Table Of Content1 Basic Game Theory
Game theory is the study of mathematical modeling of strategic interactions among
agents. Agents, or players in the game, are usually considered rational (i.e., seeking
theirownmaximumbenefitfromthegame’soutcome).Theoutcomesofthegame,or
theconsequencesoftheactionsofagents,dependontheir(strategic)interactionsand
(nonstrategic)environmentalfactors.Thus,arationalplayershouldobserve,analyze,
andpredicttheactionsofotherplayersinthegameinordertoselecttheappropriate
actionthatleadstothemostdesiredoutcome.Ingametheory,themaingoalistopre-
dicttheoutcomeofthegamegiventherationalityofplayersanddefinedenvironments.
Agameiscomposedofatleastthefollowingelements:players,actions,outcomes,
andutilityfunctions.Players,whicharedenotedasN={1,2,3,...,N},aretheagents
who would rationally maximize their utilities in the game. The way for Player i to
maximizethecorrespondingutilityistochoosetheactiona fromtheactionsetA .
i i
An action profile a = {a ,a ,...,a } represents the actions that each player may
1 2 N
selectinthegame.Whentheactionprofileaisdetermined,theoutcomeofthegame
can then be derived by the outcome function O(a). Given the outcome, each player
may receive a utility according to the utility function U (O(a)). In sum, we may
i
describeagameasatupleG={N,A,U,O}.
1.1 Strategic-FormGamesandNashEquilibrium
Thestrategic-formgameisoneofthemostbasicgamestructures,wheretherelation
between outcome and action can be represented in a matrix form. A well-known
strategic-form game is the prisoner’s dilemma. Two players are questioned by the
prosecutorregardingtheevidenceofacrime.Theymaychoosetostaysilentorbetray
theotherforashorterprisonsentence.Theprisoner’sdilemmacanbeexplainedwith
thepayoffmatrixinTable1.1.Inthematrix,eachcellrepresentstheutilityreceived
byPlayersAandB,respectively,iftheychoosetheactionprofile.
In a strategic-form game, we seek the Nash equilibrium, which represents the
expectedactionprofile(i.e.,theoutcomeofthegame)iftheplayersarerational.
2 BasicGameTheory
Table1.1.Prisoner’sdilemma:astrategic-formgame
(AB) StaySilent Betray
StaySilent (−1,−1) (−8,0)
Betray (0,−8) (−5,−5)
definition 1.1 (Nash equilibrium) Nash equilibrium is the action profile a∗ =
{a∗,a∗,...,a∗}∈A that
1 2 N
U (O(a∗,a∗ ))≥U (O(a,a∗ )),∀i ∈N,a ∈A,
i i −i i i −i i i
where ai is the action of Player i and a−i is the action profile of all players except
Playeri.
The concept of the Nash equilibrium states that every player is satisfied with the
selectedactionintheprofilegiventheactionsselectedbyotherplayersintheprofile.
In other words, no player can receive higher utility by changing their own action.
Therefore, rational players have no incentive to change their actions when the game
falls into Nash equilibria. Notice that the Nash equilibrium or pure-strategy Nash
equilibriumwedefinedinDefinition1.1maynotalwaysbeuniqueorevenmaynot
exist.ItisnecessarytoanalyzetheexistenceanduniquenessofNashequilibriainthe
definedgamemodel.
Intheprisoner’sdilemmagameillustratedinTable1.1,readersmayobservethat,
given any action selected by one player, the other player will have higher utility if
they choose to betray. Therefore, {Betray,Betray} is the unique Nash equilibrium.
Given that a Nash equilibrium exists and is unique in the prisoner’s dilemma, we
could predict that the final outcome of the game, if all players are rational, will be
{Betray,Betray}.
One may notice that the best choice for the players, if they cooperate, would be
the {StaySilent,StaySilent}, for the utility of (−1, −1). Nevertheless, this better
outcomeisimpossibleintheprisoner’sdilemma,sincebothplayershavetheincentive
tobetrayforashorterprisonsentence.Thisrationalchoiceeventuallyleadsthemtoa
worseoutcome,whichexplainswhythisisa“dilemma.”Thisexamplesuggeststhat
rationaldecisionsinthegamemayleadtosuboptimaloutcomesfromtheperspective
ofoverallsystemefficiency.
1.2 Extensive-FormGamesandSubgame-PerfectNashEquilibrium
Inastrategic-formgame,playersdonothaveknowledgeoftheactionsselectedbythe
otherplayers.Suchgamesaresuitableforproblemsinvolvingsimultaneousdecision-
making or if the decisions are privately made. For scenarios in which the actions of
players can be observed, either fully or partially, by other players, extensive-form
gameswouldbemoresuitable.
1.2 Extensive-FormGamesandSubgame-PerfectNashEquilibrium 3
Figure1.1 Ultimatumgame
Anextensive-formgamecanberepresentedbyagametree,inwhichtheterminal
nodes are the outcomes of the game with payoffs to the players, and the rest of the
nodes are the decision timings when the players (and/or the nature) may select the
actions(and/orexternalinfluence)todirectthegametowardcertainoutcomes.
Theultimatumgame,whichisillustratedinFigure1.1,isafamousexampleofan
extensive-formgame.Intheultimatumgame,Player1isrequestedtoproposeanoffer
forsharingacake,whilePlayer2canchoosetoacceptorrejecttheoffer.IfPlayer2
acceptstheoffer,thecakewillbeallocatedasis.Nevertheless,ifPlayer2choosesto
rejecttheoffer,neitherplayerwillreceivethecake.
Inthisgame,wemayidentifythreeNashequilibriaaccordingtoDefinition1.1:
(1) Player1offerstofairlysharethecakeandPlayer2onlyacceptswhentheoffer
isfair,or{Fair,Fair|Accept,Unfair|Reject}.
(2) Player 1 offers to unfairly share the cake and Player 2 only accepts when the
offerisunfair,or{Unfair,Fair|Reject,Unfair|Accept}.
(3) Player 1 offers to unfairly share the cake and Player 2 accepts any offer, or
{Unfair,Fair|Accept,Unfair|Accept}.
Thefirstoneistheofferthatwesometimesseewhenpeoplearefacingthesituation
similartoPlayer2;thatis,theywillmakeanultimatumorthreatthattheywilldamage
bothofthemiftheyaretreatedunfairly.Thisiswhythegameiscalledtheultimatum
game. Nevertheless, although such a claim can be supported by a Nash equilibrium,
itmaynotbeasuitablesolutionconceptfortheextensive-formgame,asitdoesnot
considerthefactthatPlayer2alreadyknowswhichactionPlayer1hasselectedatthe
timeoftheirdecision.
The subgame-perfect Nash equilibrium is a refined solution concept for the
extensive-form game. A subgame is a subtree of the game tree starting from any
nodeandincludingallbranchesfollowingthestartingnode.Asubgame-perfectNash
equilibriumisdefinedasfollows:
definition 1.2 (Subgame-perfect Nash equilibrium) A Nash equilibrium is a
subgame-perfectNashequilibriumifitisaNashequilibriumineverysubgame.
4 BasicGameTheory
The concept of the subgame-perfect Nash equilibrium captures the rationality of
players when they know the actions of other players beforehand. In the ultimatum
game,forinstance,Player2alreadyknowswhetherPlayer1hasofferedafairshare
when they choose to accept or not. In both subgames, after Player 1 proposes the
offer, the only Nash equilibrium in the subgames would be Player 2 choosing to
accept the offer, since rejecting will give Player 2 zero utility. With this refinement,
the only subgame-perfect Nash equilibrium would be {Unfair,Fair|Accept,
Unfair|Accept}.
1.3 IncompleteInformation:SignalandBayesianEquilibrium
In some problems, the outcome of the game involves uncertainty. The uncertainty
maycomefromtheexternalfactorsthatcouldnotbeobserveddirectlyortheprivate
preferencesandactionsoftheplayers.Theexactoutcomeofthegamethereforemay
notbeknownatthetimeofthedecision-making.Insuchagame,rationalplayersmust
estimatetheinfluenceofthisuncertainty.Insteadofseekingutilitymaximization,they
aimtomaximizetheirexpectedutility.
LetusconsideragamewithplayersetNandactionsetA.Thegameisinastate
θ ∈ (cid:2),which isunknown tosomeor all players. The utilityof Player i isgiven by
U (a,θ),a∈C,whichdependsontheactionselectedbyPlayeri,theactionsselected
i
bytheotherplayers,andthestateθ.
Theuncertaintyinthestatecanbeestimatedifthedistributionofthestateisknown.
Itcaneitherbeknowninadvance(signalinggame)orlearnedaboutfromthereceived
information (social learning). Let us assume that the probability of the state p (or
i
belief), on state θ over state space (cid:2) is known or derived after strategic thinking.
Playersthenmaymaximizetheirexpectedutilitiesbasedonthebeliefasfollows:
(cid:2)
pi(Ii)={pi,θ|θ ∈(cid:2)}, pi,θ(Ii)=1,
θ∈(cid:2)
whereI istheinformationreceivedbyPlayeri inthegame.
i
Prob(I |θ)
pi,θ(Ii)= (cid:3)θ(cid:6)∈(cid:2)Proib(Ii|θ(cid:6))
Giventhisnewobjectiveoftheplayers,wemayextendtheequilibriumconcepttoa
BayesianNashequilibrium.
definition1.3(BayesianNashequilibrium) ABayesianNashequilibriumisthe
∗
actionprofilea ,where
(cid:2) (cid:2)
pi,θ(Ii)Ui(ai∗,a∗−i,θ)≥ pi,θ(Ii)Ui(ai,a∗−i,θ),∀i ∈N,ai ∈Ai.
θ∈(cid:2) θ∈(cid:2)
Thereputationgameisafamousexampleofagamewithincompleteinformation.
In the game we have two firms. Firm 1 is in the market and prefers a monopoly.
Firm 2 is new and would like to enter the market. There are two possible types of
1.4 RepeatedGamesandStochasticGames 5
Table1.2.Reputationgame
Stay Exit
Sane/Prey (2,5) (X,0)
Sane/Accommodate (5,5) (10,0)
Crazy/Prey (0,−10) (0,0)
Firm 1: Sane and Crazy, each with 0.5 probability. Their actions and corresponding
utilitiesareillustratedinthegamematrixinTable1.2.
Insuchagame,therearetwopossibleBayesianNashequilibria,whiletheirexis-
tencedependsonthevalueofX.
Pooling equilibrium: When X = 8, both Sane and Crazy Firm 1 will choose to
prey.Insuchacase,Firm2hasnowaytodistinguishbetweenthesetwotypes.Given
thedistributionofthetype,Firm2willchoosetoexitinsteadofstay.
Separatingequilibrium:WhenX=2,SaneFirm1willaccommodateandCrazy
Firm 1 will prey. Firm 2 will stay when seeing accommodate and exit when seeing
prey.Inthisequilibrium,Firm2canjudgeFirm1’stypethroughtheobservedaction.
Inotherwords,Firm1’sactioncanbetreatedasasignaltoimprovetheestimationof
theunknowntypes.Therulesofsignalinginincompleteinformationwillbediscussed
indetailinPartIII.
1.4 RepeatedGamesandStochasticGames
Thepreviousgamemodelsarebasedontheassumptionthatthegamewillbeplayed
onlyonce(i.e.,aone-shotgame).Inpractice,agentsmayfacethesameproblemmul-
tipletimes,eachtimewiththesameordifferentplayers.Insuchascenario,wemay
formulatetheproblemasarepeatedgame.Arepeatedgameconsistsofaseriesofbase
gamesinwhichtheplayersplaythesamebasegamesequentially.Itcanbewrittenas
anextensive-formgamebyexpandingthebasegameinagametreerepeatedly.
Therearetwokindsofrepeatedgames:finiteandinfiniterepeatedgames.Forfinite
repeatedgames,thenumberofroundsofthebasegameisfinite.Thissuggeststhatan
ending base game exists, and the game tree in the extensive form is finite. In such a
scenario,thegamecanbeanalyzeddirectlywithintheconceptofasubgame-perfect
Nashequilibrium.Fortheothercase,wheretheroundsareinfinite,thereisnoending
gameandthereforeasubgame-perfectNashequilibriumcannotbeapplieddirectly.
TheutilityofPlayeri inarepeatedgamecanbewrittenasfollows:
(cid:2)T
Ui = lim δtui(ai(t),a−i(t)), (1.1)
T→∞
t=0
where ai(t) and a−i(t) are the actions selected by Player i and the other players at
roundt,respectively,and0< δ <1isthediscountfactorforevaluatingtheutilityin
thefuturetotheplayersinthepresent.
6 BasicGameTheory
Takingtheprisoner’sdilemmaasanexample,wemayconsiderarepeatedversion
oftheprisoner’sdilemma,whichiscalledtheiteratedprisoner’sdilemma.Whenthe
rounds are finite, it can be easily shown that the original {Betray,Betray} equi-
librium still holds as the unique Nash equilibrium of the game. Nevertheless, when
the rounds are infinite, the equilibrium becomes nonunique as cooperation between
playersbecomespossible.Forinstance,atit-for-tatstrategy(i.e.,staysilentinthefirst
round and then choose the action selected by the opponent in the previous round)
is also a Nash equilibrium, since any deviation from such a strategy will lead to
{Betray,Betray}ineveryround,whilecontinuingtousethestrategywillkeepthem
at{StaySilent,StaySilent}andeventuallyleadtohigherutilityinthelongrun.This
examplesuggeststhatcooperationislikelytoemergeinarepeatedgame.Readerswill
findmorerelatedexamplesinPartI.
In practice, even if the agent is facing the same problem and applying the same
action multiple times, the results could be different due to external uncertainty or
randomnessinthesystem.Wemaycapturethischaracteristicwithstochasticgames,
whicharerepeatedgameswithuncertainty.
The uncertainty is captured through adding a state θ ∈ (cid:2) to the game. The state
isobservablebytheplayersbutitmaychangeinthenextroundwiththeprobability
described by P(θ(cid:6)|θ,a), which depends on the current state and the action profiles
selected by the players. The utility of the players depends on not only the action
profile, but also the state. Therefore, the utility of Player i in the stochastic game
canbewrittenasfollows:
Ui = lim ui(θ(0),ai(0),a−i(0))
T→∞
(cid:2)T
+ P(θ(t)|θ(t −1),a(t −1))δtui(θ(t),ai(t),a−i(t)). (1.2)
t=1
Giventhatthestatetransitionsdependonthepreviousstateandtheactionprofile,
thesystemcanbeformulatedasaMarkovdecisionprocesswhentheactionprofileis
given,whichhelpsustoderivetheexpectedutilitygivencertainactionprofiles.Then,
a Bayesian Nash equilibrium can be applied to derive the equilibrium of the game.
ReaderswillfindmoreexamplesinPartsIIandIII.
5 Indirect Reciprocity Data Fusion
Game and Application to Cooperative
Spectrum Sensing
Encouraging sensors to share their data is a crucial issue due to the importance of
data sharing in data fusion. In this chapter, we discuss a reputation-based incentive
framework in which an indirect reciprocity game is used to model the data sharing
stimulation problem. Within this framework, sensors decide on how to report their
resultstothefusioncenter(FC)andearnareputation.Onthebasisofthisreputation,
they are capable of gaining benefits in the future. For accuracy of fusion and sens-
ing,reputationdistributionisintroducedwithinthegame,inwhichthegame’sNash
equilibrium(NE)isderivedandthecorrespondinguniquenessisproventheoretically.
Furthermore,theschemeisappliedtocooperativespectrumsensing.Itisshownthat
with an appropriate cost-to-gain ratio, when the average received energy exceeds
a given threshold, the optimal strategy for the secondary users (SUs) is to report;
otherwise, they would remain silence. It is also verified that this kind of optimal
strategyisadesirableevolutionarilystablestrategy(ESS).
5.1 Introduction
Today, in the big data era, with diversified measurement techniques in various
domains, information about a system of interest or a phenomenon can be acquired
through different types of sensors (as well as data suppliers). In order to extract
knowledge for various purposes and analyze information from multiple sensors
jointly, concepts of data fusion were developed and corresponding techniques were
introduced [1]. The data fusion process’s analytic outcomes enable users to acquire
acomprehensive view and amore united picture of the system, make more accurate
decisions,andanswerquestionsaboutthesystem.
Data fusion methodologies were first created for military applications and then
evolved intononmilitary fields suchas driver assistancesystems[2],cognitive radio
networks(CRNs)[3],andsmartgrids[4].Numerousalgorithmshavebeenproposed
to exploit the diversity of multiple sensors effectively, including those based on soft
fusionandhardfusion.Withinthesoftfusion-basedalgorithms,aquantizedversionof
alocaldecisionstatisticsuchasanysuitablysufficientstatisticsorthelog-likelihood
ratiowouldbesenttotheFC,whilewithinthehardfusion-basedalgorithms,aone-bit
hardlocaldecisionwouldbesentbyeverysensortotheFC.
5.1 Introduction 81
It is verified that the existing fusion algorithms could improve system reliability,
butmostofthemassumethateverysensorisaltruisticandwillingtounconditionally
share data for fusion, which may not be true in reality. Actually, if cooperation is
not capable of bringing benefits to sensors on account of the extra energy and cost
requiredincollectingandreportingdata,thesensorsmightnotchoosethecooperation.
Accordingly,thefusionalgorithmswouldnotrunproperlyduetoalackofsufficient
data.Therefore,stimulatingdifferentdataproviderstocollaborateandsharetheirdata
isacrucialtopic.
As for cooperation stimulation schemes, reputation-based and payment-based
schemes are two most common types. Reputation-based cooperation stimulation
schemes have been broadly discussed within wireless data networks [5], ad hoc
networks [6], and peer-to-peer (P2P) networks [7] to guarantee trustworthy commu-
nications,orinCRNs[8]andwirelesssensornetworks[9]toguaranteedatafusion.
However, few reputation-based schemes have been developed in order to stimulate
sensors to share their sensing results until recently, and this kind of approach has
no theoretical justification yet. Payment-based schemes such as auctions and virtual
currencieshavebeenproposedtoguaranteeparticipatorysensing[10]withinwireless
communicationsystems,filesharingwithinP2Pnetworks[11],anddynamicspectrum
sharing within CRNs [12]. Though these schemes have produced promising results,
they are applied with limitations such as central banking server(s) for the sake of
securitycontrolortherequirementoftamper-proofhardware.
Another way of analyzing the cooperation problem is game theory. Specifically,
a number of cooperation schemes on the basis of cooperative game theory such as
the coalition game [13,14] and the bargaining game [15,16] have been introduced.
The focus of cooperative game-theoretic frameworks is allocating and acquiring net
incomewithinanalliance.Incooperativegames,playershaveaninterestinmaximiz-
ing their outcome, though when considering the benefits on both sides, they might
want to accept a solution that involve bargaining. Thus, a stimulation mechanism
or an enforceable contract is needed to guarantee cooperation. Furthermore, most
game-theoretic frameworks arebasedonthedirectreciprocitymodel inwhichother
players would not evaluate all players’ behaviors, but only their opponents’ behav-
iors.Accordingtothebackwardinductionprincipleandtheprisoner’sdilemma,two
directlyparticipatingplayerswouldcooperateonlywhenrepeatingthegameinfinite
times. However, players could alter their partners periodically to obtain better per-
formance due to mobility or changes of environment; thus, for them, playing non-
cooperatively is the only optimal strategy. In this case, they also need a stimulation
mechanism.
Indirectreciprocityhasbeenbroadlyadoptedwithinsocialscienceandevolution-
ary biology [17,18] and has recently drawn a lot of attention in applications such as
energyexchange[19],packageforwarding[20]andcooperativetransmission[21,22].
Its concept is “I help you not because you have helped me but because you have
helped others,” which means that the current donor could give help to the recipient
sinceithelpedothersasadonorbefore.Thissignifiesthattheevaluationsfromboth
other observers and opponents would be taken into consideration, and this endows
82 IndirectReciprocityDataFusionGameandApplicationtoCooperativeSpectrumSensing
playerswiththeincentivetocooperateeventhoughthegamewouldnotbereplayed
limitlessly.
Within the chapter, first, we discuss a framework that achieves successful data
fusion through incorporating an incentive mechanism on the basis of indirect reci-
procityandstimulatingcooperationamongsensorsthatarenotrequiredtoplaywith
the same group of sensors all of the time. Then, the application to the cooperative
spectrumsensing(CSS)systemwouldbediscussed.
5.2 IndirectReciprocityDataFusionGame
5.2.1 SystemModel
AdatafusionsystemisconsideredasshowninFigure5.1,wherethereareM sensors
sensingandobtaininglocalobservations.Itisassumedthatthelocalobservationsare
obtainedforclassichypothesistests.Presumethatthehypothesescouldbeevaluated
fromtheobservationofwhichsensorreceivesitssignalenergy(i.e.,energydetection).
(cid:3)
Let S = 1 N |r (t)|2 denote the signal’s average energy that is received by
m N t=1 m
sensor m, where N represents the signal sample’ number and r (t) represents the
m
sampleofobservedsignalsatsensorm’sreceiver,whichcouldbesignifiedas
(cid:26)
h s(t)+n (t), ifH ,
r (t)= m m 1 (5.1)
m
n (t), ifH .
m 0
Figure5.1 Thesystemmodel.
