Volume 1, issue 1 PDF
Preview Volume 1, issue 1
“jMID-vol1(1)-01” — 2016/12/19 — 7:59 — page i — #1 Journal of Mechanism and Institution Design Volume 1, Issue 1 ZaifuYang,TommyAndersson,VinceCrawford,YuanJu,PaulSchweinzer UniversityofYork,UniversityofKlagenfurt,SouthwesternUniversityofEconomicsand Finance,2016 “jMID-vol1(1)-01” — 2016/12/19 — 7:59 — page ii — #2 JournalofMechanismandInstitutionDesign,1(1),2016,ii Editorial board Editor ZaifuYang,UniversityofYork,UK Co-editors TommyAndersson,LundUniversity,Sweden VincentCrawford,OxfordUniversity,UK YuanJu,UniversityofYork,UK PaulSchweinzer,Alpen-Adria-UniversitätKlagenfurt,Austria AssociateEditors PeterBiro,HungarianAcademyofSciences,Hungary RandallCalvert,WashingtonUniversityinSt.Louis,USA Kim-SauChung,TheChineseUniversityofHongKong,HongKong MichaelSuk-YoungChwe,UniversityofCalifornia,LosAngeles,USA XiaotieDeng,ShanghaiJiaoTongUniversity,China LarsEhlers,UniversitédeMontréal,Canada AytekErdil,UniversityofCambridge,UK RobertEvans,UniversityofCambridge,UK AlexGershkov,HebrewUniversityofJerusalem,Israel SayantanGhosal,UniversityofGlasgow,UK Claus-JochenHaake,UniversitätPaderborn,Germany JohnHatfield,UniversityofTexasatAustin,USA PaulHealy,OhioStateUniversity,USA Jean-JacquesHerings,MaastrichtUniversity,Netherlands SergeiIzmalkov,NewEconomicSchool,Russia IanJewitt,OxfordUniversity,UK FuhitoKojima,StanfordUniversity,USA GlebKoshevoy,RussianAcademyofSciences,Russia DinardvanderLaan,TinbergenInstitute,Netherlands StephanieLau,CommodityFuturesTradingCommission,USA JingfengLu,NationalUniversityofSingapore,Singapore JinpengMa,RutgersUniversity,USA DavidManlove,UniversityofGlasgow,UK DebasisMishra,IndianStatisticalInstitute,India RudolfMüller,MaastrichtUniversity,Netherlands TymofiyMylovanov,UniversityofPittsburgh,USA SérgioParreiras,UniversityofNorthCarolina,USA “jMID-vol1(1)-01” — 2016/12/19 — 7:59 — page iii — #3 JournalofMechanismandInstitutionDesign,1(1),2016,iii AssociateEditors(continued) DavidPérez-Castrillo,UniversitatAutònomadeBarcelona,Spain NeilRankin,UniversityofYork,UK FrankRiedel,UniversitätBielefeld,Germany JózsefSákovics,UniversityofEdinburgh,UK AlejandroSaporiti,UniversityofManchester,UK MichaelSchwarz,GoogleResearch,USA EllaSegev,Ben-GurionUniversityoftheNegev,Israel ArunavaSen,IndianStatisticalInstitute,India AkiyoshiShioura,TokyoInstituteofTechnology,Japan NingSun,ShanghaiUniversityofFinanceandEconomics,China DolfTalman,TilburgUniversity,Netherlands AkihisaTamura,KeioUniversity,Japan JaccoThijssen,UniversityofYork,UK WalterTrockel,UniversitätBielefeld,Germany UtkuÜnver,BostonCollege,USA JamesWalker,UniversityofReading,UK DavidWettstein,Ben-GurionUniversityoftheNegev,Israel CharlesZheng,WesternUniversity,Canada Publishedby TheSocietyforthePromotionofMechanismandInstitutionDesign Editorialoffice,CentreforMechanismandInstitutionDesign UniversityofYork,Heslington,YorkYO105DD UnitedKingdom http://www.mechanism-design.org ISSN:2399-844X(Print),2399-8458(Online),DOI:10.22574/jmid Thefoundinginstitutionalmembersare UniversityofYork,UK Alpen-Adria-UniversitätKlagenfurt,Austria SouthwesternUniversityofEconomicsandFinance,China. Cover&LogoArtwork@JasmineYang LATEXEditor&JournalDesign@PaulSchweinzer(using‘confproc’) PrintedinYorkandKlagenfurtbytheCentreforMechanismandInstitutionDesign “jMID-vol1(1)-01” — 2016/12/19 — 7:59 — page iv — #4 JournalofMechanismandInstitutionDesign,1(1),2016,iv A Letter from the Editor H istory, illuminated by theoretical, empirical, and experimental studies, has shown that the institutions chosen by or imposed upon a society have a profound impact on its performance. The theory of mechanism and institution design is about how to devise new mechanisms or institutions, or improveexistingones,tobetterachievedesiredeconomicorsocialoutcomes. The challenge of design lies in the fact that individuals have different prefer- encesabouthowsocietyallocatesitsscarceresources,andprivateinformation mustimplicitlyorexplicitlyberevealedtorealizesociety’sgoals. The Journal of Mechanism and Institution Design seeks to provide an independentandpeer-reviewedopen-accessonlinejournalthatwillbeanatu- ralEnglish-languagehomefororiginalanalysesofmechanismandinstitution design. There are three compelling reasons for founding this new Journal. First, over the past few decades, mechanism and institution design has been oneofthemostflourishingandinfluentialresearchareas,andwebelievethat it will continue to grow in importance. Second, we believe that mechanism and institution design can serve as a common language to bridge fields rang- ing from economics, politics and law, to computer science, mathematics, and engineering, improving communication and productivity. Third, we believe that open access journals are the future of scholarly publishing, and that an independent,non-commercialjournalsuchasourshasevenmoreadvantages. The Internet has greatly enhanced the free and wide dissemination of knowl- edge,butitismosteffectivewhenaccessisopen. The Journal aims to publish original articles that deal with the issues of designing, improving, analyzing and testing economic, financial, political, or social mechanisms and institutions. It seeks scientifically important and so- cially relevant research, whether theoretical or applied, and whether empiri- cal,experimental,historical,orpractical. Itstrivestomaintainahighstandard forclarityofthoughtandexpression. The Journal is published and owned by the Society for the Promotion of Mechanism and Institution Design, a not-for-profit, unincorporated associa- tion devoted solely to the development of mechanism and institution design and the dissemination of scientific knowledge of the field. The Society tries tobeself-supportingandruntheJournalataminimumcostbyrequiringonly one author of each submitted paper to pay a modest membership fee of the Society to cover the cost of running the Journal. We are committed to han- dling every submitted paper as quickly as possible through a consistent and fairevaluation. Working together with our dedicated Editorial Board and professional col- leagues, we are confident that this journal will find a secure home in the sci- entificcommunitiesthatcontributetomechanismandinstitutiondesign. ZaifuYang,York,December16,2016 “jMID-vol1(1)-01” — 2016/12/19 — 7:59 — page 1 — #5 “p˙01” — 2016/12/18 — 10:04 — page 1 — #1 JournalofMechanismandInstitutionDesign ISSN:2399-844X(Print),2399-8458(Online) DOI:10.22574/jmid.2016.12.001 CONVERGENCE OF PRICE PROCESSES UNDER TWO DYNAMIC DOUBLE AUCTIONS JinpengMa RutgersUniversity,USA [email protected] QionglingLi RiceUniversity,USA [email protected] ABSTRACT Westudytheconvergenceoftwopriceprocessesgeneratedbytwodynamic doubleauctions(DA)andprovideconditionsunderwhichthetwopricepro- cessesconvergetoaWalrasianequilibriumintheunderlyingeconomy. When theconditionsarenotsatisfied,thepriceprocessesmayresultinabubbleor crash. Keywords: Double auction mechanisms, incremental subgradient methods, networkresourceallocations. JELClassificationNumbers: D44,D50. 1. INTRODUCTION A doubleauction(DA)mechanismisamarket-clearingsystembywhich dispersedprivateinformationfeedsintothesystemsequentiallythrough bilateraltrading. Withlittle concentratedinformationabouttotaldemandand supply of an asset or good available to all participants in the marketplace, it Both authors declare there are no conflicts of interest. This paper supersedes the paper “Bubbles,CrashesandEfficiencywithDoubleAuctionMechanisms”(Ma&Li,2011),which hasbeendistributedandpresentedinvariousconferences. WethankMarkSatterthwaitefor introducingustothetopic. Anyerrorsareourown. Copyright c JinpengMa,QionglingLi/1(1),2016,1–44. (cid:13) LicensedundertheCreativeCommonsAttribution-NonCommercialLicense3.0,http://creativecommons.org. “jMID-vol1(1)-01” — 2016/12/19 — 7:59 — page 2 — #6 “p˙01” — 2016/12/18 — 10:04 — page 2 — #2 2 Dynamicdoubleauction isnaturaltoaskwhetherthepriceprocessgeneratedbythisDAmechanism convergestoanequilibriumoftheunderlyingeconomyornot. BothA.Smith(1776)andHayek(1945)raiseasimilarquestionhowamar- ketmechanisminalaissez-faireeconomy,whereindividualparticipantswith littleinformationabouttotaldemandandsupplyactsolelyintheirself-interests, isabletointegrate“dispersedbitsof[incomplete]information”correctlyinto prices. A.Smith(1776)useshisfamous“invisiblehand”metaphortodescribe its magnificence of a price mechanism. Hayek (1945, p. 519) has further exploredtheidea: “The peculiar character of the problem of a rational economic order is determined precisely by the fact that the knowledge of the circumstances of which we must make use never exists in concentratedor integrated form,butsolely asthedispersed bitsof incompleteandfrequentlycontradictoryknowledgewhichallthe separate individuals possess. The economicproblem of society is thus not merely a problem of how to allocate “given” resources [ ], it is rather a problem of the utilization of knowledge not ··· giventoanyoneinitstotality.” Hegoesonbysaying: “Thismechanismwouldhavebeenacclaimedasoneof the greatest triumphs of the human mind” if “It were the result of deliberate humandesign”(Hayek,1945,p.527). ItshouldbenotedthatDAmechanisms employedinrealexchangemarketsacrosstheworldaredeliberatelydesigned byhumans. Ananswertothequestionisimportantforunderstandingpricedetermina- tioninanexchangemarket,sinceDAmechanismshavebeenwidelyusedin equity,commodityandcurrencymarkets,amongothers. Forexample,anans- wertothequestionisvitalforunderstandingtheefficientmarketshypothesis (Fama,1965)andtheexcessvolatilitypuzzle(Shiller,1981). Nonetheless,it isnoteasytocomeupwithananswer. Indeed,doesaDAmechanismmatter for the price determination of an asset? According to the efficient markets hypothesis,theanswershouldbenosincethepriceofanassetinanexchange market should always follow its fundamental, with no systematic disparity betweenthetwothatcanbedetectedwithfundamentalortechnicalanalysis. Ontheotherhand,excessvolatilitysuggeststhattheanswermaybeyes,since thepriceofanequitycandeviatefromitsfundamentaltoagreatdegreeand suchadeviationhasbeenrealizedbyaDAmechanismthroughasequenceof JournalofMechanismandInstitutionDesign1(1),2016 “jMID-vol1(1)-01” — 2016/12/19 — 7:59 — page 3 — #7 “p˙01” — 2016/12/18 — 10:04 — page 3 — #3 JinpengMa,QionglingLi 3 tradingbetweenbuyerandsellerpairs. But,ifaDAmechanismreallymatters, how is it possible for an equity with fundamental value of 100 to be traded, say,at300or50? ThemainobjectiveofthispaperistoinvestigateifaDAmechanismcan generateasequenceofpricesthatconvergestoanequilibriumoftheunderlying economy whenindividual demandsand supplies areonly privately known. To achievethisgoal,westudyabenchmarkmodelgivenbelow: m n P minimizeF(y)= ∑ f (y)+ ∑g (y) i j i=1 j=1 subject to y Y, a nonempty convex subset of Rd, where f and g are + i j ∈ real-valued (possibly non-differentiable) convex functions defined on the d- dimensionalEuclideanspaceRd. Alargeclassofquasilineareconomieswith msellersandnbuyerscanberepresentedbythisform(see Section 2.1). For these economies, the quantity demanded and supplied at prices y for buyer j=1,2, ,nandselleri=1,2, ,marejustsubsetsof thesubdifferentials ··· ··· ∂g (y)and∂ f (y),respectively,usingtheFenchelduality(Ma&Nie,2003). j i − Thus, an equilibrium of the underlying economy studied in this paper is an optimalsolutiontotheproblemP. AnIllustrative Example. Forsimplicity, consideranexchange economy wherethereisasingleobjector asset withafinitenumberofidenticalcopies forsale. Inadynamicdoubleauction,abuyersubmitsabidorderconsisting ofabidpriceandabidsize,and asellersubmitsanaskorderconsistingofan ask price and an ask size, with the bid price at least as high as the ask price. Thebidsizeisthequantitythebuyeriswillingtobuyatthebid priceandthe asksizeisthequantitytheselleriswillingtosellattheaskprice. Thepriceof anobjectisaweightedaverageofthebidpriceandtheaskprice,withweight α (0,1), as in a static double auction in Chatterjee & Samuelson (1983), ∈ Myerson & Satterthwaite (1983), Wilson (1985), and Gresik (1991). Thus, givenasequenceofpairsofonebuyerandoneseller,asequenceofpricesis generatedbyadoubleauction. Thenexttwoquestionsare,atagiveniteration, who will be the buyer and the seller pair and how are bid and ask prices are determined? Weprovidetwospecificexamplesofdoubleauctiontoaddress thetwoquestions. Inthefirstdoubleauction,weassumethatthenumberofbuyersequalsthe number ofsellers, andbuyersand sellersform two cyclicrings: abuyerring andasellerring(Figure1). Thissystemcanberealizedifthebuyerandseller JournalofMechanismandInstitutionDesign1(1),2016 “jMID-vol1(1)-01” — 2016/12/19 — 7:59 — page 4 — #8 “p˙01” — 2016/12/18 — 10:04 — page 4 — #4 4 Dynamicdoubleauction ringsconsistoftwopermutationsofagents. Apairofabuyerandaselleris selectedaccordingtothetwocyclicrings,onepairatatime. Theprocessstarts withpriceX atk. Thenbuyerπ (1)andsellerπ(1)arethefirstpairtosubmit k 0 theirbidandask,respectively,basedontheobservedX . Aftertheiterationof k thepair(π (1),π(1)),thenextpairwillbebuyerπ (2)andsellerπ(2). This 0 0 iterationprocessendswiththepair(π (m),π(m))andthepriceX . 0 k+1 Figure1. Iterationsunderadoubleauction,whereπ andπ aretwo 0 permutationsofagents. ? Sellers’ring π(1) - -π(i) - -π(m) ··· ··· Buyers’ringπ (1) - -π (i) - -π (m) 0 0 0 ··· ··· 6 We need to determine how a buyer bids and a seller asks. A buyer’s bid equalsthe newly updatedpricefrom thepreviouspairalong thetworings plus a priceincrement that equalsthe product of thebid step size andthe bid size. Thebidstepsizeisthepriceincrementforoneunitoftheobjectthatthebuyer iswillingtobuy. Thus,the moreabuyerwantstobuy,thehigherthe bidprice increment. Anaskpriceisdeterminedsimilarly. Aseller’saskpriceequalsthe newlyupdatedpricefromthepreviouspairalongthetworingsminusaprice thatequalstheproductoftheaskstepsizeandtheasksize. Theaskstepsize isnowthepricedecrementforoneunitoftheobjectforsale. Thus,themorea sellerwantstosell,thelowertheaskprice. Tobemoreprecise,letΦ be i 1,k − thepriceatiterationk anddesignatethenextselectedpairas(π(i),π (i)). The 0 askpriceψ andthebidpriceϕ aredeterminedby,respectively, π(i),k π (i),k 0 ψ =Φ a S (Φ ),ϕ =Φ +b D (Φ ), (1) π(i),k i 1,k k π(i) i 1,k π (i),k i 1,k k π (i) i 1,k − − · − 0 − · 0 − where S (Φ ) and D (Φ ) are the quantity supplied (i.e. the π(i) i 1,k π (i) i 1,k − 0 − ask size) and demanded (i.e. the bid size) at price Φ , respectively. If i 1,k − S (Φ ) and D (Φ ) are set-valued maps, equation (1) should be π(i) i 1,k π (i) i 1,k − 0 − JournalofMechanismandInstitutionDesign1(1),2016 “jMID-vol1(1)-01” — 2016/12/19 — 7:59 — page 5 — #9 “p˙01” — 2016/12/18 — 10:04 — page 5 — #5 JinpengMa,QionglingLi 5 understoodwithtwoselectionsfromthedemandandsupply. a and b are k k { } { } theaskandbidstepsizes,respectively. ThepriceΦ ,whichiscommunicated i,k tothenextpair,isdeterminedbyaweightedaverageofthebidandaskprices, withweightα (0,1): ∈ Φ =αψ +(1 α)ϕ . (2) i,k π(i),k π (i),k − 0 Equations (1)and (2)provide the ruleon howthe price atan iterationevolves from one pair to the other along the two rings. The price process starts at Φ = X and ends with X = Φ at time k. Then this process repeats 0,k k k+1 m,k withtwodifferentpermutationsofagents. Thus,asequenceofpricesX ,k= k 0,1,2, ,isgenerated. Notethatweconsiderthecasewheremispotentially ··· alargenumber. Inoursecondrandomizeddoubleauctionapairmadeupofabuyeranda seller is independently selected. Here we do not need the condition that the numberofbuyersequalsthenumberofsellersbecausesuchanauctioncanbe seenasaspecialcase,inwhichthebuyerringandthesellerringinFigure1 eachconsistof asingleagent. Thus,equations (1)and(2) provide asequence ofprices X onceagain. k { } Results. AssumethattheunderlyingeconomyhasaWalrasianequilibrium b and the limit lim k exists for two diminishing step sizes a and b . k→∞ ak { k} { k} Supposethereisapositivescalarλ suchthat(seeAssumption3.2) ∞ b a ∑ k λ k <+∞. (3) | n − m| k=0 Then we show that λ must be lim bk.1 Our first main result Theorem k ∞ a → k 4.4 demonstrates that the price process X must converge to a Walrasian k { } equilibriumprice vectorof theunderlyingeconomy aslongas theweight α satisfiestheequalityα = λ . BeyondtheexistenceofWalrasianequilibrium, 1+λ this convergence result does not depend on privately known demands and supplies. Instead it depends on the two parameters α and λ related to the auction form. If the weight α does not satisfy the equality α = λ , then 1+λ thepriceprocess X stillconvergestoapricebutitmaybehigherorlower k { } than the equilibrium price(s). A higher than equilibrium price (i.e. bubble) is obtained when α < λ and a lower than equilibrium price (i.e. crash) is 1+λ 1 Theconverseofthisclaimisnottrue. SeeExample4.9. JournalofMechanismandInstitutionDesign1(1),2016 “jMID-vol1(1)-01” — 2016/12/19 — 7:59 — page 6 — #10 “p˙01” — 2016/12/18 — 10:04 — page 6 — #6 6 Dynamicdoubleauction obtainedwhenα > λ bythedoubleauction. Forexample,α mustberight 1+λ at 1 forλ =1inorderfortheauctiontoarriveataWalrasianequilibrium. 2 For ourrandomized double auction,our second majorresult Theorem 4.7 showsthattheaboveresultstillholdswhenλ isdefinedbyλ = mlim bk, n k ∞ a → k wherenisthenumberofbuyers(oragents)andmisthenumberofsellers(or b objects), under Assumption 3.3. For example, if lim k =1 and m=2n, k ∞a → k thenα mustberightat 2 fortheauctiontoarriveataWalrasianequilibrium. 3 Onceagain,thisconclusiondoesnotdependon f andg ,whichareunknown i j tothemechanismdesigner. Theabovetworesultsareprovedforageneralcasewheretherearemultiple heterogeneous objects, with each object having a finite number of identical copies. In the general case, X is a sequence of price vectors rather than a k { } sequence of prices. The condition on λ is also stated for the case where the b limitlim k maynotexist. k ∞ a → k Noisesareidentifiedasakeyfactorintheformationofbubblesandcrashes (Shleifer,1999). Soitisofinteresttoseehownoisesmaychangeourresults. To examine this issue, we follow Ram et al. (2009) to introduce stochastic noisesintobuyandsellordersunderthetwoDAmechanisms. Interestingly, our main results still hold for certain noises. This means that not all noises can affectthe informationalefficiency ofa DA mechanism. The relationship between α and λ is still the key for the convergence of the price processes underthetwoDAmechanismswithstochasticnoises. Literature. The benchmark model is the dual problem of the linear pro- gramming relaxationin Bikhchandani &Mamer (1997) thatcan be reformula- ted as a convex optimization problem in the price space without constraints by P. An optimal solution to this dual P (i.e., a minimizer) is a Walrasian equilibriumpricevectoroftheoriginaleconomyifthezerodualitygapcon- dition holds.2 They also mention that the ascending price auction designed in Kelso & Crawford (1982) for a noted many-to-one job matching market can be usedto achievea Walrasian equilibrium pricevector for their economy underthegrosssubstitutescondition. AsalientfeatureoftheEnglishauction inKelso&Crawford(1982)istheconstantincrease(i.e. stepsize)ofprices forthoseworkersinexcessdemandateachiteration. Milgrom(2000)studies 2 Thisdualapproachanditsrelatedgradientmethodinsearchforanequilibriumingamescan betracedbacktoArrow&Hurwicz(1957). Beyonditsapplicationsineconomicsandgame theory, thedualapproachhasmanyapplicationsinotherareas(see, e.g.,Bertsekas,2009; Nedic´ &Ozdaglar,2009). JournalofMechanismandInstitutionDesign1(1),2016
Description:The list of books you might like
