2 s Spence-Mirrlees condition (supermodularity): ∂ x 1 ∂ x 2 > 0 – leads to positive assortative matching. ∂ 2 s ∂ x 1 ∂ x 2 < 0 (submodularity) is also twisted – leads to negative assortative matching. When x 1 , x 2 , ..., x m ∈ R , twist on splitting sets is essentially equivalent to: ∂ 2 s ∂ 2 s ∂ 2 s ] − 1 [ > 0 ∂ x i ∂ x j ∂ x k ∂ x j ∂ x k ∂ x i for all distinct i , j , k . Brendan Pass (joint with Y.-H. Kim (UBC)) Uniqueness and purity in multi-agent matching problems

Example: One dimensional case m = 2: Recall classical (two marginal, one dimension) ∂ 2 s Spence-Mirrlees condition (supermodularity): ∂ x 1 ∂ x 2 > 0 – leads to positive assortative matching. ∂ 2 s ∂ x 1 ∂ x 2 < 0 (submodularity) is also twisted – leads to negative assortative matching. When x 1 , x 2 , ..., x m ∈ R , twist on splitting sets is essentially equivalent to: ∂ 2 s ∂ 2 s ∂ 2 s ] − 1 [ > 0 ∂ x i ∂ x j ∂ x k ∂ x j ∂ x k ∂ x i for all distinct i , j , k . ∂ 2 s Satisfied for supermodular costs: ∂ x i ∂ x j > 0 for all i � = j . These surpluses were studied by Carlier (2003) – lead to positive assortative matching. Brendan Pass (joint with Y.-H. Kim (UBC)) Uniqueness and purity in multi-agent matching problems

Example: One dimensional case m = 2: Recall classical (two marginal, one dimension) ∂ 2 s Spence-Mirrlees condition (supermodularity): ∂ x 1 ∂ x 2 > 0 – leads to positive assortative matching. ∂ 2 s ∂ x 1 ∂ x 2 < 0 (submodularity) is also twisted – leads to negative assortative matching. When x 1 , x 2 , ..., x m ∈ R , twist on splitting sets is essentially equivalent to: ∂ 2 s ∂ 2 s ∂ 2 s ] − 1 [ > 0 ∂ x i ∂ x j ∂ x k ∂ x j ∂ x k ∂ x i for all distinct i , j , k . ∂ 2 s Satisfied for supermodular costs: ∂ x i ∂ x j > 0 for all i � = j . These surpluses were studied by Carlier (2003) – lead to positive assortative matching. ∂ 2 s Violated for submodular costs: ∂ x i ∂ x j < 0 for all i � = j . Brendan Pass (joint with Y.-H. Kim (UBC)) Uniqueness and purity in multi-agent matching problems

Example: Hedonic surplus � m s ( x 1 , x 2 , ..., x m ) = max y i =1 b i ( x i , y ) Brendan Pass (joint with Y.-H. Kim (UBC)) Uniqueness and purity in multi-agent matching problems

Example: Hedonic surplus � m s ( x 1 , x 2 , ..., x m ) = max y i =1 b i ( x i , y ) Motivation (Carlier-Ekeland (2010), Chiappori-McCann-Nesheim (2010)): agents of type x i have a surplus b i ( x i , y ) for a particular contract y - total joint utility s comes from maximizing the sum over all feasible contracts. Brendan Pass (joint with Y.-H. Kim (UBC)) Uniqueness and purity in multi-agent matching problems