When might markets be self- converging? A local, quickly convergent tatonnement algorithm Richard Cole (joint work with Lisa Fleischer)

When might markets be self-When might markets be self-converging? A local, quickly converging? A local, quickly

convergent tatonnement algorithm convergent tatonnement algorithm

Richard ColeRichard Cole

(joint work with Lisa Fleischer)(joint work with Lisa Fleischer)

... it seems likely that if the concept of the ... it seems likely that if the concept of the competitive equilibrium is to be useful in competitive equilibrium is to be useful in economic analysis, we need a mechanism for economic analysis, we need a mechanism for explaining how agents in the economy may act explaining how agents in the economy may act to generate competitive prices.to generate competitive prices.

Sean Crockett, Stephen Spear, Shyam SunderSean Crockett, Stephen Spear, Shyam Sunder

A simple decentralized institution for learning competitive A simple decentralized institution for learning competitive equilibriumequilibrium

November 2002November 2002

TatonnementTatonnement

Leon Walrus, 1834-1910

Prices adjust as follows: Excess supply: prices decrease Excess demand: prices increase (1874, Elements of Pure Economics)

Problem definitionProblem definition

Goods Goods gg11, , gg22, …, , …, ggnn

Prices Prices pp11, , pp22, …, , …, ppnn

Agents Agents aa11, , aa22, …, , …, aamm

Utilities Utilities uu11, , uu22, …, , …, uumm

uujj gives agent gives agent aajj’s preferences’s preferences

wwijij: initial allocation of : initial allocation of ggii to to aajj; ; wwii = ∑ = ∑jj wwijij = 1, by assumption = 1, by assumptionxxijij((pp,,wwjj,u,ujj)):: demand of demand of aajj for good for good ggii; ; xxii = = ∑∑jj xxijij, demand for , demand for ggii

Problem: Find prices Problem: Find prices pp such that such that xxii ≤ ≤ wwii for all for all ii..

A differential formulationA differential formulation

iii zt

p d

dSamuelson, 1947

zi = xi - 1, the excess demand.

Theorem: [Arrow, Block, Hurwitz, 1959; Nikaido, Uzawa, 1960] Assuming Gross Substitutes, the above differential equation converges to an equilibrium price

Discrete price updatesDiscrete price updates

Value of Value of λλii matters. matters.

Uzawa: there exist Uzawa: there exist λλii for which for which convergence occurs with the update rule:convergence occurs with the update rule: ppii ← ← ppii + + λλzzii

Large Large λλii can lead to cyclic or chaotic can lead to cyclic or chaotic behaviorbehavior

(e.g. (e.g. J.K. Goeree et alJ.K. Goeree et al. J. of Economic Behavior & Org., . J. of Economic Behavior & Org.,

1998)1998)

Conclusion: Avoid overreactingConclusion: Avoid overreacting

Our modelOur model

At each time stepAt each time step1.1. An arbitrary subset of the prices is updatedAn arbitrary subset of the prices is updated

2.2. The agents recalculate their demands The agents recalculate their demands thereby yielding a new total demand for thereby yielding a new total demand for each goodeach good

Complexity ModelComplexity Model

Rounds Rounds (from asynchronous distributed computing)(from asynchronous distributed computing)

– A round is the minimal time interval in which A round is the minimal time interval in which every price updates at least onceevery price updates at least once

bb-bounded asynchronicity-bounded asynchronicity– In each round any price updates at most In each round any price updates at most bb

timestimes

Round 1 Round 2

p1P1p2 p2 p2 p2

p1

Our goalsOur goals

A price update procedure, with one instance per A price update procedure, with one instance per price, satisfying:price, satisfying:

1.1. LocalLocal• The instances are independentThe instances are independent• Each instance uses limited "local" information Each instance uses limited "local" information

2.2. The procedure is simple The procedure is simple

3.3. It is asynchronous It is asynchronous

4.4. It yields quick convergence      It yields quick convergence     

Recent Work (in Computer Science)Recent Work (in Computer Science)

Many P-time algorithms for finding exact and Many P-time algorithms for finding exact and approximate equilibria for restricted markets and approximate equilibria for restricted markets and hardness results for other marketshardness results for other markets

X. Deng, C.H. Papdimitriou, M. Safra, On the complexity of equilibria, X. Deng, C.H. Papdimitriou, M. Safra, On the complexity of equilibria, STOC 02.STOC 02.

N.R. Devanur, C.H. Papadimitriou, A. Saberi, V. V. Vazirani, Market N.R. Devanur, C.H. Papadimitriou, A. Saberi, V. V. Vazirani, Market equilibrium via a primal-dual-type algorithm, FOCS 02.equilibrium via a primal-dual-type algorithm, FOCS 02.

N.R. Devanur, V. V. Vazirani, The spending constraint model for N.R. Devanur, V. V. Vazirani, The spending constraint model for market equilibrium: algorithm, existnece and uniqueness results, market equilibrium: algorithm, existnece and uniqueness results, STOC 04.STOC 04.

R. Garg and S. Kapoor, Auction algorithms for market equilibrium, R. Garg and S. Kapoor, Auction algorithms for market equilibrium, STOC 04.STOC 04.

B. Codenotti, S. Pemmaraju, K. Varadarajan, On the polynomial time B. Codenotti, S. Pemmaraju, K. Varadarajan, On the polynomial time computation of equilibria for certain exchange econmies, SODA 05.computation of equilibria for certain exchange econmies, SODA 05.

Recent tatonnement algorithmsRecent tatonnement algorithms

S. Crockett, S. Spear, S. Sundar, A simple decentralized S. Crockett, S. Spear, S. Sundar, A simple decentralized institution for learning competitive equilibrium, Technical institution for learning competitive equilibrium, Technical Report, 2002.Report, 2002.

M. Kitti, An iterative tatonnement process, Unpublished M. Kitti, An iterative tatonnement process, Unpublished manuscript, 2004.manuscript, 2004.

B. Codenotti, B. McCune, K. Varadarajan. Market B. Codenotti, B. McCune, K. Varadarajan. Market equilibrium via the excess demand function, STOC 05.equilibrium via the excess demand function, STOC 05.

They are non-local algorithms. Also, there is no complexity They are non-local algorithms. Also, there is no complexity analysis in the first two.analysis in the first two.

Reminder: Our GoalReminder: Our Goal

A local price update procedure with fast A local price update procedure with fast asynchronous convergence.asynchronous convergence.

To the best of our knowledge, no prior results of To the best of our knowledge, no prior results of this type.this type.

How to measure convergenceHow to measure convergence

Seek prices Seek prices pp such that: such that:

*

* ||max

i

iii p

pp

p* denotes the equilibrium solution

Usual approachUsual approach

uj(Actual) ≥ uj(Opt)/(1 + ε) for all j

Difficulty: in our setting, consider only cummulative demand; no allocation of supply shortfall or excess.

Further, no direct knowledge of utility functions.

Further standard definitionsFurther standard definitions

Homogeneity (of degree 0): demand at Homogeneity (of degree 0): demand at prices prices pp and and λλpp, , λλ ≠≠ 0, is the same for 0, is the same for all goods.all goods.

Normalize: Choose one good, the Normalize: Choose one good, the numerairenumeraire, to have price 1. We call , to have price 1. We call this good this good moneymoney. Write . Write gg$$, , pp$$..

Standard utility functionsStandard utility functions

Cobb-DouglasCobb-DouglasEach agent Each agent aaii spends preset fractions of wealth spends preset fractions of wealth

on each good: fraction on each good: fraction ααijij on good on good jj..

Yielded by utility Yielded by utility ππjj ( (xxijij))ααijij

or by ∑or by ∑jj ααijij log log xxijij

Standard utility functionsStandard utility functions

CES (Constant Elasticity of Substitution)CES (Constant Elasticity of Substitution)For each agent the fractional rate of change of For each agent the fractional rate of change of

demand with respect to the fractional rate of demand with respect to the fractional rate of change in the price is constant, i.e.change in the price is constant, i.e.

Yielded by utility function [Yielded by utility function [∑∑jj ( (bbijijxxijij))ρρ]]1/1/ρρ

Only consider Only consider ρρ s.t. 0 < s.t. 0 < ρρ < 1. < 1.

j

ikijk

j

ik

p

xc

p

x

Standard utility functionsStandard utility functions

Weak Gross SubstitutesWeak Gross SubstitutesIf price If price ppii increases then increases then

xxjljl stays the same or increases for stays the same or increases for jj ≠ ≠ ii, ,

xxilil stays the same or decreases, stays the same or decreases,

and something changes.and something changes.

Could also view as a market conditionCould also view as a market conditionxxjj stays the same or increases for stays the same or increases for jj ≠ ≠ ii, ,

xxii stays the same or decreases, stays the same or decreases,

and something changes.and something changes.

Our price update ruleOur price update rule

pp′′ii ← ← ppii(1 + (1 + λλ min{1, min{1, zzii})})

zzii = = xxii – 1, the excess demand – 1, the excess demand

By contrast, Uzawa used the ruleBy contrast, Uzawa used the rule

pp′′ii ← ← ppii + + λλ z zii

Our resultsOur results

Cobb-Douglas Utilities:Cobb-Douglas Utilities:

Take Take λλ = 1. Initial prices = 1. Initial prices ppII, final prices , final prices ppFF. . Converge in:Converge in:

rounds. ||

||maxlogmaxlog

1O

*

**

k

Fk

kIk

kIj

jj pp

pp

p

p

Our ResultsOur Results

CES Utilities:CES Utilities:Let Let ρρ = max = maxii ρρii

Take Take λλ ≤ 4(1- ≤ 4(1-ρρ). Initial prices ). Initial prices ppII, final prices, final prices p pFF. . Converge in:Converge in:

rounds. ||

||maxlogmaxlog

1O

*

**

k

Fk

kIk

kIj

jj pp

pp

p

p

Our resultsOur results

GS with the wealth effect (parameter GS with the wealth effect (parameter ββ ≤≤ 1) and 1) and bounded (bounded (ss ≥ 1) elasticity of demand. ≥ 1) elasticity of demand.

For CES, For CES, ββ = 1, = 1, ss = 1/(1 - = 1/(1 - ρρ); for Cobb-Douglas, ); for Cobb-Douglas, ββ = 1, = 1, ss = = 1.1.

Take Take λλ ≤ 1/(4 ≤ 1/(4ss); Initial prices ); Initial prices ppII, final prices , final prices ppFF. .

Converge in:Converge in:

rounds. ||

||maxlogmaxlog

1O

*

**

k

Fk

kIk

kIj

jj pp

pp

p

p

Bounded Elasticity of DemandBounded Elasticity of Demand

ss

px

px

i

i

i

i

somefor

Halve the price: demands increase by at most 2s.

Wealth EffectWealth Effect

There is a parameter There is a parameter ββ, 0 < , 0 < ββ ≤ 1, with the ≤ 1, with the following property:following property:Suppose the wealth of Suppose the wealth of bbjj increases from increases from ww to to ww′, with ′, with

no change in prices. Let no change in prices. Let xxijij, , x′x′ijij be the be the

corresponding demands. Then:corresponding demands. Then:

w

w

x

x

ij

ij ''

Double the wealth: demand for each good increasesby at least 2β.

What if What if λλ is not known? is not known?

Have a second algorithm, in which each price Have a second algorithm, in which each price setter gradually reduces its setter gradually reduces its λλ..Assuming Assuming bb-bounded asynchronicity, converge -bounded asynchronicity, converge

in:in:

rounds. ||

||maxlogmaxlog

1 O

2

*

**

k

Fk

kIk

kIj

jj pp

pp

p

psb

What is What is αα??

Running example: Cobb-Douglas utilities, 2 Running example: Cobb-Douglas utilities, 2 goods plus money.goods plus money.

p2

p1

z1 = 0

z2 = 0

X P

p2

p1

z1 = 0

z2 = 0

X

“correct” adjustment for p1

What if the cone is narrow?What if the cone is narrow?

p2

p1

It takes lots of steps to halve p1 – p1*, p2 – p2*

What determines the cone’s width?What determines the cone’s width?

The convergence parameter The convergence parameter αα; what is it?; what is it?

Roughly: minimum (over all goods) of fraction Roughly: minimum (over all goods) of fraction of spending on that good provided by money of spending on that good provided by money at equilibriumat equilibrium

Formally:Formally:

Claim: The parameter Claim: The parameter αα in the complexity is tight. in the complexity is tight.

il k lkl

lili pw

wx min ;$*

Elements of the analysisElements of the analysis

WLOG let 1 = argminWLOG let 1 = argminii ppii//ppii*. Then:*. Then:

1.1. If If pp11 ≥ ≥ pp11*, then *, then p′p′ii ≥ ≥ ppii* for all i.* for all i.

2.2. If If pp11 < < pp11*, then *, then

*11

**

'

1

1

1

1

p

p

p

p

p

p

i

i

Or δ′ ≤ δ[1 – α(1 – δ)] where δ = (p1* - p1)/p1*

And analogously for maxmaxjj p pjj//ppjj*.*.

Proof outline for Proof outline for pp′′11

))(1(

)})(,1min{1(

)})1(,1min{1('

)1(

]1[(

*1

1*1

1

*1

1*1

1

1

*1

11

1

*1

1

1

*1*

11

*11

p

ppp

p

ppp

p

ppp

p

pz

p

pxx

pp

Proof outline for Proof outline for pp′′11

*1

*1

*1

*11

1

*1

11

1

*1

1

1

*1*

11

*11

)1(

)1(

))1(1('

)1(

]1[(

ppp

pp

p

ppp

p

pz

p

pxx

pp

Reason for the wealth effectReason for the wealth effect

At prices At prices ppii*/*/ff, for , for ff > 1, > 1, ii ≠ $, want all ≠ $, want all zzii to be to be substantial and certainly not negative.substantial and certainly not negative.

Wealth effect yields: Wealth effect yields: xxii ≥ ≥ xxii*(1 + *(1 + θθ((αβαβ)) = 1 + )) = 1 + θθ((αβαβ))

Otherwise, one low price, Otherwise, one low price, pp11*/2 say, may induce a */2 say, may induce a very low price, very low price, ppnn*/*/ccnn for some constant for some constant cc > 1. > 1.

An analogous claim holds at prices An analogous claim holds at prices f pf pii*, for *, for f f > 1.> 1.

λλ unknown unknown

Keep separate Keep separate λλii for each price updater. for each price updater.

Initially, Initially, λλii = 1. After 4 = 1. After 4hh updates to updates to ppii, , λλii set set to 1/2to 1/2hh..

Once all Once all λλii ≤ ≤ λλ, then prices improve. Analysis , then prices improve. Analysis largely as before except have to undo largely as before except have to undo damage done by damage done by λλii when too large. when too large.

Analysis, Analysis, λλ unknown unknown

Mega-round: maximal collection of Mega-round: maximal collection of rounds in which smallest rounds in which smallest λλii unchangedunchanged

Analysis idea: Analysis idea: 1.1. Progress doubles in successive mega-Progress doubles in successive mega-

roundsrounds

2.2. Successive mega-rounds comprise 4 times Successive mega-rounds comprise 4 times as many roundsas many rounds

Result, Result, λλ unknown unknown

rounds. ||

||maxlogmaxlog

1 O

2

*

**

k

Fk

kIk

kIj

jj pp

pp

p

psb

Converge in:

ConclusionsConclusions

Theorem: There is a simple, local, asynchronous, quickly Theorem: There is a simple, local, asynchronous, quickly convergent price setting algorithm, assuming a bound on convergent price setting algorithm, assuming a bound on the elasticity of demand is known to each price setter. the elasticity of demand is known to each price setter. This algorithm works in GS markets with a wealth effect.This algorithm works in GS markets with a wealth effect.

Observation: The rate of convergence depends on the Observation: The rate of convergence depends on the desire for money at equilibrium.desire for money at equilibrium.

Parameters: Parameters: αα and and ss are tight for this algorithm. are tight for this algorithm.Open Question: Is Open Question: Is ββ tight? tight?Open Question: Are these parameters tight in any local, Open Question: Are these parameters tight in any local,

asynchronous algorithm?asynchronous algorithm?

ConclusionsConclusions

Theorem: Even if the bound Theorem: Even if the bound ss is not known, is not known, there is a fairly quickly convergent there is a fairly quickly convergent algorithm.algorithm.

It is not as natural (would really want more It is not as natural (would really want more adaptive control of adaptive control of λλ).).

Open Question: Devise natural Open Question: Devise natural conditions allowing adaptive control of conditions allowing adaptive control of λλ..

Ongoing and Future WorkOngoing and Future Work

• Algorithms for indivisible markets (Cole Algorithms for indivisible markets (Cole and Rastogi)and Rastogi)• Also, how well can one do?Also, how well can one do?

• More realistic dynamic settingMore realistic dynamic setting