Convergence of Ttonnement in Fisher Markets P R E S E N T E D B Y N - - PowerPoint PPT Presentation

Convergence of Tâtonnement in Fisher Markets

P R E S E N T E D B Y N O A A V I G D O R - E L G R A B L I ｜ A P R I L 9 , 2 0 1 4 B A S E D O N J O I N T W O R K W I T H Y U V A L R A B A N I

Overview

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• ▪ Fisher markets

› Definition › Market equilibrium › Eisenberg-Gale convex 


program ▪ Tâ tonnement ▪ Our results

Fisher Markets

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AGENTS GOODS (Utility func., budget) Prices =?

Fisher Markets -- Notation

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▪ Consider a marketplace with:

• n selfish agents
• k perfectly divisible goods

▪ Each agent i =1, 2, …, n has:

• Initial budget bi
• Utility function

ui :

• ▪ Fisher markets are a special case of the exchange market model of

Walras (1874) ∈ R+

Defines the “value” agent i gains from a basket of goods

Xi ⊆ Rk

+ → R+

Market Equilibrium

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▪ Given prices p for the goods, 
 the best basket of goods for player i satisfies the following: 
 
 
 
 
 ▪ Prices p are called market equilibrium iff there exists optimal xi(p) for every agent i such that
 X

i

xi(p) ≤ X

i

ei xi(p) = max ui(x) s.t : xi · p ≤ e · p xi ∈ Xi

Total demand

Agents maximize their utility within the budget constraint

bi

Total supply

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Arrow-Debreu Theorem (1954)

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The proof is not constructive

▪ In exchange markets with continuous, strictly monotone, and quasi- concave utility functions:
 a market equilibrium always exists.

• ▪ Moreover : every good with positive price is fully consumed
• ▪ In particular this is true for Fisher markets

Utility Functions

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▪ Linear utilities:
 
 
 ▪ Leontief utilities:
 
 
 ▪ Cobb-Douglas utilities:
 
 
 
 


u(x) =

m

X

i=1

aixi

u(x) = min

i {aixi}

Buy only goods with minimum . 
 Goods are perfect substitutes

pi/ai

Split the money on all goods (s.t >0) proportional to . Goods are perfect complements

ai pi/ai

u(x) = Πk

i=1xai i , s.t k

X

i

ai = 1

Spend ajb on good j

Utility Functions

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▪ CES (Constant Elasticity of substitution)

• ▪ 0 < 𝜍 ≦ 1 : weak gross substituts

▪ 𝜍 < 0 : complementary goods

u(x) = @X

j

(ajxj)ρ 1 A

1/ρ

Raising the price of one good does not decrease the demand for other goods Raising the price of one good does not increase the demand for other goods

▪ 𝜍 =1 → linear utilities

▪ 𝜍→ - → Leontief utilities ▪ 𝜍→ 0 → Cobb-Douglas utilities

∞

Eisenberg-Gale’s Convex Program [EG ’59, E ’61, KV ’02, 


CV ’05, JV ‘10]


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Primal:
 
 
 
 
 Dual:
 
 
 
 


▪ Eisenberg-Gale markets [JV ’10]:
 Fisher markets that their equilibrium allocation and prices can be computed using this form of a convex program.
 max Pn

i=1 bi ln ui(xi)

s.t. Pn

i=1 xij ≤ 1

∀j = 1, 2, . . . , m xij ≥ 0 ∀i = 1, 2, . . . , n; ∀j = 1, 2, . . . , m. min Pm

j=1 pj + Pn i=1 g∗ i (µi)

s.t. pj ≥ −µij i = 1, 2, . . . , n; ∀j = 1, 2, . . . , m pj ≥ 0 ∀j = 1, 2, . . . , m,

The KKT conditions show that an optimal solution must satisfy the equilibrium conditions

Summarize

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• ▪ Warlas (1874) - Formal definition of a market and a market equilibrium

▪ Arrow-Debreu (1954) - Existence of equilibrium in exchange markets 
 (and in Fisher markets in particular) ▪ Eisenberg-Gale (1959) - Constrictive proof of equilibrium.

Tâtonnement [Warlas 1874]

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▪ How do prices converge to equilibrium ? 
 The tâtonnement process: Prices are lowered for goods with excess supply. 
 Prices are raised for goods with excess demand. The agents always response by demanding an optimal basket at the current prices.

• ▪ The tâtonnement process we consider:

∀j = 1, 2, . . . , k :

pt+1

j

= pt

j(1 + ✏zj(pt))

• the excess demand
• f good j with prices p

zj(p)

This is considered to be a computational process and not a natural market process

Tâtonnement Convergence - Previous Results

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*WGS - weak gross substitutes

ρ ∈ (−1, 0)

[Arrow, Block, Hurwitz ’59]

time-continuous process for WGS utilities. [Scarf ’60] examples of non-WGS utilities for which 
 tâtonnement does not converge. poly-time convergence of a discrete tâtonnement-like process for WGS

[Fleisher, Garg, Kapoor, Khandekar ’08]

discrete tâtonnement-like process for a wide range of markets but with weak form of convergence. [Cole, Fleisher ‘08] discrete tâtonnement (similar to ours) for non-linear CES utilities that satisfy WGS.

[Cheung, Cole, Rastogi ’12]

analysis for some non-WGS utilities including complimentary CES utilities with , and multi level nested CES utilities (with some restrictions).

[Cheung, Cole, Devanaur ’13]

discrete tâtonnement converge fast for CES utilities with

• convergence
• f the average
• f the prices.
• satisfies only

the sum of budget constraint

ρ ∈ [− inf, 0) ρ ∈ (−1, 0) [Codenotti, McCune, Varadarjan ’05]

Approximate Equilibrium

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A price-demand pair (p,x) is δ-approximate equilibrium iff :

• for every agent i :
• For every good j :
• For every good j :

xi = xi(p)

zj(p) < δ

Each agent’s demand is optimal given p The demand for each good doesn’t exceed the supply by much Goods with small demand must have a small price

zj(p) < −δ ⇒ pj ≤ δ

Our Main Theorem

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β-smooth markets : Prices with dual cost close to optimal have small excess demand for every good

• Theorem:

For β-smooth markets (given δ>0, for small ε, and large T) if: C1. is monotonically non-increasing

• C2. For every T’> T- f(δ,ε):

then: the pair is δ-approximate equilibrium.

∀p φ(p) ≤ φ(p∗) + β(α) ⇒ ∀j |zj(p) − zj(p∗)| ≤ α

φ(p0), φ(p1), . . . , φ(pT )

1 T 0 + 1 ·

T 0

X

t=0

zj(pt) ≤ β(δ/3)

3) ∀j (pT , xT +1) Which markets are β-smooth and satisfy conditions C1, C2?

▪ Which markets are β smooth and satisfy conditions C1, C2?

• Applying the Theorem :
• Given such a market.
• Fix a sufficiently small constant ε
• For every δ>0, if


Then is δ-approximate

Eisenberg-Gale Markets Convergence

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Sufficient condition: ɸ(p) is twice continuously differentiable and strongly convex.

(pT , xT +1)

T ≥ Ω ✓− ln p0

min

✏23 ◆

Examples

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▪ Using the sufficient condition: ▪ non-linear CES utilities ▪ Exponential utilities:

• ▪ Using an extended argument:

▪ Leontief ▪ Nested CES-Leontief

u(x) =

k

X

j=1

aj(1 − e−θxj)

Interesting Questions

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▪ Find tight bounds on the convergence rate ▪ Analyze the convergence of an asynchronous process ▪ Extend this to some Arrow-Debreu markets

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Thank you!