Computing Walrasian Equilibrium Renato Paes Leme - PowerPoint PPT Presentation

Computing Walrasian Equilibrium Renato Paes Leme Sam Wong (Google) (Berkeley)

supplies: demand: flour, bakeries, milk, hospitals, vegetables, households, medicine, schools, paper, … …

supplies: demand: flour, Task: Allocate supplies bakeries, milk, e ffi ciently to hospitals, vegetables, satisfy the demands households, medicine, of the city. schools, paper, … …

supplies: demand: flour, Task: Allocate supplies bakeries, milk, e ffi ciently to hospitals, vegetables, satisfy the demands households, medicine, of the city. schools, paper, … … Invisible Hand of the market

Theory of Market Equilibrium • Adam Smith: “Wealth of the Nations” (1776): invisible hand 
 • Leon Walras: “Elements of Pure Economics” (1874): mathematical 
 theory of market equilibrium 
 • Arrow-Debreu (1950’s): general equilibrium theory 
 • Kelso-Crawford (1982): discrete and combinatorial theory of market equilibr.

Market equilibrium n goods m buyers

Market equilibrium n goods m buyers v 3 v 2 v 1 v 4 v i : 2 N → R • Valuations

Market equilibrium p 3 p 5 p 1 p 2 p 4 p 6 n goods m buyers v 3 v 2 v 1 v 4 v i : 2 N → R • Valuations

Market equilibrium p 3 p 5 p 1 p 2 p 4 p 6 n goods m buyers v 3 v 2 v 1 v 4 v i : 2 N → R • Valuations • Demands D ( v i , p ) = argmax S ⊆ N [ v i ( S ) − P i ∈ S p i ]

Market equilibrium p 3 p 5 p 1 p 2 p 4 p 6 n goods S 1 ∈ D ( v 1 , p ) m buyers v 3 v 2 v 1 v 4 v i : 2 N → R • Valuations • Demands D ( v i , p ) = argmax S ⊆ N [ v i ( S ) − P i ∈ S p i ]

Market equilibrium p 3 p 5 p 1 p 2 p 4 p 6 n goods S 1 ∈ D ( v 1 , p ) S 2 ∈ D ( v 2 , p ) m buyers v 3 v 2 v 1 v 4 v i : 2 N → R • Valuations • Demands D ( v i , p ) = argmax S ⊆ N [ v i ( S ) − P i ∈ S p i ]

Market equilibrium p 3 p 5 p 1 p 2 p 4 p 6 n goods S 1 ∈ D ( v 1 , p ) S 2 ∈ D ( v 2 , p ) ∅ ∈ D ( v 3 , p ) m buyers v 3 v 2 v 1 v 4 v i : 2 N → R • Valuations • Demands D ( v i , p ) = argmax S ⊆ N [ v i ( S ) − P i ∈ S p i ]

Market equilibrium p 3 p 5 p 1 p 2 p 4 p 6 n goods S 1 ∈ D ( v 1 , p ) S 2 ∈ D ( v 2 , p ) S 4 ∈ D ( v 4 , p ) ∅ ∈ D ( v 3 , p ) m buyers v 3 v 2 v 1 v 4 v i : 2 N → R • Valuations • Demands D ( v i , p ) = argmax S ⊆ N [ v i ( S ) − P i ∈ S p i ]

Market equilibrium • Market equilibrium: prices s.t. 
 S i ∈ D ( v i , p ) p ∈ R n i.e. each good is demanded by exactly one buyer. First Welfare Theorem : in equilibrium the welfare 
 P i v i ( S i ) is maximized. (proof: LP duality) How do markets converge to equilibrium prices ? How to compute a Walrasian equilibrium ?

How to access the input Microscopic Macroscopic Telescopic

How to access the input Microscopic Macroscopic Telescopic Value oracle: given i and S: 
 v i ( S ) query .

How to access the input Microscopic Macroscopic Telescopic Value oracle: Demand oracle: given i and S: 
 given i and p: 
 v i ( S ) query . query . S ∈ D ( v i , p )

How to access the input Microscopic Macroscopic Telescopic Value oracle: Demand oracle: Aggregate Demand: given p, query. given i and S: 
 given i and p: 
 v i ( S ) query . query . S ∈ D ( v i , p ) P i S i ; S i ∈ D ( v i , p )

Algorithms for computing equilibria (general case) Algorithm Oracle Access Running time tatonnement (trial-and-error) [Walras, Kelso-Crawford, …]

Walrasian tatonnement p 3 p 5 p 1 p 2 p 4 p 6 n goods m buyers v 3 v 2 v 1 v 4

Walrasian tatonnement p 3 p 5 p 1 p 2 p 4 p 6 n goods m buyers v 3 v 2 v 1 v 4

Walrasian tatonnement p 3 p 5 p 1 p 2 p 4 p 6 n goods m buyers v 3 v 2 v 1 v 4

Walrasian tatonnement p 3 p 5 p 1 p 2 p 4 p 6 n goods m buyers v 3 v 2 v 1 v 4

Walrasian tatonnement p 3 p 5 p 1 p 2 p 4 p 6 n goods m buyers v 3 v 2 v 1 v 4

Walrasian tatonnement +1 +1 − 1 p 3 p 5 p 1 p 2 p 4 p 6 n goods m buyers v 3 v 2 v 1 v 4

Walrasian tatonnement +1 +1 − 1 p 3 p 5 p 1 p 2 p 4 p 6 n goods m buyers v 3 v 2 v 1 v 4

Walrasian tatonnement +1 +1 − 1 p 3 p 5 p 1 p 2 p 4 p 6 n goods m buyers v 3 v 2 v 1 v 4

Walrasian tatonnement +1 +1 − 1 p 3 p 5 p 1 p 2 p 4 p 6 n goods m buyers v 3 v 2 v 1 v 4

Walrasian tatonnement +1 +1 − 1 p 3 p 5 p 1 p 2 p 4 p 6 n goods m buyers v 3 v 2 v 1 v 4

Walrasian tatonnement +1 +1 − 1 p 3 p 5 p 1 p 2 p 4 p 6 n goods m buyers v 3 v 2 v 1 v 4

Walrasian tatonnement +1 +1 − 1 p 3 p 5 p 1 p 2 p 4 p 6 n goods m buyers v 3 v 2 v 1 v 4

Walrasian tatonnement +1 +1 − 1 +1 p 3 p 5 p 1 p 2 p 4 p 6 n goods m buyers v 3 v 2 v 1 v 4

Walrasian tatonnement +1 +1 − 1 +1 p 3 p 5 p 1 p 2 p 4 p 6 n goods m buyers v 3 v 2 v 1 v 4

Walrasian tatonnement +1 +1 − 1 +1 p 3 p 5 p 1 p 2 p 4 p 6 n goods m buyers v 3 v 2 v 1 v 4

Gradient Descent Interpretation • [Kelso-Crawford] analyzes it and shows convergence under a condition called gross substitutes. • pseudo poly algorithm


 
 
 Gradient Descent Interpretation • [Kelso-Crawford] analyzes it and shows convergence under a condition called gross substitutes. • pseudo poly algorithm • [Ausubel] defined the potential: 
 f ( p ) = P i max S [ v i ( S ) − p ( S )] + p ([ n ]) such that gradient descent is exactly tatonnement: 
 ∂ j f ( p ) = 1 − [total demand for j ] • If equilibrium exists then equil prices = argmin f ( p )

Algorithms for computing equilibria (general case) Algorithm Oracle Access Running time tatonnement / gradient descent pseudo poly aggregate demand [Walras, Kelso-Crawford, …]

Algorithms for computing equilibria (general case) Algorithm Oracle Access Running time tatonnement / gradient descent pseudo poly aggregate demand [Walras, Kelso-Crawford, …] Linear programming demand + value 
 poly time [Nisan-Segal] oracle

Algorithms for computing equilibria (general case) Algorithm Oracle Access Running time tatonnement / gradient descent pseudo poly aggregate demand [Walras, Kelso-Crawford, …] Linear programming demand + value 
 poly time [Nisan-Segal] oracle poly time aggregate demand this paper ˜ O ( n 2 · T AD + n 5 )

From LP to convex optimization • Nisan and Segal LP: min P i u i + p ([ n ]) u i ≥ v i ( S ) − p ( S ) , ∀ i, S

From LP to convex optimization • Nisan and Segal LP: • demand oracle finds separating constraint min P i u i + p ([ n ]) • value oracle to add the u i ≥ v i ( S ) − p ( S ) , ∀ i, S hyperplane

From LP to convex optimization • Nisan and Segal LP: • demand oracle finds separating constraint min P i u i + p ([ n ]) • value oracle to add the u i ≥ v i ( S ) − p ( S ) , ∀ i, S hyperplane • Idea: using cutting plane method to minimize f ( p ) = P i [max S v i ( S ) − p ( S )] + p ([ n ])

From LP to convex optimization • Nisan and Segal LP: • demand oracle finds separating constraint min P i u i + p ([ n ]) • value oracle to add the u i ≥ v i ( S ) − p ( S ) , ∀ i, S hyperplane • Idea: using cutting plane method to minimize f ( p ) = P i [max S v i ( S ) − p ( S )] + p ([ n ]) • Two issues with black box application: f ( p ) , ∂ f ( p ) • Evaluate f : ellipsoid and cutting plane need • Approximation : give only approximate solutions

From LP to convex optimization • Optimizing only using the gradient 
 We adapt the cutting plane algorithm of 
 Lee-Sidford-Wong’15 to optimize f using only 
 ∂ f ( p ) • Obtaining exact solutions • Exact solution is only known for LPs [Khachiyan] • idea: explore the connection of this program and LP • But we have restricted access to constraints 
 (only via aggregate demand oracle) • Only a restricted perturbation is enough.

Gross substitutes case

Gross substitutes case necessary and “su ffi cient” condition for tatonnement to converge gross substitutes “increase in the price for one [Kelso-Crawford] good doesn’t decrease demand for other good.”

Gross substitutes case necessary and “su ffi cient” condition for tatonnement to converge gross substitutes valuated matroids “increase in the price for one [Kelso-Crawford] [Dress-Wenzel] good doesn’t decrease demand for other good.” generalization of Grassman-Plucker relations, when can be v ( S ) − P S p j optimized using Greedy algo

Gross substitutes case necessary and “su ffi cient” condition for tatonnement to converge gross substitutes valuated matroids “increase in the price for one [Kelso-Crawford] [Dress-Wenzel] good doesn’t decrease demand for other good.” generalization of Grassman-Plucker relations, when can be v ( S ) − P S p j optimized using Greedy algo (if those v ( S ) ∈ { 0 , −∞ } are matroids).