Dr Thanassis Tiropanis t.tiropanis@southampton.ac.uk

COMP6217SocialNetworkingTechnologies

GameTheoryandSocialNetworks

Thenarrative

• Modellinghowindividualsrespondtoeachothers’ actions

• Predictingbehaviourwhenindividualsinteract

• Predictingbehaviourspreadandevolutioninagroup(nextsession)

• Predictingbehaviourspreadinanetwork(nextsession)

Thenarrative

Modellinghowindividualsrespondtoeachothers’ actions

WhatisaGame• Individualscanactaccordingtotheirself-interestwhenpresentedwithchoices

• Butwhenmorethanoneindividualsinteractwitheachothertheirchoicescanleadtodifferentoutcomes

• Actingaccordingtoselfinterestdoesnotalwaysyieldthemaximumprofitinsuchcases

• Howcanwereasonaboutbehaviour?

• Howcanwepredictoutcomes?

PresentationorExam?• Youandyourpartnerneedtoworkonyourcommonprojectandyourexamatthesametime

• Youneedtomakeachoicebetweenthetwo

• Yourgradeswillbedeterminedbasedonhowwellyoudoonboth

SOURCE: http://www.cs.cornell.edu/home/kleinber/networks-book

WhatisaGame

• Agameistheenvironmentwheresuchinteractionstakeplaceanditconsistsof:– Asetofparticipants:players– Optionsperparticipant:strategies– Benefitperchoiceofoption:payoff

• Payoffscanbebasedonthechoicesnotofoneparticipantbutofallparticipants

• Theyareshowninapayoffmatrix

Prisoner’sDilemma• Twohavebeentakenprisonersandare

questionedbythepolice• Theyarebothguilty• Whenquestionedtheyareofferedtheoptionto

confess– Shouldbothofthemconfesstheywillbeconvictedto

serveinprisonfor5years– Shouldjustoneofthemconfess,theconfessorwillbe

letfree,whiletheotheronewillserve10years– Shouldnoneofthemconfess,theywillbothservea

yearforresistingarrest.

• Prisonerscannotcommunicatewitheachother

Prisoner’sDilemma

ConfessStrategy NotConfessStrategy

ConfessS

trategy

-5, -5 0, -10

Not

ConfessS

trategy

-10, 0 -1,-1

Prisoner’sDilemma

ConfessStrategy NotConfessStrategy

ConfessS

trategy

P1(S,T),P2(S,T) P1(S’,T),P2(S’,T)

Not

ConfessS

trategy

P1(S,T’),P2(S,T’) P1(S’,T’),P2(S’,T’)

Bestresponses• Let’sassumewehaveaplayer1andaplayer2withstrategiesSandTrespectively.– P1(S,T)andP2(S,T)arethepayoffsforeachplayergiventheirstrategies.

• Foraplayer,abestresponseisthebestchoicetheycanmakegivenacertainexpectationofachoicefromtheotherplayer

• GivenachoiceofastrategyTbyplayer2,abestresponseforplayer1isstrategyS,whenforeveryotheravailablestrategyS’– P1(S,T)≥ P1(S’,T)

Strictlybestresponses

• GivenachoiceofastrategyTbyplayer2,astrictbestresponseforplayer1isstrategyS,whenforeveryotheravailablestrategyS’

– P1(S,T)> P1(S’,T)

DominantStrategies• AdominantstrategySforPlayer1isonethatisthebestresponsetoeverystrategyofPlayer2.

• AstrictlydominantstrategySforPlayer1isonethatisthestrictlybestresponsetoeverystrategyofPlayer2

• Thereistheassumptionthatplayershavecomecommonknowledgeofpossiblepayoffsofeachother,etc

ConfessStrategy NotConfessStrategy

ConfessS

trategy

-5, -5 0, -10

Not

ConfessS

trategy

-10, 0 -1,-1

Prisoner’sDilemma

Dominant strategy for 2

Dom

inan

t stra

tegy

for 1

Best outcome for both is out of their dominant strategies

Thenarrative

Predictingbehaviourwhenindividualsinteract

Predictingoutcomes

• Ingameswithstrictlydominantstrategies,weexpectplayerstochosethosestrategies– Thisbasicassumptionhasbeendebatedbutitisabasiconeingametheory

• Ingameswithoutstrictlydominantstrategies,howcanwepredictthechoicesoftheplayers?– SEEEQUILIBRIA

Example- equilibria• Firm1andFirm2arecompetingforclientsA,BandC• Firm1toosmall,Firm2islarge• Theyneedtodecidewhichclienttoapproach

– Iftheyapproachthesameclienttheygethalftheclient’sbusinesseach– IfFirm1approachesaclientonitsowntheywillget0business– IfFirm2approachesBorConitsown,theywillgettheirfullbusiness– Aisalargeclientandwilldobusinessonlywithbothofthemandtheypayoff

willbehigher(4each)– BusinesswithBorCisworth2

SOURCE: http://www.cs.cornell.edu/home/kleinber/networks-book

Example- equilibria

SOURCE: http://www.cs.cornell.edu/home/kleinber/networks-book

• (A,A)istheonlyNashEquilibrium

NashEquilibrium• Inagamewhereplayer1chosesstrategySandplayer2chosesstrategyT,thepairofstrategies(S,T)isaNashEquilibrium if– SisabestresponsetoT,and– TisabestresponsetoS.

• Theexpectationisthatevenwhentherearenodominantstrategies,ifthereareNashequilibria,playerswillchosethestrategiesoftheequilibria

• Thisisbasedonthebeliefthateachpartywillmakethischoice

• ButhowcanwepredictbehaviourwhentherearemorethanoneNashEquilibriainagame?– Andtheyyieldthesamepayoffs?

Is there a Nash equilibrium in the prisoner’s dilemma game?

MultipleEquilibria

• ACoordinationGame– Whatcanyouandyourpartnerchoosewhenpreparingacommonpresentation?KeynoteorPowerPoint?• Weassumethatyoucannotconvertfromonetotheother

SOURCE: http://www.cs.cornell.edu/home/kleinber/networks-book

Two Nash Equilibria:(P, P) (K, K)

MultipleEquilibria:FocalPoints

• Topredictwhichofthemultipleequilibriaplayerswillchoseonecanarguethattherecanbe“naturalreasons” notshowninthepayoffmatrixthatwillcreateabiasforoneequilibrium– Thiswillbeafocalpoint– E.g.ifPowerPointismorefrequentlyusedintheUniversitymaybebothplayerswillchosethisinsteadofKeynote

• Reference:Schelling,T.(1960)AStrategyofConflict.HarvardUniversityPress

MultipleEquilibria

• Anti-coordinationgames:– Hawk-DoveGame– Chicken

Dovestrategy Hawk Strategy

DoveStrategy

3,3 1,5

HawkStrategy

5,1 0, 0

MatchingPennies• WhataboutgameswithnoNashEquilibria?• Twoplayersholdapennyeachandtheydecidewhichsideto

showtoeachothereachtime• Player1loosesher/hispennyiftheymatch• Player2looseshis/herpennyiftheydon’tmatch

HeadStrategy TailStrategy

Head

Strategy

-1,+1 +1,-1TailStrategy

+1,-1 -1,+1

MixedStrategies

• Whentherearenoequilibria(asinthematchingpenniesgame)wecanassignaprobabilityoneachstrategy– E.g.Player1willchooseHeadwithaprobabilityp• andTailwithwithprobability1-p• Player1ischoosingapurestrategyHeadifp=1

MixedStrategiesandEquilibria

• AnequilibriumwithmixedstrategiesisonewhereprobabilitiesofstrategiesforPlayer1isthebestresponsetoaprobabilityofstrategiesbyPlayer2

• Inthematchingpenniesgame,wehaveanequilibriumforprobability½foreachstrategyforeachplayer– Incaseswherepayoffsareless‘symmetric’ equilibriaarebasedonunequalprobabilities

StrategyOptimisation• Purestrategiesvs.Mixedstrategies

– MixedstrategiescanhelpfindadditionalNashequilibriaortheonlyNashequilibria

• Individualoptimisationvs.groupoptimisation– Dominantstrategies,Nashequilibria,focalpointsrefertoindividualoptimisation

– Paretooptimalityandsocialoptimalityrefertogroupoptimisation

ParetoOptimality• Takeachoiceofstrategies;itisPareto-optimalifthereisnootherchoiceinwhichallplayersreceivepayoffsthat– areatleastashigh,and– Atleastoneplayerreceivesastrictlyhigherpayoff

• Itcouldbethatauniquenashequilibriumisnotpareto-optimal;abindingagreementisrequiredtoensurethatapareto-optimalsetofstrategiesischoseninthatcase

Confess NotConfess

Confess

-5, -5 0, -10

Not

Confess

-10, 0 -1,-1Which pairs of strategies are pareto-optimal?

x

vv

v

SocialOptimality• Achoiceofstrategiesbytheplayersthatmaximizesthesumoftheplayers’ payoffs

• IfapairofstrategiesissociallyoptimalisalsoPareto-optimal– Discuss:why?

• Of,course,addingpayoffstoestablishsocialwelfarehastobemeaningful

Which pair of strategies here is socially-optimal?

SOURCE: http://www.cs.cornell.edu/home/kleinber/networks-book

ParetoOptimality• Takeachoiceofstrategies;itisPareto-optimalifthereisnootherchoiceinwhichallplayersreceivepayoffsthat– areatleastashigh,and– Atleastoneplayerreceivesastrictlyhigherpayoff

• Itcouldbethatauniquenashequilibriumisnotpareto-optimal;abindingagreementisrequiredtoensurethatapareto-optimalsetofstrategiesischoseninthatcase

Confess NotConfess

Confess

-5, -5 0, -10

Not

Confess

-10, 0 -1,-1

x

vx

x

Which pairs of strategies are socially-optimal?

MultiplayerGames

• Theycanbeusedtomodelgameswithmorethanoneplayers

• Nashequilibriuminamultiplayergamewithplayers1,…,n– Asetofstrategies(S1,S2,…,Sn)inwhicheachstrategyisthebestresponsetoalltheothers

– Forplayeri,strategySi isabestresponseifforanyotheravailablestrategyS’iPi(S1,…,Si,Si+1,…,Sn)≥Pi(S1,…,S’i,Si+1,…,Sn)

GameTheory&SocialNetworks

• Howdopeopledecidetoestablishconnections?

• ModellingandunderstandingprivacyandtrustinSocialNetworksReference:Buskens.Socialnetworksandtrust.(2002)

• Givenanetworkstructureandthatinteractioncanhappenalongestablishededgeswhatisthebehaviourondifferenttypesofnetworks?

• Discuss:Otherproblems?

ResearchCase

• HawksandDovesinsmall-worldnetworks

• “TheroleofnetworkclusteringoncooperationintheHawk-Dovegame”

• Assumingstaticnetworkstructures

• “Dovelikebehaviourisadvantagedifsynchronousupdateisused”

SOURCE: Tomassini et al. Hawks and Doves on small-world networks. Physical Review E (2006) vol. 73 (1) pp. 016132

able to propagate !having a gain exactly equal to that of theirneighboring doves". The system is thus found locked in aconfiguration of a very high proportion of doves with a sig-nificant number of isolated hawks.

If r!0, lone hawks always have a higher payoff than thedoves in their surroundings and will thus infect one of theirneighbors with its strategy. However for 0"r#0.1, once thepair of hawks is established, their payoff is lower than theone of any of the doves connected to either one them. Evena dove that interacts with both hawks has an average payoffstill greater than what a hawk composing the pair receives.Consequently, when 0"r#0.1, clusters of hawks first startby either disappearing or reducing to single hawks, as previ-ously explained for the r=0 case, but then these lone hawkswill become pairs of hawks. If the updates are done synchro-nously, a pair of hawks will either vanish or reduce back to asingle hawk. One can clearly see that in the long run, hawkswill become extinct. Now if the updates are done asynchro-nously, a pair cannot totally disappear because only oneplayer is updated at a time. However, this mechanism of apair reducing to a single hawk and turning back into a pairagain will cause the small groups of two hawks to moveacross the network and “collide” with each other, forminglarger groups that reduce back to a single-pair hawk forma-tion. Therefore, after a large number of time steps, only avery few hawks will survive.

If we take another look at Figs. 1 and 2, we note thatwhen the population of players is constrained to a latticelikestructure, the proportion of doves is reduced to zero for val-ues of the gain-to-cost ratio greater than or equal to #0.8,whereas this not the case when the topology is a randomgraph. Let us try to give a qualitative explanation of the twodifferent behaviors. The first thing to be pointed out is that,in the case of the replicators dynamics, if a dove is sur-rounded by eight hawk neighbors, it is condemned to die forvalues of r! 79 , whatever the topology may be. However, thisdoes not explain why for these same values, doves no longerexist on square lattices or small worlds but are able to sur-vive on random graphs. If the population were mixing, r=0.8 would induce a proportion of doves equal to 20%.Therefore, let us suppose that at a certain time step, there isapproximately 20% of doves in our population. Furthermore,as pointed out by Hauert and Doebeli $6%, in the Hawk-Dovegame on lattices, the doves are usually spread out and formmany small isolated patches. Thus, we will also suppose20% of doves in the population implies that in a set of play-ers comprising an individual and its immediate eight neigh-bors, there are about two doves. Hence, a D-player has onaverage one dove and seven hawks in its neighborhood. Inthe lattice network, this pair of doves can be linked in twodifferent manners !see Fig. 8", having either two or fourcommon neighbors, thus, an average of three.

More generally, if we denote $ the clustering coefficientof the graph and k̄ the average degree, a pair of doves willhave on average $!k̄−1" common neighbors. Let us denote xone of the two doves composing the pair as Hx, a hawklinked to x but not to the other dove of the pair, and Hx,y, onethat is connected to both doves. If 23 "r"

78 and, assuming

that the hawks surrounding the pair of doves are not inter-

acting with any other doves !this gives the pair of doves amaximum chance of survival", we have

GHx " Gx " GHx,y ,

where G% is the average payoff of player %.Consequently, according to Eq. !1", x can infect Hx, and

Hx,y can infect x.Let us now calculate for what values of r the probability

that x invades the site of at least one Hx is less than an Hx,yinfecting x. To do so, let us distinguish the case of the asyn-chronous updating policy from the synchronous one.

A. Asynchronous dynamics

The probability that an Hx neighbor is chosen to be up-dated and adopts strategy D is given by

!2"

where N is the size of the population, !!" the probability anHx hawk is chosen to be updated !among the N players", !!!"the probability the chosen Hx hawk compares its payoff withplayer x, and finally & is the function defined in Eq. !1".

The probability that x is chosen to be updated and is in-fected by one of the Hx,y hawks is given by

!3"

where !!" is the probability x is chosen to be updated, !!!"the probability it measures itself against an Hx,y neighbor,and & the function defined by Eq. !1".

For a square lattice with a Moore neighborhood !$= 37 andk̄=8", expressions !2" and !3" give us r! 4659 &0.78, whereas

FIG. 8. !Color online" Lattice: two possible configurations.

TOMASSINI, LUTHI, AND GIACOBINI PHYSICAL REVIEW E 73, 016132 !2006"

016132-8

PredictingbehaviourwithGameTheory

• Arethere(strictly)dominantstrategies?• OranyNashequilibria?• IftherearemanyNashequilibriacanwepredictwhichonewillbeachievedbasedonhigherpayoffsorfocalpoints?

• Aretherepareto-optimalpairsofstrategies?– AreNashequilibriaamongthem?Abindingagreementwouldberequiredifnot.

• Isthereasocially-optimalpairofstrategies?

Lessonslearned• UnderstandingofthemainconceptsofGameTheory.Givenapayoffmatrixbeabletoidentifyandexplainbestresponses,dominantstrategies,equilibria,focalpoints,paretooptimality,socialoptimality.

• AbilitytoexplainhowGameTheorycanapplytospecificproblemsinsocialnetworksandoutlinehow.

• Easley,D.andKleinberg,J.NetworksCrowdsandMarkets.CambridgeUniversityPress,2010.http://www.cs.cornell.edu/home/kleinber/networks-book (chapters6and7)