SynthesisLectureson CommunicationNetworks
Editor JeanWalrand, UniversityofCalifornia,Berkeley
SynthesisLecturesonCommunicationNetworksisanongoingseriesof50-to100-pagepublications ontopicsonthedesign,implementation,andmanagementofcommunicationnetworks.Eachlectureis aself-containedpresentationofonetopicbyaleadingexpert.Thetopicsrangefromalgorithmsto hardwareimplementationsandcoverabroadspectrumofissuesfromsecuritytomultiple-access protocols.Theseriesaddressestechnologiesfromsensornetworkstoreconfigurableopticalnetworks. Theseriesisdesignedto:
•Providethebestavailablepresentationsofimportantaspectsofcommunicationnetworks.
•Helpengineersandadvancedstudentskeepupwithrecentdevelopmentsinarapidlyevolving technology.
•Facilitatethedevelopmentofcoursesinthisfield
NetworkGames:Theory,Models,andDynamics
IshaiMenacheandAsumanOzdaglar 2011
AnIntroductiontoModelsofOnlinePeer-to-PeerSocialNetworking
GeorgeKesidis 2010
StochasticNetworkOptimizationwithApplicationtoCommunicationandQueueing Systems
MichaelJ.Neely 2010
SchedulingandCongestionControlforWirelessandProcessingNetworks
LibinJiangandJeanWalrand 2010
PerformanceModelingofCommunicationNetworkswithMarkovChains JeonghoonMo 2010
CommunicationNetworks:AConciseIntroduction
JeanWalrandandShyamParekh 2010
PathProblemsinNetworks
JohnS.BarasandGeorgeTheodorakopoulos 2010
PerformanceModeling,LossNetworks,andStatisticalMultiplexing RaviR.Mazumdar 2009
NetworkSimulation
RichardM.Fujimoto,KalyanS.Perumalla,andGeorgeF.Riley 2006
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NetworkGames:Theory,Models,andDynamics IshaiMenacheandAsumanOzdaglar www.morganclaypool.com
ISBN:9781608454082paperback
ISBN:9781608454099ebook
DOI10.2200/S00330ED1V01Y201101CNT009
APublicationintheMorgan&ClaypoolPublishersseries SYNTHESISLECTURESONCOMMUNICATIONNETWORKS
Lecture#9
SeriesEditor:JeanWalrand, UniversityofCalifornia,Berkeley SeriesISSN SynthesisLecturesonCommunicationNetworks Print1935-4185Electronic1935-4193
NetworkGames
Theory,Models,andDynamics
IshaiMenache
MicrosoftResearchNewEngland
AsumanOzdaglar
MassachusettsInstituteofTechnology
SYNTHESISLECTURESONCOMMUNICATIONNETWORKS#9
ABSTRACT
Traditionalnetworkoptimizationfocusesonasinglecontrolobjectiveinanetworkpopulatedby obedientusersandlimiteddispersionofinformation.However,mostoftoday’snetworksarelargescalewithlackofaccesstocentralizedinformation,consistofuserswithdiverserequirements,and aresubjecttodynamicchanges.Thesefactorsnaturallymotivateanewdistributedcontrolparadigm, wherethenetworkinfrastructureiskeptsimpleandthenetworkcontrolfunctionsaredelegatedto individualagentswhichmaketheirdecisionsindependently(“selfishly").Theinteractionofmultiple independentdecision-makersnecessitatestheuseofgametheory,includingeconomicnotionsrelated tomarketsandincentives.
Thismonographstudiesgametheoreticmodelsofresourceallocationamongselfishagentsin networks.Thefirstpartofthemonographintroducesfundamentalgametheoretictopics.Emphasis isgiventotheanalysisof dynamics ingametheoreticsituations,whichiscrucialfordesignand controlofnetworkedsystems.Thesecondpartofthemonographappliesthegametheoretictools fortheanalysisofresourceallocationincommunicationnetworks.Wesetupageneralmodelof routinginwirelinenetworks,emphasizingthecongestionproblemscausedbydelayandpacket loss.Inparticular,wedevelopasystematicapproachtocharacterizingtheinefficienciesofnetwork equilibria,andhighlighttheeffectofautonomousserviceprovidersonnetworkperformance.We thenturntoexaminingdistributedpowercontrolinwirelessnetworks.Weshowthattheresulting Nashequilibriacanbeefficientifthedegreeoffreedomgiventoend-usersisproperlydesigned.
KEYWORDS
gametheory,Nashequilibrium,dynamics,communicationnetworks,routing,power control
ToHelena,Sophia,andRami
IshaiMenache
ToDaronandmyparentsfortheir unconditionalloveandsupport
AsumanOzdaglar
1.2SolutionConcepts
1.2.1DominantandDominatedStrategies
1.2.2IteratedEliminationofStrictlyDominatedStrategies
1.2.3NashEquilibrium
1.2.4CorrelatedEquilibrium
1.3ExistenceofaNashEquilibrium
1.3.1GameswithFinitePureStrategySets
1.3.2GameswithInfinitePureStrategySets
1.3.3ContinuousGames
1.3.4DiscontinuousGames
1.4UniquenessofaNashEquilibrium
1.AAppendix:MetricSpacesandProbabilityMeasures
2.2LearningDynamicsinGames–FictitiousPlay
2.2.1ConvergenceofFictitiousPlay
2.2.2Non-convergenceofFictitiousPlay
2.2.3ConvergenceProofs
2.3GameswithSpecialStructure
2.3.1SupermodularGames
2.3.2PotentialGames
2.AAppendix:Lattices
3.1SelfishRouting,WardropEquilibriumandEfficiency
3.1.1RoutingModel ..................................................73
3.1.2WardropEquilibrium
3.1.3InefficiencyoftheEquilibrium
3.1.4MultipleOrigin-DestinationPairs
3.2PartiallyOptimalRouting
3.2.1BackgroundandMotivation
3.2.2TheModel
3.2.3EfficiencyofPartiallyOptimalRouting
3.2.4Extensions ......................................................87
3.3CongestionandProviderPriceCompetition
3.3.1PricingandEfficiencywithCongestionExternalities
3.3.2Model
3.3.3MonopolyPricingandEquilibrium
3.3.4OligopolyPricingandEquilibrium
3.3.5EfficiencyAnalysis ..............................................92
3.3.6Extensions
3.4ConcludingRemarks
4.1NoncooperativeTransmissionSchedulinginCollisionChannels
4.1.1TheModelandPreliminaries
4.1.2EquilibriumAnalysis
4.1.3AchievableChannelCapacity
4.1.4Best-ResponseDynamics
4.1.5Discussion
4.2NoncooperativePowerControlinCollisionChannels
4.2.1TheModel
4.2.2EquilibriumAnalysis
4.2.3Best-ResponseDynamicsandConvergencetothePowerEfficient Equilibrium
4.2.4Equilibrium(In)EfficiencyandBraess-LikeParadoxes
4.2.5Discussion .....................................................128
4.3RelatedWorkandExtensions
4.4FutureDirections
Preface
Traditionalnetworkoptimizationfocusesonawell-definedcontrolobjectiveinanetwork populatedbyobedientusersandlimiteddispersionofinformation,andusesconvexoptimization techniquestodetermineefficientallocationofresources(see[8],[20],[22],[102]).Mostoftoday’s networkedsystems,suchastheInternet,transpor tationnetworks,andelectricitymarkets,differ fromthismodelintheirstructureandoperation.First,thesenetworksarelarge-scalewithlackof accesstocentralizedinformationandsubjecttodynamicchanges.Hence,controlpolicieshavetobe decentralized,scalable,androbustagainstunexpecteddisturbances.Second,thesenetworksconsist ofinterconnectionofheterogeneousautonomousentitiesandserveuserswithdiverserequirements, sothereisnocentralpartywithenforcementpoweroraccurateinformationaboutuserneeds. Thisimpliesthatselfishincentivesandprivateinformationofusersneedtobeincorporatedintothe controlparadigm.Finally,thesenetworksaresubjecttocontinuousupgradesandinvestmentsinnew technologies,makingeconomicincentivesofserviceandcontentprovidersmuchmoreparamount.
Thesenewchallengeshavenaturallymotivatedanewdistributedcontrolparadigm,wherethe networkinfrastructureiskeptsimpleandthenetworkcontrolfunctionsaredelegatedtoindividual agents,whichmaketheirdecisionsindependently(“selfishly"),accordingtotheirownperformance objectives.Thekeyaspectofthisapproachistoviewthenetworkasaresourcetobesharedby anumberofheterogeneoususerswithdifferentservicerequirements.Theinteractionofmultiple independentdecision-makersnecessitatestheuseofgametheory(thestudyofmulti-agentproblems) andalsosomeideasfromeconomicsrelatedtomarketsandincentives.Consequently,therecent engineeringliteratureconsidersavarietyofgame-theoreticandeconomicmarketmodelsforresource allocationinnetworks.
Thismonographstudiesgametheoreticmodelsforanalysisofresourceallocationamong heterogeneousagentsinnetworks.Giventhecentralrolethatgametheoryplaysintheanalysis ofnetworkedsystems,thefirstpartofthemonographwillbedevotedtoasystematicanalysis offundamentalgametheoretictopics.Emphasiswillbeplacedongametheoretictoolsthatare commonlyusedintheanalysisofresourceallocationincurrentnetworksorthosethatarelikelyto beusedinfuturenetworks.Westartinthe nextchapterwithstrategicformgames,whichconstitute thefoundationofgametheoreticanalysisandenableustointroducethecentralconceptofNash equilibrium,whichmakespredictionsaboutequilibriumbehaviorinsituationsinwhichseveral agentsinteract.Wealsointroduceseveralrelatedconcepts,suchascorrelatedequilibria,bothto clarifytheconditionsunderwhichNashequilibriaarelikelytoprovideagoodapproximationto behaviorinvariousdifferentcircumstancesandasalternativeconceptsofequilibriumthatmightbe usefulinthecommunicationareainfutureresearch.
xivPREFACE
TheNashequilibriumisbyitsverydefinitionastaticconcept,andassuch,thestudyofits propertiesdoesnotcovertheanalysisof dynamics,namely,ifandhowanequilibriumisreached.Issues ofdynamicresourceallocationandchangesinbehaviorofusersinwirelineandwirelessnetworks areofcentralimportanceintheanalysisofcommunicationnetworks.WepresentinChapter 2 two complementarywaysofintroducingdynamicanalysisingametheoreticsituations.First,westudy extensiveform(dynamic)gameswheretherelevantconceptof(subgameperfect)Nashequilibrium willexhibitsomedynamicsitself.Second,welookatdynamicsinducedbytherepeatedplayof thesamestrategicformgamewhenagentsarenotsosophisticatedtoplayNashequilibrium,but followsimplemyopicrulesorrulesofthumb.Thegametheoryliteraturehasestablishedthatthis typeofmyopicplayhasseveralinterestingpropertiesinmanygamesand,infact,convergesto Nashequilibriumforcertainclassesofgames.Interestingly,theseclassesofgames,whichinclude potentialgamesandsupermodulargames,havewidespreadapplicationsfornetworkgamemodels, sothelastpartofthischapterincludesadetailedanalysisofpotentialandsupermodulargamesand thedynamicsofplayundervariousmyopicandreactiverules.
Thesecondpartofthemonographappliesgametheoretictoolstotheanalysisofresource allocationinwirelineandwirelesscommunicationnetworks.Chapter 3 focusesonwirelinenetworks, withspecialattentiondrawntotheconsequencesofselfishroutingincommunicationnetworks.We firstshowhowthenotionsofNashequilibriumandsubgameperfectNashequilibriumenable ustodevelopsimplemodelsofroutingandresourceallocationinwirelinenetworks.Wesetup ageneralmodelofroutinginwirelinenetworksemphasizingthecongestionproblemscausedby delayandpacketloss.Weshowhowavarietyofdifferentmodelsofcongestioncanbemodeled asastaticstrategicformgamewiththecostofcongestioncapturedbylatencyfunctions.Wefirst establishexistenceofequilibriaandprovidebasiccharacterizationresults.Wethenturntothe questionofefficiencyofequilibriainwirelinecommunicationproblems.Itiswellknownsince theworkofAlfredPigouthatequilibriainsuchsituationswithcongestionproblemscaninvolve significantexternalities.Weprovideexamplesillustratingtheseinefficiencies,bothdemonstrating thatinefficienciescouldbesignificantlyverylarge(unbounded)andthepossibilitiesofparadoxical resultsuchastheBraess’paradox.Wethen developasystematicapproachtocharacterizingthe inefficienciesoftheseequilibria.
Classicmodelsofwirelinecommunicationignorethefactthatautonomousserviceproviders areactiveparticipantsintheflowsofcommunicationanddosobyeither(i)redirectingtraffic withintheirownnetworkstoachieveminimumintradomaintotallatencyor(ii)chargingprices formaximizingtheirindividualrevenues.Wethenshowhowthepresenceofautonomousservice providerscanbeincorporatedintothegeneralframeworkofwirelinegames.Wedemonstratehow existenceofpurestrategyandmixedstrategyequilibriacanbeestablishedinthepresenceofserviceproviderroutingandpricing,anddevelopanalternativemathematicalapproachtoquantifying inefficiencyofequilibriainseveralnetworkingdomains.Theinterestingsetofresultsisthatina varietyofnetworktopologies,thepresenceofpricesamelioratesthepotentialinefficienciesthat existinwirelinecommunicationnetworks.However,wealsoshowthatincomplexcommunication
PREFACExv networks,eveninthepresenceofpricesandoptimalintradomainroutingdecisions,inefficiencies couldbesubstantial,partlybecauseofthedoublemarginalizationprobleminthetheoryofoligopoly.
Chapter 4 dealswithwirelesscommunicationgames.Perhapsthemostlucidexamplefor theconsequencesofselfishbehaviorinwirelessnetworksisthecasewhereamobilecapturesa sharedcollisionchannelbycontinuouslytransmitting packets,henceeffectivelynullifyingother users’throughput.Weshowthatthisundesiredscenariocanbeavoidedunderanaturalpowerthroughputtradeoffassumption,whereeachuserminimiz esitsaveragetransmissionrate(whichis proportionaltopowerinvestment)subjecttominimum-throughputdemand.Animportantelement inourmodelsistheincorporationoffadingeffects,assumingthatthechannelqualityofeachmobile istime-varyingandavailabletotheuserpriortothetransmissiondecisions.Ourequilibriumanalysis revealsthatthereareatmosttwoNashequilibriumpointswhereallusersobtaintheirdemands, withonestrictlybetterthantheotherintermsofpowerinvestmentforallusers.Furthermore,we suggestafullydistributedmechanismthatleadstothebetterequilibrium.Theabovemodelisthen extendedtoincludewirelessplatformswheremobilesareallowedtoautonomouslycontroltheir transmissionpower.Wedemonstratetheexistenceofapower-superiorequilibriumpointthatcan bereachedthroughasimpledistributedmechanism.Onthenegativeside,however,wepointto thepossibilityofBraess-likeparadoxes,wheretheuseofmultiplepowerlevelscandiminishsystem capacityandalsoleadtolargerper-userpowerconsumption,comparedtothecasewhereonlya singlelevelispermitted.
WeconcludethemonographinChapter 5 byoutlininghigh-leveldirectionsforfuturework intheareaofnetworkgames,incorporatingnovelchallengesnotonlyintermsofgame-theoretic modelingandanalysis,butalsowithregardtotheproperexploitationoftheassociatedtoolsin currentandfuturenetworkingsystems.
LargepartsofChapters 3 and 4 arebasedonourindividualresearchinthearea.Weare naturallyindebtedtoourcoauthorswhoseadvice,knowledge,anddeepinsighthavebeeninvaluable. IshaiMenachewishestoparticularlythankNahumShimkinforhisguidanceandcollaboration. AsumanOzdaglarwouldliketothankDaronAcemogluforhiscollaborationandsupport.
WearealsogratefultoErminWeiformaking detailedcommentsondraftsofthismonograph. Finally,wewishtoacknowledgetheresearchsupportofNSFgrantsCMMI-0545910andSES0729361,AFOSRgrantsFA9550-09-1-0420andR6756-G2,AROgrant56549NS,theDARPA ITMANETprogram,andaMarieCurieInternationalFellowshipwithinthe7thEuropeanCommunityFrameworkProgramme.
IshaiMenacheandAsumanOzdaglar
March2011
StaticGamesandSolution Concepts
Thischapterpresentsthefundamentalnotionsandresultsinnoncooperativegametheory.InSection 1.1,weintroducethestandardmodelforstaticstrategicinteractions,thestrategicformgame.In Section 1.2,wedefinevarioussolutionconceptsassociatedwithstrategicformgames,including theNashequilibrium.WethenproceedinSections 1.3–1.4 toaddresstheissuesofexistenceand uniquenessofaNashequilibrium.
1.1STRATEGICFORMGAMES
Wefirstintroducestrategicformgames(alsoreferredtoasnormalformgames).A strategicform gameisamodelforastaticgameinwhichallplayersactsimultaneouslywithoutknowledgeofother players’actions.
Definition1.1 (StrategicFormGame)Astrategicformgameisatriplet I ,(Si )i ∈I ,(ui )i ∈I where
1. I isafinitesetofplayers, I ={1,...,I }
2. Si isanon-emptysetofavailableactionsforplayer i
3. ui : S → R isthepayoff(utility)functionofplayer i where S = i Si 1
Forstrategicformgames,wewillusetheterms action and (pure)strategy interchangeably.2 Wedenoteby si ∈ Si anactionforplayer i ,andby s i =[sj ]j =i avectorofactionsforallplayers except i .Werefertothetuple (si ,s i ) ∈ S asan action(strategy)profile,or outcome.Wealsodenote by S i = j =i Sj thesetofactions(strategies)ofallplayersexcept i .Ourconventionthroughout willbethateachplayer i isinterestedinactionprofilesthat“maximize"hisutilityfunction ui
1 Hereweimplicitlyassumethatplayershavepreferencesovertheoutcomesandthatthesepreferencescanbecapturedbyassigning autilityfunctionovertheoutcomes.Notethatnotallpreferencerelationscanbecapturedbyutilityfunctions(seeChapters1,3 and6of[60]formoreonthisissue).
2 Wewilllaterusetheterm“strategy"moregenerallytorefertorandomizationsoveractions,orcontingencyplansoveractionsin thecontextofdynamicgames.
41.STATICGAMESANDSOLUTIONCONCEPTS
Thenexttwoexamplesillustratestrategicformgameswithfiniteandinfinitestrategysets.
Example1.2 FiniteStrategySets Atwo-playergamewherethestrategyset Si ofeachplayeris finitecanberepresentedinmatrixform.Weadopttheconventionthattherows(columns)ofthe matrixrepresenttheactionsetofplayer1(player2).Thecellindexedbyrow x andcolumn y contains apair, (a,b),where a isthepayofftoplayer1and b isthepayofftoplayer2,i.e., a = u1 (x,y) and b = u2 (x,y).Thisclassofgamesissometimesreferredtoas bimatrixgames.Forexample,consider thefollowinggameof“MatchingPennies.”
Matching Pennies.
Thisgamerepresents“pureconflict,”inthesensethatoneplayer’sutilityisthenegativeof theutilityoftheotherplayer,i.e.,thesumoftheutilitiesforbothplayersateachoutcomeis“zero.” Thisclassofgamesisreferredtoas zero-sumgames (or constant-sumgames )andhasbeenextensively studiedinthegametheoryliterature[15].
Example1.3 InfiniteStrategySets Thestrategysetsofplayerscanalsohaveinfinitelymanyelements.Considerthefollowinggameof CournotCompetition, whichmodelstwofirmsproducingthe samehomogeneousgoodandseekingtomaximizetheirprofits.Theformalgame G = I ,(Si ),(ui ) consistsof:
1.Asetoftwoplayers, I = 1, 2
2.Astrategyset Si =[0, ∞) foreachplayer i ,where si ∈ Si representstheamountofgoodthat theplayerproduces.
3.Autilityfunction ui foreachplayer i givenbyitstotalrevenueminusitstotalcost,i.e., ui (s1 ,s2 ) = si p(s1 + s2 ) ci si where p(q) representsthepriceofthegood(asafunctionofthetotalamountofgood q ),and ci istheunitcostforfirm i .
Forsimplicity,weconsiderthecasewherebothfirmshaveunitcost, c1 = c2 = 1,andtheprice functionispiecewiselinearandisgivenby p(q) = max{0, 2 q } Weanalyzethisgamebyconsideringthe best-responsecorrespondences foreachofthefirms. Forfirm i ,thebest-responsecorrespondence Bi (s i ) isamappingfromtheset S i intoset Si such that Bi (s i ) ={si ∈ Si | ui (si ,s i ) ≥ ui (si ,s i ), forall si ∈ Si }
1.2.SOLUTIONCONCEPTS5
Notethatforthisexample,thebestresponsecorrespondencesareunique-valued(hencecanbe referredtoasbest-responsefunctions),sinceforall s i ,thereisauniqueactionthatmaximizesthe utilityfunctionforfirm i .Inparticular,wehave:
Bi (s i ) = argmax si ≥0 (si p(si + s i ) si ).
When s i > 1,theactionthatmaximizestheutilityfunctionoffirm i ,i.e.,itsbestresponse,is 0. Moregenerally,itcanbeseen(byusing firstorderoptimalityconditions,seeAppendix 1.B)thatfor any s i ∈ S i ,thebestresponseoffirm i isgivenby
Figure1.1: BestresponsefunctionsfortheCournotCompetitiongame.
Figure 1.1 illustratesthebestresponsefunctionsasafunctionof s1 and s2 .Intuitively,we expecttheoutcomeofthisgametobeatthepointwherebothofthesefunctionsintersect.Inthe nextsection,wewillarguethatthisintersectionpointisareasonableoutcomeinthisgame.
1.2SOLUTIONCONCEPTS
Thissectionpresentsthemainsolutionconceptsforstrategicformgames:strictlyandweaklydominantanddominatedstrategies,pureandmixedNashequilibrium,andcorrelatedequilibrium.
1.2.1DOMINANTANDDOMINATEDSTRATEGIES
Insomegames,itmaybepossibletopredicttheoutcomeassumingthatallplayersarerationaland fullyknowledgeableaboutthestructureofthegameandeachother’srationality.Thisisthecase, forinstance,forthewell-studiedPrisoner’sDilemmagame.Theunderlyingstoryofthisgameisas follows:Twopeoplearearrestedforacrime,placedinseparaterooms,andquestionedbyauthorities tryingtoextractaconfession.Ifthey bothremainsilent(i.e.,cooperatewitheachother),thenthe authoritieswillnotbeabletoprovechargesagainstthemandtheywillbothserveashortprison term,say2years,forminoroffenses.Ifonlyoneofthemconfesses(i.e.,doesnotcooperate),his termwillbereducedto1yearandhewillbeusedasawitnessagainsttheotherperson,whowillget asentenceof5years.Iftheybothconfess,theybothgetasmallersentenceof4years.Thisgame canberepresentedinmatrixformasfollows:
2, 2 5, 1 1, 5
Prisoner’sDilemma.
Inthisgame,regardlessoftheotherplayersdecision,playing Don’tCooperate yieldsa higherpayoffforeachplayer.Hence,thestrategy Don’tCooperate is strictlydominant,i.e.,no matterwhatstrategytheotherplayerchooses,thisstrategyalwaysyieldsastrictlybetteroutcome. Wecanthereforeinferthatbothplayerswillchoose Don’tCooperate andspendthenextfouryears injailwhileiftheybothchosethestrategy Cooperate,theycouldhaveendedupinjailonlyfortwo years!Prisoner’sDilemmaisaparadigmaticexampleofself-interestedrationalbehaviornotleading tojointly(socially)optimaloutcomes.WewillseeinChapters 3 and 4 thatmanynetworkgames exhibitsuchinefficienciesinequilibriumduetoselfishnatureofplayers.
Acompellingnotionofequilibriumingameswouldbethe dominantstrategyequilibrium, whereeachplayerplaysadominantstrategyasformalizedinthenextdefinition.
Definition1.4 DominantStrategy Astrategy si ∈ Si isa dominantstrategy forplayer i if ui (si ,s i ) ≥ ui (si ,s i ) forall si ∈ Si andforall s i ∈ S i Itis strictlydominant ifthisrelationholdswithastrictinequality.
Definition1.5 DominantStrategyEquilibrium Astrategyprofile s ∗ isa (strictly)dominant strategyequilibrium ifforeachplayer i , s ∗ i isa(strictly)dominantstrategy.
InthePrisoner’sDilemmagame,(Don’tCooperate, Don’tCooperate)isastrictlydominantstrategyequilibrium.Thoughcompelling,dominantstrategyequilibriadonotalwaysexist,
1.2.SOLUTIONCONCEPTS7 asillustratedbytheMatchingPenniesgame(cf.Example 1.2)andthenextexample.Considera slightlymodifiedPrisoner’sDilemmagameinwhichplayersalsohavethestrategy Suicide leading tothefollowingpayoffstructure:
COOPERATE
DON’T COOPERATE
Prisoner’s Dilemma with Suicide.
Thispayoffmatrixmodelsascenarioinwhichifoneplayerchoosesthestrategy Suicide, then,duetolackofwitnesses,theotherplayergetsofffreeifheremainssilent(cooperates).Inthis game,thereisnodominantstrategyequilibriumbecauseoftheadditionalstrategy Suicide.Notice, however,thatthestrategy Suicide isthe worst possibleoptionforaplayer,nomatterwhattheother playerdoes.Inthissense, Suicide isstrictlydominatedbytheothertwostrategies.Moregenerally, wesaythatastrategyis strictlydominated foraplayerifthereexistssomeotherstrategythatyields astrictlyhigherpayoffregardlessofthestrategiesoftheotherplayers.
Definition1.6 StrictlyDominatedStrategy Astrategy si ∈ Si is strictlydominated forplayer i ifthereexistssome si ∈ Si suchthat
ui (si ,s i )>ui (si ,s i ) forall s i ∈ S i
Next,wedefineaweakerversionofdominatedstrategies.
Definition1.7 WeaklyDominatedStrategy Astrategy si ∈ Si is weaklydominated forplayer i ifthereexistssome si ∈ Si suchthat
ui (si ,s i ) ≥ ui (si ,s i ) forall s i ∈ S i , and ui (si ,s i )>ui (si ,s i ) forsome s i ∈ S i .
Itisplausibletoassumethatnoplayerchoosesastrictlydominatedstrategy.Moreover,commonknowledgeofpayoffsandrationalityleads playerstodoiteratedeliminationofstrictlydominatedstrategies,asillustratednext.
81.STATICGAMESANDSOLUTIONCONCEPTS
1.2.2ITERATEDELIMINATIONOFSTRICTLYDOMINATEDSTRATEGIES
InthePrisoner’sDilemmawithSuicidegame,thestrategy Suicide isastrictlydominatedstrategy forbothplayers.Therefore,norationalplayerwouldchoose Suicide.Moreover,ifplayer1iscertain thatplayer2isrational,thenhecaneliminateheropponent’s Suicide strategy.Wecanuseasimilar reasoningforplayer2.Afteroneroundofeliminationofstrictlydominatedstrategies,wearebackto thePrisoner’sDilemmagame,whichhasadominantstrategyequilibrium.Thus,iteratedelimination ofstrictlydominatedstrategiesleadstoauniqueoutcome,(Don’tCooperate, Don’tCooperate) inthismodifiedgame.Wesaythatagameis dominancesolvable ifiteratedeliminationofstrictly dominatedstrategiesyieldsauniqueoutcome. Considernextanothergame.
3,09,62,8 LEFT UP MIDDLE MIDDLE DOWN RIGHT
4,35,16,2 2,18,43,6
Exampleforiteratedeliminationofstrictlydominatedstrategies.
Inthisgame,therearenostrategiesthatarestrictlydominatedforplayer1(therowplayer).On theotherhand,thestrategy Middle isstrictlydominatedbythestrategy Right forplayer2(the columnplayer).Thus,weconcludethatitis notrationalforplayer2toplay Middle andwecan thereforeremovethiscolumnfromthegame,resultinginthefollowingreducedgame.
4,36,2 2,13,6 3,02,8 LEFT UP MIDDLE DOWN RIGHT
Gameafteroneremovalofstrictlydominatedstrategies.
Now,notethatbothstrategies Middle and Down arestrictlydominatedbythestrategy Up for player1,whichmeansthatbothoftheserowscanberemoved,resultinginthefollowinggame.
4,36,2 LEFT UP RIGHT
Gameafterthreeiteratedremovalsofstrictlydominatedstrategies.
1.2.SOLUTIONCONCEPTS9
Weareleftwithagamewhereplayer1doesnothaveanychoiceinhisstrategies,whileplayer2can choosebetween Left and Right.Since Left willmaximizetheutilityofplayer2,weconcludethat theonlyrationalstrategyprofileinthegameis(Up,Left).3
Moreformally,wedefinetheprocedureof iteratedeliminationofstrictlydominatedstrategies (oriteratedstrictdominance) asfollows:
• Step0: Foreach i ,let S 0 i = Si .
• Step1: Foreach i ,define
S 1 i ={si ∈ S 0 i | theredoesnotexist si ∈ S 0 i suchthat ui si ,s i >ui (si ,s i ) forall s i ∈ S 0 i } ...
• Stepk: Foreach i ,define
S k i ={si ∈ S k 1 i | theredoesnotexist si ∈ S k 1 i suchthat ui si ,s i >ui (si ,s i ) forall s i ∈ S k 1 i }.
•Foreach i ,define
S ∞ i =∩∞ k =0 S k i
Itisimmediatethatthisprocedureyieldsanonemptysetofstrategyprofilesundersome assumptionsstatedinthenexttheorem.
Theorem1.8 Supposethateither(1)each Si isfinite,or(2)each ui (si ,s i ) iscontinuousandeach Si iscompact.Then S ∞ i isnonemptyforeach i
WenextapplyiteratedeliminationofstrictlydominatedstrategiestotheCournotCompetitiongame(cf.Example 1.3).Recallthatweusethenotation S k i todenotethesetofstrategiesof player i thatsurviveiteratedeliminationofstrictlydominatedstrategiesatstep k .Inthefirststep, wenotethatbothfirmsmustchooseaquantitybetween [0, ∞),i.e.,
S 1 1 =[0, ∞), S 1 2 =[0, ∞).
Sincetherangeofthebestresponsefunctionofplayer1is [0, 1/2],anystrategyoutsidethisrange isneverabestresponseandthereforeisstrictlydominated.Thesamereasoningholdsforplayer2. Thus,atthesecondstep,wehave
S 2 1 =[0, 1/2], S 2 2 =[0, 1/2]
3 Onemightworrythatdifferentordersfortheremovalofdominatedstrategiescanyielddifferentresults.However,itcanbe shownthattheorderinwhichstrategiesareeliminateddoesnot affectthesetofstrategiesthatsurviveiteratedeliminationof strictlydominatedstrategies.
101.STATICGAMESANDSOLUTIONCONCEPTS
Giventhatplayer2onlychoosesactionsintheinterval [0, 1/2],thenplayer1can restrict thedomain ofhisbestresponsefunctiontoonlythesevalues.Usinghisbestresponsefunction,thisimpliesthat thestrategiesoutsidetheinterval [1/4, 1/2] arestrictlydominatedforplayer1.Applyingthesame reasoningforplayer2,weobtain
(seeFigure 1.2).Itcanbeshownthatinthelimit,theendpointsoftheintervalsconvergetothe pointwherethetwobestresponsefunctionsintersect.Hence,theCournotCompetitiongameis anotherexampleofadominancesolvablegame.Mostgames,however,arenotsolvablebyiterated strictdominance;therefore,weneedastrongerequilibriumnotiontopredicttheoutcome.
Figure1.2: EliminationofstrictlydominatedstrategiesfortheCournotcompetitiongame.
1.2.3NASHEQUILIBRIUM
Wenextintroducethefundamentalsolutionconceptforstrategicformgames,thenotionofa Nash equilibrium.ANashequilibriumcapturesasteadystateoftheplayinastrategicformgamesuch thateachplayeractsoptimallyandformscorrectconjecturesaboutthebehavioroftheotherplayers.
Definition1.9 NashEquilibrium A (purestrategy)Nashequilibrium ofastrategicformgame I ,(Si ),(ui )i ∈I isastrategyprofile s ∗ ∈ S suchthatforall i ∈ I ,wehave ui (s ∗ i ,s ∗ i ) ≥ ui (si ,s ∗ i ) forall si ∈ Si .
1.2.SOLUTIONCONCEPTS11
Hence,aNashequilibriumisastrategyprofile s ∗ suchthatnoplayer i canprofitbyunilaterally deviatingfromhisstrategy s ∗ i ,assumingeveryotherplayer j followshisstrategy s ∗ j .Thedefinition ofaNashequilibriumcanberestatedintermsofthebest-responsecorrespondences.
Definition1.10 NashEquilibrium-Restated Let I ,(Si ),(ui )i ∈I beastrategicgame.Forany s i ∈ S i ,considerthebest-responsecorrespondenceofplayer i , Bi (s i ),givenby
Wesaythatanactionprofile s ∗ isa Nashequilibrium if
Thisimpliesthatfortwoplayergames,thesetofNashequilibriaisgivenbytheintersection ofthebestresponsecorrespondencesofthetwoplayers(e.g.,recalltheCournotCompetitiongame). BelowwegivetwootherexamplesofgameswithpurestrategyNashequilibria.
Example1.11 BattleoftheSexes
Consideratwoplayergamewiththefollowingpayoffstructure:
2,10,0 0,01,2 Battle of the Sexes. BALLET BALLET SOCCER SOCCER
Thisgame,referredtoastheBattleoftheSexesgame,representsascenarioinwhichthetwo playerswishtocoordinatetheiractions,buthavedifferentpreferencesovertheiractions.Thisgame hastwopureNashequilibria,i.e.,thestrategyprofiles(Ballet,Ballet)and(Soccer,Soccer).
Example1.12 SecondPriceAuction–withCompleteInformation
Weconsiderasecondpriceauction:Thereisasingleindivisibleobjecttobeassignedtoone of n players.Player i ’svaluationoftheobjectisdenotedby vi .Weassumewithoutlossofgenerality that v1 ≥ v2 ≥···≥ vn > 0 andthateachplayerknowsallthevaluations v1 ,...,vn ,i.e.,itisa completeinformationgame.4 Therulesofthisauctionmechanismaredescribedasfollows:
•Theplayerssimultaneouslysubmitbids, b1 ,..,bn .
4 Theanalysisoftheincompleteinformationversionofthisgame,inwhichthevaluationsofotherplayersareunknown(or probabilisticallyknown),issimilar.
121.STATICGAMESANDSOLUTIONCONCEPTS
•Theobjectisgiventotheplayerwiththehighest bid(ortoarandomplayeramongtheones biddingthehighestvalue).
•Thewinnerpaysthe second highestbid.
Thismechanisminducesagameamongtheplayersinwhich thestrategyofeachplayerisgivenby herbid,andherutilityforabidprofileisgivenbyhervaluationoftheobjectminusthepriceshe pays,i.e.,ifplayer i isthewinner,herutilityis vi bj where j istheplayerwiththesecondhighest bid;otherwise,herutilityiszero.
Wefirstshowthatthestrategyprofile (b1 ,..,bn ) = (v1 ,..,vn ) isaNashequilibrium.First notethatifindeedeveryoneplaysaccordingtothisstrategyprofile,thenplayer1receivestheobject andpaysaprice v2 .Hence,herpayoffwillbe v1 v2 > 0,andallotherpayoffswillbe0.Now, player1hasnoincentivetodeviatesinceherutilitycannotincrease.Similarly,forallotherplayers i = 1,inorderforplayer i tochangeherpayoff,sheneedstobidmorethan v1 ,inwhichcaseher payoffwillbe vi v1 < 0.Therefore,noplayerhasanincentivetounilaterallydeviate,showingthat thisstrategyprofileisaNashequilibrium.
Wenextshowthatthestrategyprofile (v1 , 0, 0,..., 0) isalsoaNashequilibrium.Asbefore, player1willreceivetheobject,andwillhaveapayoffof v1 0 = v1 .Usingthesameargumentas before,weconcludethatnoneoftheplayershaveanincentivetodeviate,andthisstrategyprofileis aNashequilibrium.Weleavethisasanexercisetoshowthatthestrategyprofile (v2 ,v1 , 0, 0,..., 0) isalsoaNashequilibrium.
Sofar,wehaveshownthatthegameinducedbythesecondpriceauctionhasmultipleNash equilibria.Wefinallyshowthatforeachplayer i ,thestrategyofbiddinghervaluation,i.e., bi = vi , infact,weaklydominatesallotherstrategies.Givenabidprofile,let B ∗ denotethemaximumofall bidsexcludingplayer i ’sbid,i.e.,
Assumethatplayer i ’svaluationisgivenby v ∗.Figure 1.3 illustratestheutilityofplayer i asafunction of B ∗ ,whenshebidshervaluation, bi = v ∗ ,lessthanhervaluation, bi <v ∗ ,andmorethanher valuation bi >v ∗ .Inthesecondgraph,whichrepresentsthecasewhenshebids bi <v ∗ ,noticethat whenever bi ≤ B ∗ ≤ v ∗ ,player i receiveszeroutilitysinceshelosestheauctiontowhoeverbid B ∗ Ifshewouldhavebidhervaluation,shewouldhavepositiveutilityinthisregion(asdepictedinthe firstgraph).Thesefiguresshowthatforeachplayer,biddingherownvaluationweaklydominates allherotherstrategies.
AnimmediateimplicationoftheprecedinganalysisisthatthereexistNashequilibria(e.g., thestrategyprofile (v1 , 0, 0,..., 0))thatinvolvetheplayofweaklydominatedstrategies.
GiventhatNashequilibriumisawidelyusedsolutionconceptinstrategicformgames,a naturalquestioniswhyoneshouldexpecttheNashequilibriumoutcomeinastrategicformgame. Onejustificationisthatsinceitrepresentsasteady statesituation,rationalplayersshouldsomehow reasontheirwaytoNashequilibriumstrategies;thatis,Nashequilibriummightarisethrough
1.2.SOLUTIONCONCEPTS13
Figure1.3: Theutilityofplayer i asafunctionof B ∗ ,themaximumofallbidsexcept i ’sbid.
introspection.Thisjustificationrequiresthatplayersarerationalandknowthepayofffunctionsof allplayers,thattheyknowtheiropponentsarerationalandknowthepayofffunctions,andsoon.A secondjustificationisthatNashequilibriaareself-enforcing.Thatis,ifplayersagreeonastrategy profilebeforeindependentlychoosingtheiractions,thennoplayerhasanincentivetodeviateifthe agreedstrategyprofileisaNashequilibrium.AfinaljustificationoftheNashequilibriumoutcome isthroughlearningdynamicsorevolution,whichwillbediscussedinChapter 2.
1.2.3.1MixedStrategyandMixedStrategyNashEquilibrium RecallthegameofMatchingPennies:
1, 1 1,1 1,11, 1 Matching Pennies. HEADS HEADS TAILS TAILS
ItiseasytoseethatthisgamedoesnothaveapureNashequilibrium,i.e.,foreverypurestrategy inthisgame,oneofthepartieshasanincentivetodeviate.However,ifweallowtheplayersto randomize overtheirchoiceofactions,wecandetermineasteadystateoftheplay.Assumethat player1picks Heads withprobability p and Tails withprobability 1 p ,andthatplayer2picks both Head and Tail withprobability 1 2 .Then,withprobability 1 2 p + 1 2 (1 p) = 1 2 player1willreceiveapayoff1.Similarly,shewillreceiveapayoff-1withthesameprobability.This impliesthatifplayer2playsthestrategy ( 1 2 , 1 2 ),thennomatterwhatstrategyplayer1chooses,
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The Project Gutenberg eBook of Kommunistija bolshevikkipakinoita
This ebook is for the use of anyone anywhere in the United States and most other parts of the world at no cost and with almost no restrictions whatsoever. You may copy it, give it away or re-use it under the terms of the Project Gutenberg License included with this ebook or online at www.gutenberg.org. If you are not located in the United States, you will have to check the laws of the country where you are located before using this eBook.
Title: Kommunisti- ja bolshevikkipakinoita
Author: Ilmari Kivinen
Release date: January 16, 2024 [eBook #72733]
Language: Finnish
Original publication: Helsinki: Kust.Oy Kirja, 1928
Credits: Juhani Käkkäinen and Tapio Riikonen *** START OF THE PROJECT GUTENBERG EBOOK KOMMUNISTI- JA BOLSHEVIKKIPAKINOITA ***
OHJELMA JA TYÖJÄRJESTYS
PÖLLÖLÄN KYLÄN
KANSANKÄRÄJILLE
Pöytäkirja, pidetty Hölmölän pitäjän Pöllölän kylän kansankäräjäin valmistuslautakunnan kokouksessa Iso-Hölön pirtissä viime sunnuntaina klo 8 a.p. Läsnä oli 11 henkilöä sekä Sinkkosen akka, ynnä Pussisen poika, joka makasi uunilla, jota ei merkitty pöytäkirjaan.
* * * * *
1:si§:llä:
Hyväksyttiin, että Pöllölän kylän kansankäräjät kokoutuvat tänä päivänä klo 3 i.p. tässä Ison-Hölön pirtissä, koska ei suutari Näppinen päässyt eduskuntaan, vaikka sai 18:toista äänilippua.
* * * * *
2:nenpyk.
Merkittiin, että kansankäräjäin kanslia on tuolla kyökissä, mutta pitää piika-Reetan siivota sitä ennen tiskit pois pöydältä ja pyyhkiä pöytä.
Kysymyksen johdosta, onko oltava myöskin piikakirjoituskanslia, tiedusteli lautakunnan puheenjohtaja lois Mikko Tarjus, osaako Reeta kirjoittaa. Kun Reeta sanoi, ettei oikein muuten kuin mallin jälkeen, ja että Sinkkoskan pennut olivat hävittäneet mallin, päätettiin, ettei tarvitse olla piikakirjoituskansliaa.
Sinkkoska sanoi, että eikö ne lie olleet Reetan omat pennut, mikä merkittiin pöytäkirjaan.
Sihteeri kysyi, että oliskos tästä asiasta vielä muuta, johon kokous yksiäänisesti vastasi, että liekkös tuossa sitten muuta.
Israel Huttunen ilmiantoi, että Helsingin eduskunnassa on ravintolakin, johon Reeta sanoi, että kyllä hän pitää kahvipannun tulella, mutta pitäisi olla jokaisella omat sokerit, mikä hyväksyttiin.
Kysyttiin, että kutka ovat oikeutetut edustamaan Pöllölän kyläläisiä kansankäräjillä ja hyväksyttiin ilman äänestystä, että tulkoot ne, jotka töiltään joutavat.
* * * * * Kolmas§.
Keskusteltiin ja päätettiin kansankäräjäin avajaisohjelma kuin seuraa.
Että kansankäräjät kokoutuvat aika ja paikka kuin yllä.
Että aluksi lauletaan yksiäänisesti pelimannin sävellystä »Voi minua poika raukkaa».
Että kokouksen avaa lautakunnan puheenjohtaja lois Mikko Tarjus ilman ikämiespuhetta ja käskee valitsemaan suutari Näppisen puhemieheksi ja itsensä ensimmäiseksi ja Israel Huttusen toiseksi varapuhemieheksi, mikä hyväksyttiin.
Että, sitten kun puhemiesmiehistö on valittu ja suutari Näppinen istunut puhemiespaikalle
jättää tämän valmistuslautakunnan esimies Mikko Tarjus Issakaisen rengin seuraamana kansankäräjille kertomuksen valmistuslautakunnan toiminnasta sekä tämän pöytäkirjan, kuin myöskin lautakunnan esitykset kansankäräjille.
* * * * *
4:s§:lä.
Israel Huttunen ilmineerasi, että kun Mikko Tarjus ja Issakaisen renki tulevat kyökistä jättämään pöytäkirjan ja esitykset, niin pitää kansankäräjien nousta seisomaan, mikä hyväksyttiin yksinkertaisella äänten enemmistöllä.
Sinkkosen akka sanoi, ettei se ole kyökki, vaan kanslia, mikä merkittiin pöytäkirjaan.
* * * * *
5§:lä.
Päätettiin, että äänestykset kansankäräjillä toimitetaan avonaisella lippuäänestyksellä, joka on oleva sinivalkoinen lippu, mutta jos sitten vaaditaan huutoäänestystä, niin on se toimitettava.
Sinkkosen akka käski merkitä pöytäkirjaan, että jos hän ei saa äänestää punaisella lipulla, niin saa olla äänestämättä.
Kokous hyväksyi yksimielisellä ääntenenemmistöllä, että pyydetään kansakoululta lippu lainaksi lippuäänestyksiä varten.
* * * * *
6:uudes§.
Merkittiin pöytäkirjaan, että kansankäräjät valitsevat seuraavat valiokunnat:
perustuslakivaliokunnan;
kielikysymysvaliokunnan; sotilasvaliokunnan;
kirkko- ja kouluvaliokunnan
sekä tupakkavaliokunnan, jos osuuskauppaan tulee huomenna tupakkoja, mikä hyväksyttiin.
* * * * *
7:mäs§:lä.
Sihteeri sanoi, että pitäisi lopettaa tämä kokous, koska pöytäkirjapaperi rupeaa loppumaan, johon kokous sanoi, että olet
tainnut kirjoittaa liian suuria puustaimia, etkä olisi tarvinnut kirjoittaa kaikkia päätöksiä.
Sihteeri vastasi, ettei hän ole kirjoittanut puoliakaan, mikä hyväksyttiin.
Puhemiehen välikysymykseen, lopetetaanko kokous, vastattiin huutoäänestyksellä, että lopetetaan vain.
Israel Huttunen pani vastalauseen sitä vastaan että Pussisen poika oli ottanut osaa uunin päältä huutoäänestykseen, vaikka ei ollut vielä ripillä käynyt, mikä merkittiin pöytäkirjaan.
Puhemiehen ehdotuksesta huudettiin Pussisen poika alas.
Pussisen poika sanoi, ettei hän voi tulla alas, kun hänen housunsa ovat pesussa, mikä hyväksyttiin.
Pöytäkirjan tarkastajaksi valittiin allekirjoittanut sihteeri.
Puhemiehen ehdotuksesta kohotettiin kaksinkertainen eläköönhuuto.
Lopuksi laulettiin moniäänisesti »Hiljaa juuri kuin lammen laine».
Aika ja paikka kuin yllä. (1919)
