Convergence of T\^atonnement in Fisher Markets

0.24 MB·

by  Noa Avigdor-Elgrabli

Convergence of Taˆtonnement in Fisher Markets
Noa Avigdor-Elgrabli§ Yuval Rabani¶ Gala Yadgar}
April 21, 2015
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0 Abstract
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r Analyzing simple and natural price-adjustment processes that converge to amarket equilibrium is
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a fundamental question in economics. Such an analysis may have implications in economic theory,
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computational economics, and distributed systems. Taˆtonnement, proposed by Walras in 1874, is a

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process by which prices go up in response to excess demand, and down in response to excess supply.
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This paper analyzes the convergence of a time-discrete taˆtonnement process, a problem that recently

] attracted considerable attention of computer scientists. Weprove that the simple taˆtonnement process
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that we consider converges (efficiently) to equilibrium prices and allocation in markets with nested
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CES-Leontiefutilities, generalizing someofthepreviousconvergence proofsformorerestrictedtypes
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s ofutilityfunctions.
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§Yahoo!LabsHaifa,MATAM,Haifa31095,Israel. Email:[email protected]
¶The Rachel and Selim Benin School of Computer Science and Engineering, The Hebrew University of Jerusalem,
Jerusalem 91904, Israel. Email: [email protected]. Research supported by Israel Science Foundation grant
number856-11and by the Israeli Center of Excellence on Algorithms. Part of this work was done while visiting Microsoft
Research.
}Computer Science Department, Technion—Israel Institute of Technology, Haifa 32000, Israel. Email:
[email protected]. This work was supported in part by the Israel Ministry of Industry, Trade, and Labor un-
dertheNet-HDConsortium,andbyMicrosoft.
1 Introduction
Motivation. Generalequilibriumtheory,acornerstoneofmicroeconomics,dealswithmarketsthatcon-
sist of agents that are endowed with goods (or money). Each agent wants to maximize its utility. This
mayinvolveexchangingtheinitialendowmentofgoodsforgoodsthatotheragentshold. Theexchangeof
goodsagainstothergoods(ormoney)isgovernedbyprices thatare settoguaranteeamarketequilibrium
where supply equals demand. In this context,an issuethat was raised already by thefounders of this the-
ory is that justifying market equilibrium as a likely outcome of exchange requires suggesting a plausible
marketdynamicthatleadstoequilibrium. Inotherwords,provingthatsomesimple,natural,decentralized
price-adjustmentprocessconvergesquicklytoamarketequilibriumisafundamentalchallengeofgeneral
equilibrium theory. Indeed, Walras [22] proposed such a process which he named taˆtonnement. In this
process, prices are adjusted upwards in response to excess demand and downwards in response to excess
supply. (Notice that while buyers are expected to react to price changes by maximizing their utility sub-
ject to their budget constraints, the price changes themselves cannot be justified based on the incentives
of the sellers.) In modern economic thought, taˆtonnement is regarded as a model for a centrally planned
economy rather than a decentralized market economy. In other words, it is considered a method of com-
putation rather than a spontaneous market process. A scenario where this view of taˆtonnement might be
useful arises in thecontextofdistributedlarge-scale computersystems,suchas clouds. Artificial markets
can be set to facilitiate cooperative sharing of computing, storage, and communication resources among
selfish agents (see, for example, [6, 25, 23]). Often, a service provider either controls the dispersed re-
sources, orat leastregulatesthepeer-to-peer interactionamongtheusers,thusenforcing adherence tothe
priceadjustmentprotocol(see, for instance,[1, 24]). On theotherhand, thebuyingagents are stillfreeto
pursuetheirgoalswithouthavingtoformulateanddisclosetheirutilityfunctionsbeyondtheirreactionsto
the changing prices. (One of thereasons that this is importantis that the conventionaleconomicinterpre-
tation of utility functions is that they are a convenient expression of a preference order on the continuum
ofconsumptionbaskets,ratherthana meaningfuland consciousnumericalevaluationofeach basket.)
Our results. We consider in this paper the following simple discrete taˆtonnement rule. For some fixed
ǫ ą 0,foreverygood, itspricepptq attimet “ 1,2,3,... is givenby
pptq “ p1´ǫq¨ppt´1q`ǫ¨fptq¨ppt´1q, (1)
wherefptqisthetotaldemandforthisgoodassumingthepricesattimet´1(thesupplyisalwaysscaled
to1). WeprovethatthisprocessconvergestoequilibriumpricesandallocationinvariousEisenberg-Gale
markets, most notably markets with nested CES-Leontief utility functions. We also note that the rate of
convergence gives a polynomial time algorithm for computing an approximate equilibrium—the precise
boundsarestatedinSections4,5,and 6. Mostpreviousresultsontheconvergenceofdiscreteversionsof
taˆtonnementapplytospecialcases ofourresults. Ourresultsapplytoutilityfunctionsforwhichprevious
work did not establish convergence of taˆtonnement. We note that resource sharing in computer systems
mayrequirerathercomplicatedutilityfunctionsthatcouldnotbehandledbypreviouswork(forinstance,
users maydesireacombinationofalternativebundlesofresources).
Markets. AFishermarket(see[20])iscomposedofasetofagentsandasetofperfectlydivisiblegoods,
eachoflimitedquantity. Eachagentisendowedwithabudgetandautilityfunctiononthepossiblebaskets
ofgoods. Thegoalofanagentistospendthebudgettobuyan optimalbasketofgoodswhichmaximizes
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the agent’s utility subject to the budget constraint. An equilibrium consists of an assignment of prices
to goods and an optimal purchase of goods by each agent in which the market clears, i.e., the demand
for each good equals to its supply. Eisenberg-Gale markets [15] are a rather general special case, where
the utility functions are such that equilibrium prices and allocation can be formulated as the solutions
to dual convex programs, originally due to Eisenberg and Gale [13] and extended in [12, 18, 10, 16].
Fisher markets are a special case of the more general Walrasian model [22] of an exchange economy.
Arrow and Debreu [4] proved, using the Kakutani fixed-point theorem, that in a Walrasian market, if the
utilityfunctions are continuous, strictly monotone, and quasi-concave, then an equilibrium always exists.
However,theirproofgivesnoindicationofthemarket dynamicsthatmightleadto an equilibrium.
Marketsareoftenclassifiedaccordingtothetypeofutilityfunctionsused. Utilityfunctionsareusually
assumed to be concave and monotonically non-decreasing. (Economically, the former assumption is the
lawofdiminishingmarginalutility,andthelatterisimpliedbyassumingfreedisposal.) Autilityfunction
oftheform
1{ρ
m
upxq “ aρxρ ,
j j
˜j“1 ¸
ÿ
where ρ P p´8,0q Y p0,1s, is called a utility with constant elasticity of substitution or a CES utility.
CES utilities with ρ P p0,1s are a special case of utilities that satisfy the weak gross substitutes (WGS)
property—increasing the price of a good does not decrease the demand for any other good. If ρ “ 1, this
isalinearutility. InCESutilitieswithρ ă 0,thegoodsarecomplementary. Thelimitofupxqasρ Ñ ´8
is called a Leontief utility, and the limit as ρ Ñ 0 is called a Cobb-Douglas utility. A utility function of
theform
k m ρ1{ρ2 1{ρ1
upxq “ cρ1 aρ2 xρ2
¨ j1 j1,j2 j1,j2 ˛
jÿ1“1 ˜jÿ2“1 ¸
˝ ‚
is a (two-level) nested CES utility (see [17]). A resource allocation utility (see [18]) is a nested linear-
Leontief utility (ρ1 “ 1 and ρ2 “ ´8). An exponential utility (a.k.a. constant absolute risk aversion
utility) is a simple example of a utility that does not exhibit constant elasticity of substitution. It has the
form
m
upxq “ a 1´e´θxj .
j
j“1
ÿ ` ˘
Taˆtonnement. The idea of taˆtonnement is due to Walras [22]. Arrow et al. [3] showed that a time-
continuous version of taˆtonnement converges to equilibrium for WGS utilities. Scarf [19] gave examples
of Arrow-Debreu markets with non-WGS utilities (in particular Leontief utilities) where specific imple-
mentationsoftaˆtonnementcycleand neverconvergetoan equilibrium.
ThemorerecentcomputerscienceliteratureconsidersdiscreteversionsoftaˆtonnementinFishermar-
kets. (Discrete taˆtonnement was also discussed in the economics literature, see [21].) Codenotti et al. [9]
show that a taˆtonnement-like process involving some price coordination converges in polynomial time
for WGS utilities. Another taˆtonnement-like process that involves coordination is analyzed in Fleischer
et al. [14]. They show, for many interesting markets, a weak form of convergence of the process. They
consider the average of the prices and allocations over the steps of the process and show that they con-
verge to a weak notion of approximate equilibrium, which satisfies only the sum of budget constraints of
the agents. On the other hand, their results apply to a wide range of markets. In particular, they apply
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to all the markets where we demonstrate explicit bounds on convergence to our much stronger notion of
approximateequilibrium.
Thefirstresulttodemonstratetheconvergencetoequilibriumofatruediscretetaˆtonnementprocessis
due to Cole and Fleischer [11], who showed that the prices converge to equilibrium prices for non-linear
CES utilities that satisfy WGS (i.e., ρ P p0,1q) and for Cobb-Douglas utilities. They analyze the same
price-adjustmentprocessgiveninEquation(1). Theiranalysisrelies on astrongproperty thattheyprove:
inthecasestheyanalyzethepriceofeachgoodmovestowardstheequilibriumvalueateachiteration. This
is generally false for non-WGS utilities. Nevertheless, Cheung et al. [8] modified the analysis to apply to
some non-WGS utilities, including complementary CES utilities with ρ P p´1,0q, two-level nested CES
utilitieswithρ1,ρ2 ą ´1,andevenmulti-levelnestedCESutilitieswithsomerestrictionsontheelasticity
(which do not breach the lower bound of ´1). It should be noted that the analysis of [11, 8] applies also
to a model of asynchronous price adjustments. Asynchrony is clearly a desirable property of a plausible
dynamicforamarketeconomy(though,asmentionedabove,taˆtonnementisperhapsnottherightprocess
to consider in that case) or a true peer-to-peer system. Our results, while applicable to a wider range of
utilityfunctions,assumesynchronouspriceadjustments.
Recently, Cheung etal. [7]showthatadiscretetaˆtonnementprocess thatuses an exponentialfunction
updaterule(Equation(1)canbethoughtofasalinearizationoftheirrule)convergestotheoptimalvalueof
theEisenberg-Galeconvexprogram,forcomplementaryCESutilitiesandforLeontiefutilities,i.e. forall
ρ P r´8,0q. We notethat ourLemma 8 shows that in thecases that theyanalyze, theconvergence of the
convexprogramobjectivefunctionalsoimpliesconvergencetoequilibrium(inthesamesensethatweuse
in our results). The analysis of [7] relates their process to generalized gradient descent using a judicious
choice of a Bregman divergence. In comparison, our main result uses the multiplicative weights update
paradigm (see [2]). It is well-known that multiplicative weights update analysis and gradient descent
are dual arguments. We note that there is no Bregman divergence that yields the linear descent step of
Equation (1) (which is not to say that the analysis of [7] cannot be adapted to apply to the linear update
thatweusehere). Wefurthernotethatlikeouranalysis,theresultsin[7]onlyapplytosynchronousprice
adjustments.
Discussion and comparison. In Section 4 we give a proof of convergence via gradient descent that
applies in particular to the CES utilities analyzed in [11, 8, 7], is somewhat simpler than the proof in [7],
and gives similar bounds on the rate of convergence. Our analysis of nested CES-Leontief utilities in
Section 5 is new. Our results for nested CES-Leontief utilities give an upper bound on the convergence
rate which is proportional to 1{δ3. For Leontief utilities, the analysis in [7] combined with our Lemma 8
givesabetterboundproportionalto1{δ2.
The pervasive application of the multiplicative weights update method uses ǫ that depends on the
desiredaccuracyoftheoutcome,andprovestheconvergenceofasolutionthataveragesovertheiterations.
Nevertheless, we show convergence of the sequence of prices (and allocations), and not just convergence
of average prices (clearly implied by the former), and we show this for a fixed ǫ in Equation (1) that is
independentofthedesired closenessto equilibrium.
Thetechniquesin[14]aresomewhatsimilartoours. However,theyonlyprovetheconvergenceofthe
average, and they use a much weaker notion of approximate equilibrium. On the other hand, that allows
them to apply their results directly to utility functions for which our stronger notion of convergence is
unlikely to hold, such as linear or resource allocation utilities. By tweaking the taˆtonnement process, we
can handletheseutilitiesas well. This is briefly discussedand analyzed in Section ??. In this context,we
notethatwhetheroneisinterestedinconvergenceofthesequenceorinconvergenceoftheaveragedepends
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on the phenomenon that is being modeled. One can think of taˆtonnement as a negotiation process—the
price setters propose prices and collect purchase offers to study the market, while no exchange actually
occurs during the process. In this case, convergence of the sequence seems more suitable. An alternative
view of taˆtonnement is that of an “exploration” process—the price setters actually sell a fraction of the
supply at each step of the process at the prices of that step. In this case, one would focus on convergence
of the average. (A more rigorous treatment of this interpretation is given by the ongoing markets model
of[11,8].) Ourproofsimplythat takingtheaverageoverthestepsofthepricesand theallocationsyields
asomewhatrelaxednotionofan approximateequilibrium.
Finally,wediscusstherobustnessofourresults. Forsimplicity,weassumethatthroughouttheprocess
thepricessatisfytheconditionthattheirsumequalsthetotalbudgetavailabletotheagents. Thiscondition
is easy to satisfy initially, without explicit coordination of the prices. For example, set each price to be
arbitrary (but strictly positive), and at the first round reset the price of each good to be the total money
spentonthegood,giventheinitialarbitraryprices. Oncetheassumptionissatisfied,itismaintainedifthe
processiscarriedoutprecisely. However,wenotethateveniftheinitialpricesdeviatefromsatisfyingthis
assumption, and the price adjustment or even the responses of the agents are perturbed, the convergence
theoremswouldstillholdundertrivialmodifications. (However,notethattheinitialpricesmustbestrictly
positive—taˆtonnement cannot recover from zero prices). In particular, the proof of Proposition 6 also
implies that the prices converge very quickly towards satisfying the above assumption. Once the sum of
prices is close to the sum of budgets, the rest of the proof can be modified to the slight deviation from
equalityand alsotoslightdeviationsfromoptimalityoftheresponsesoftheagents.
2 Fisher Markets
Here we present the Fisher market model and some basic definitions. In a Fisher market there are m
perfectly divisible goods, each with quantity scaled to 1, without loss of generality. There are n agents.
Eachagentiisendowedwithabudgetofb ,andaimsatmaximizingaconcaveutilityfunctionu : Rm Ñ
i i `
R . We’llassumemonotonicity,soeachagentspendsallitsbudget. Givenmonotonicity,wemayassume
`
withoutlossofgeneralitythatforeveryagenti,u p~0q “ 0. Alsowithoutlossofgenerality,wewillassume
i
throughoutthepaperthatthebudgetsarescaled so that n b “ 1.
i“1 i
A market equilibriumis a pair pp,xq where p : t1,2,...,mu Ñ R is an assignmentof non-negative
`
prices to the goods, and x : t1,2,...,nu ˆ t1,2,...,řmu Ñ R is an allocation of goods to agents,
`
satisfyingthefollowingconditions: (i) Thetotalspend m p x ofagent i isat mostb . (ii) Thebasket
j“1 j ij i
ofgoodsx thatagentigetsmaximizestheutilityu foranybasketwhosecostisatmostb . (iii)Thetotal
i i i
demand n x forgoodj isat most1. (iv)If thetotařldemand forgoodj isless than1, thenp “ 0.
i“1 ij j
Let x ppq denote the optimal basket of goods maximizing the utility u of agent i, under the budget
i i
ř
constraint b and the market prices p. Notice that x ppq is given by a solution to the following convex
i i
program:
m
x ppq “ arg max u px q : p x ď b .
i i i j ij i
xiPRm` # j“1 +
ÿ
We assumethatcomputingan optimalbasketisatractableproblem. Furtherdenoteby
n
z ppq fl x ppq´1
j ij
i“1
ÿ
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the excess demand for good j under the prices p. Notice that an equilibrium price vector p˚ satisfies
p˚ P B, where
m n
B “ p P Rm : p “ b .
` j i
# j“1 i“1 +
ÿ ÿ
We now define the notion of approximate equilibrium. We give two alternative definitions. In the first
definition,eachagentbuysanoptimalbasketofgoodssubjecttotheprices,butthedemandforeach good
may exceed the supply by a little. In the second definition, each agent buys a near-optimal basket of
goods, and the supply constraints are satisfied. Definion 1 implies Definition 2 (possibly with a change
of the approximation guarantee δ) in many interesting markets. This is the case when a small change in
allocation results in a small change in utility. Also, Definition 2 implies Definition 1 in many interesting
markets. Specifically,ifgivenspecificprices,theoptimalallocationforeachplayerisuniqueandtheutility
functionsare stronglyconcave, then thisis the case. Definition2 is more natural when theallocationsare
verysensitivetosmallchanges intheprices (onesuchexampleis thecase oflinearutilities).
Thedefinitionthatweusein mostofthepaperis:
Definition 1. A price-demandpairpp,xq isa δ-approximateequilibriumiff
P1. For every agent i “ 1,2,...,n, the demand of i optimizes the utility of i given the prices p: x “
i
x ppq.
i
P2. Foreverygoodj “ 1,2,...,m, thedemandforj does notexceed thesupplybymuch: z ppq ď δ.
j
P3. The goods that are not purchased almost entirely do not cost much: For every j “ 1,2,...,m, if
z ppq ă ´δ then p ď δ.
j j
Thealternativedefinitionthat isoften equivalenttothefirst oneis:
Definition 2. A price-demandpairpp,xq isa δ-approximateequilibriumiff
P1. Foreveryagent i “ 1,2,...,n, theutilityofiis near-optimalgiventheprices p: u px q ě p1´δq¨
i i
u px ppqq.
i i
P2. Foreverygoodj “ 1,2,...,m, thedemandforj does notexceed thesupply: z ppq ď 0.
j
P3. The goods that are not purchased almost entirely do not cost much: For every j “ 1,2,...,m, if
z ppq ă ´δ then p ď δ.
j j
For clarity, we use Dirac’s notation to do linear algebra in a vector space over R endowed with the
standardEuclideannorm. Inparticularxu | vydenotestheinnerproductbetweentwovectorsuandv,and
xu | A | vy denotes the bilinear map of a pair of vectors u,v using a linear operator A. We denote by R
`
thesetofnon-negativereal numbers,and by R theset ofstrictlypositivereal numbers.
``
3 The Taˆtonnement Process
In this section, we discuss arbitrary Eisenberg-Gale markets [15]. In an Eisenberg-Gale market, an equi-
librium allocation of goods to agents can be computed using a specific convex program whose objective
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functionisstrictlyconcaveintheutilitiesoftheagents. TheLagrangevariablescorrespondingtothesup-
plyconstraintsgivetheequilibriumprices. Thus,an equilibriumpricevectorcan becomputedby solving
thedualprogram.
Let x denote a feasible allocation for agent i, so x is the quantity of good j that agent i gets in
i ij
the allocation x . An equilibrium allocation x˚ is given by the solution to the following Eisenberg-Gale
i
convexprogram.
max n b lnu px q
i“1 i i i
s.t. n x ď 1 @j “ 1,2,...,m
ři“1 ij
x ě 0 @i “ 1,2,...,n; @j “ 1,2,...,m.
ij
ř
An equilibriumpricevectorp˚ isgivenbythesolutiontothedualprogram:
min m p ` n g˚pµ q
j“1 j i“1 i i
s.t. p ě ´µ i “ 1,2,...,n; @j “ 1,2,...,m
j ij
ř ř
p ě 0 @j “ 1,2,...,m,
j
where g˚ is the convex conjugate of the convex function g px q “ ´b lnu px q. More specifically, g˚
i i i i i i i
satisfies g˚pµ q “ sup txµ | xy´g pxqu. Notice that g˚ is monotonically non-decreasing in each
i i xPRm i i i
`
coordinateofµ . Therefore,wecanreplaceµ by´p foreveryi,j. Forfutureuse,wedenotetheprimal
i ij j
objective function by ψpxq “ n b lnu px q, and the dual objective function by φppq “ m p `
i“1 i i i j“1 j
n g˚p´pq.
i“1 i ř ř
The followinglemma givesan explicit expression for the gradient of φ. It appears in [7, Lemma3.3].
ř
In theappendix,wegivethe(short)proofforcompleteness.
Lemma 3 ([7]). In everyEisenberg-Galemarket,
∇φppq “ ´zppq.
We alsoneed thefollowingproperty oftheprimalobjectivefunction.
1
Claim4. Foreveryα ą 0, ψ ¨x ě ψpxq´α.
1`α
Proof. Foreveryagent i, ` ˘
1 1 α 1 α 1
u ¨x “ u ¨x` ¨~0 ě ¨u pxq` ¨u p~0q “ ¨u pxq,
i i i i i
1`α 1`α 1`α 1`α 1`α 1`α
ˆ ˙ ˆ ˙
wheretheinequalityfollowsfromtheconcavityofu . Thus,
i
1 n 1 n 1
ψ ¨x “ b ln u ¨x ě b ln ¨u pxq ě ψpxq´α,
i i i i
1`α 1`α 1`α
ˆ ˙ i“1 ˆ ˆ ˙˙ i“1 ˆ ˙
ÿ ÿ
wherethesecond inequalityfollowsfrom lnp1`αq ď α and thescaling n b “ 1. (cid:4)
i“1 i
The taˆtonnement process proceeds as follows. Start with an arbitraryřassignment of strictly positive
prices p0 P Rm X B. Given the time t prices pt, each agent independently responds with its time t` 1
``
demand xt`1 fl x pptq. Given the time t ` 1 demands xt`1, the prices of goods are updated according
i i
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to the excess demand — prices increase for goods whosedemand exceeds supply and prices decrease for
goodswhosesupplyexceeds demand. Morespecifically,
pt`1 “ pt ¨ 1`εz pptq ,
j j j
whereε P p0, 1q is aconstanttobespecified later. ` ˘
2
We nowgivean alternativecharacterization oftheoptimalresponseoftheagents toapricevector.
Lemma 5. Let p P B beapricevector. Then,
n m n
1
xppq “ arg max b lnu px q : ¨ p x ď 1 .
i i i j ij
xPRn`ˆm#i“1 }p}1 j“1 i“1 +
ÿ ÿ ÿ
Proof. Theproofusesessentiallythesameargumentas in theproofofLemma3.
Givenapricevectorp P B. Considerthefollowingprogram,
n 1 m n
max b lnu px q : ¨ p x ď 1 .
i i i j ij
#i“1 }p}1 j“1 i“1 +
ÿ ÿ ÿ
By Lagrangedualityweget thatthemaximumisachievedat
n m n
p
λmax,xmax “ arg max bilnuipxiq´λ 1´ j xij .
xPRn`ˆm,λě0#i“1 ˜ j“1 }p}1 i“1 ¸+
ÿ ÿ ÿ
Wewillshowthatλmax “ }p}1andxmax “ xppqbyshowingthattheysatisfytheKKTconditions. Thefirst
conditionisthatifλmax ą 0then1 “ mj“1 }pp}j1 ni“1pxmaxqij. By thedefinitionofxppq, mj“1pjxppqij “
bi (as uipxq is monotonicallynon-decrřeasing). Třhus,xppq satisfies ni“1 mj“1pjxppqij “ř ni“1bi “ }p}1.
For the second KKT condition notice the following. In a market in which every good j has quantity
n xppq , the price-demand pair pp,xppqq is an equilibrium (eacřh ageřnt optimizes its dřemand and the
i“1 ij
market clears). Thus, xppq is a primal optimal solution to the Eisenberg-Gale convex program for this
ř
market. Therefore,
n n n
xppq “ arg max b lnu px q : x “ xppq @j
i i i ij ij
xPRn`ˆm#i“1 i“1 i“1 +
ÿ ÿ ÿ
n m n
“ arg max b lnu px q´ p pxppq ´x q
i i i j ij ij
xPRn`ˆm#i“1 j“1 i“1 +
ÿ ÿ ÿ
n m n
“ arg max b lnu px q´ p p1´x q .
i i i j ij
xPRn`ˆm#i“1 j“1 i“1 +
ÿ ÿ ÿ
The first equation follows from the definition of the Eisenberg-Gale market with quantities n xppq .
i“1 ij
The second equation follows from Lagrange duality and the fact that the prices p are a dual optimal
solution. Thethirdequationfollowsas m p n xppq isa constantindependentofx. ř
j“1 j i“1 ij
Going back to the second KKT condition, notice that putting λmax “ }p}1 gives exactly the last
equation. Therefore, xmax “ xppq, λmaxř“ }p}1řsatisfythesecond KKT conditionas well. (cid:4)
7
Proposition6. If p0 P B thenpt P B forallt.
Proof. The proof is by induction on t. For t “ 0, the claim is satisfied by assumption. For the inductive
step,noticethat
m m m n
pt`1 “ p1´ǫq pt `ǫ pt xt`1
j j j ij
j“1 j“1 j“1 i“1
ÿ ÿ ÿ ÿ
m n m n
“ p1´ǫq pt `ǫ ptx pptq “ b
j j ij i
j“1 i“1j“1 j“1
ÿ ÿÿ ÿ
Asweassumethatu px qismonotonicallynon-decreasing,bythedefinitionofx ppqforeveryi, m ptx pptq “
i i i j“1 j ij
b (every agent spends all its budget at every round). Thus the last equality follows (together with the in-
i
ductionhypothesison pt). ř (cid:4)
4 Preliminary Results
The following theorem generalizes and simplifies slightly the results of [7, Theorem 6.4] on the conver-
gence of taˆtonnement for CES utility functions (albeit for the linearized version of the process that they
consider).
Theorem 7. Suppose that there is a choice of ε ą 0 and a closed convex set P Ă Rm which includes
`
an equilibrium price vector p˚ and the sequence p0,p1,p2,... with the following properties: (i) There
are bounds pmin ď pmax such that for all p P P and for all goods j, pj P rpmin,pmaxs. (ii) In P, the
dual objective function φ is twice continuouslydifferentiable and stronglyconvex, and its gradient ∇φ is
L-Lipschitz. (iii) ε ď pmin . Then, thereexists aconstantδ ą 0 suchthatforeveryT ě 0,
p2 L
max
φppTq´φpp˚q ď p1´δqT ¨ φpp0q´φpp˚q .
` ˘
Proof. The proof closely mimics the proof of convergence of gradient descent in the case of a strongly
convexobjectivefunction.
Condition(ii)impliesthatthereexistbounds0 ă λmin ď λmax suchthatforeveryp P P theeigenval-
ues oftheHessian matrix∇2φppq lieintheintervalrλmin,λmaxs.
Set ε “ pmin . Consider a timestep t. Considera point p P P. The second order Taylor expansion
p2maxλmax
ofφppqwithrespect to φppt´1q gives
1
φppq “ φppt´1q`x∇φppt´1q,p´pt´1y` ¨xp´pt´1 | ∇φpqq | p´pt´1y,
2
for some interpolationpoint q between p and pt´1. As q P P (because P is convex), we have that the last
term satisfies
λmin ¨}p´pt´1}22 ď xp´pt´1 | ∇φpqq | p´pt´1y ď λmax ¨}p´pt´1}22
8
Thus,on theonehand,
λ
φpptq ď φppt´1q`x∇φppt´1q,pt ´pt´1y` max ¨}pt ´pt´1}2
2
2
“ φppt´1q´ǫ¨ pt´1 ∇φppt´1q 2 ` λmax ¨ǫ2 ¨ pt´1∇φppt´1q 2
j j 2 j j
j j
ÿ ` ˘ ÿ` ˘
λ
ď φppt´1q´ǫ¨pmin ¨}∇φppt´1q}22 ` max ¨ǫ2 ¨p2max ¨}∇φppt´1q}22
2
2
p 1
ď φppt´1q´ min ¨ ¨}∇φppt´1q}2.
2
p 2λ
max max
ˆ ˙
On theotherhand, foreveryp P P,
λ
φppq ě φppt´1q`x∇φppt´1q,p´pt´1y` min ¨}p´pt´1}2.
2
2
Theright-handsideis minimizedatp “ pt´1 ´ 1 ¨∇φppt´1q,so wegetthat foreveryp,
λmin
2
1 λ 1
φppq ě φppt´1q`x∇φppt´1q,´ ¨∇φppt´1qy` min ¨ ´ ¨∇φppt´1q
λ 2 λ
min › min ›2
1 › ›
“ φppt´1q´ ¨}∇φppt´1q}2. › ›
2λ 2 › ›
min
By settingp “ p˚ on theleft-handside,we get
}∇φppt´1q}22 ě 2λmin ¨ φppt´1q´φpp˚q .
Subtracting φpp˚q on bothsidesinthefirst boundand` combiningthet˘woboundswegetthat
2
p λ
φpptq´φpp˚q ď 1´ min ¨ min¨ ¨ φppt´1q´φpp˚q .
p λ
˜ max max ¸
ˆ ˙
` ˘
Thisrecurrence relation impliesthetheorem. (cid:4)
The followinglemma relates the proximity to the dual objectiveoptimum to the proximityto equilib-
rium.
Lemma8. UnderthesameassumptionsandnotationofTheorem 7,foreveryδ ą 0,foreveryp P P,and
for the dual optimal solution p˚, if φppq ď φpp˚q `min 1, λ2m1ax ¨ 12λminδ2, then }p´ p˚}2 ď λmδax and
|zppq´zpp˚q| ď δ. ! )
8
Proof. Consider the second order Taylor expansion of φppq with respect to φpp˚q. For an interpolation
pointp1 “ γp`p1´γqp˚,
1
φppq “ φpp˚q`xp´p˚ | ∇φpp˚qy` xp´p˚ | ∇2φpp1q | p´p˚y.
2
9