Multi-marginal optimal transportation and hedonic pricing Brendan - PowerPoint PPT Presentation

Multi-marginal optimal transportation and hedonic pricing Brendan Pass University of Alberta June 4, 2012 Brendan Pass Multi-marginal optimal transportation and hedonic pricing

Introduction Probability measures µ i on X i ⊆ R n , i = 1 , 2 , ... m . distribution of types i . Space Z ⊆ R n of contracts . Utility functions f i : X i × Z → R . f i ( x i , z ) = preference of type x i ∈ X i for contract z ∈ Z . Brendan Pass Multi-marginal optimal transportation and hedonic pricing

Equilibrium: two formulations Carlier-Ekeland (2010): Look for a measure ν on Z and couplings π i of µ i and ν maximizing: m � � f i ( x i , z ) d π i X i × Z i =1 Proved: 1 Uniqueness when µ 1 << dx 1 and z �→ D x 1 f 1 ( x 1 , z ) is injective. 2 Purity when µ i << dx i and z �→ D x i f i ( x i , z ) is injective. Question: What does ν look like? Brendan Pass Multi-marginal optimal transportation and hedonic pricing

Equilibrium: two formulations Carlier-Ekeland (2010): Look for a measure ν on Z and couplings π i of µ i and ν maximizing: m � � f i ( x i , z ) d π i X i × Z i =1 Proved: 1 Uniqueness when µ 1 << dx 1 and z �→ D x 1 f 1 ( x 1 , z ) is injective. 2 Purity when µ i << dx i and z �→ D x i f i ( x i , z ) is injective. Question: What does ν look like? Chiappori-McCann-Nesheim (2010): Set � m b ( x 1 , x 2 , ..., x m ) = max z i =1 f i ( x i , z ) and look for a coupling γ of µ 1 , µ 2 , ..., µ m maximizing: � b ( x 1 , x 2 , ..., x m ) d γ, X 2 × X 2 × ... × X m A multi-marginal optimal transportation problem. Questions: Uniqueness? Structure? Purity? Brendan Pass Multi-marginal optimal transportation and hedonic pricing

A theorem Theorem (P 2011) Assume 1 For all i, f i is C 2 and the matrix D 2 x i z f i of mixed, second order partial derivatives is everywhere non-singular. 2 For each ( x 1 , x 2 , ..., x m ) the maximum is attained by a unique z ( x 1 , x 2 , ..., x m ) ∈ Z. 3 � m i =1 D 2 zz f i ( x i , z ( x 1 , x 2 , ..., x m )) is non-singular. ⇒ spt( γ ) is contained in an n-dimensional, Lipschitz submanifold S ⊆ X 1 × X 2 × ... × X m ⊆ R nm . Special case of a more general result. Hedonic pricing surpluses b work very nicely here. Brendan Pass Multi-marginal optimal transportation and hedonic pricing

Geometry of spt( γ ) Set D 2 D 2 D 2  0 x 1 x 2 b x 1 x 3 b x 1 x m b  ... D 2 D 2 D 2 x 2 x 1 b 0 x 2 x 3 b ... x 2 x m b    D 2 D 2 D 2  M = x 3 x 1 b x 3 x 2 b 0 x 3 x m b ...     ... ... ... ... ...   D 2 D 2 D 2 x m x 1 b x m x 2 b x m x 3 b 0 ... A symmetric, ( nm ) × ( nm ) matrix. Signature: n -positive eigenvalues. m ( n − 1)-negative eigenvalues. spt( γ ) is spacelike : � x T M � x ≥ 0 for all tangent vectors � x ∈ X 1 × X 2 × ... × X m to spt( γ ). Brendan Pass Multi-marginal optimal transportation and hedonic pricing

Work in Progress Understanding the geometry of spt( γ ) lets us prove things about ν : Theorem (P 2012) ν is absolutely continuous with respect to Lebesgue measure. Brendan Pass Multi-marginal optimal transportation and hedonic pricing

More work in progress This, in turn, lets us prove more things about γ : Theorem (P 2012) Assume 1 µ 1 << dx 1 . 2 z �→ D x 1 f 1 ( x 1 , z ) is injective. 3 x i �→ D z f i ( x i , z ) is injective. ⇒ γ is unique and pure. Brendan Pass Multi-marginal optimal transportation and hedonic pricing