Evidence combination from an evolutionarygame theory perspectiveXinyang Deng, Deqiang Han, Jean Dezert, Yong Deng*, and Yu ShyrAbstractDempster-Shafer evidence theory is a primary methodology for multi-source information fusionsinceit allows todeal withuncertaininformation. Thistheoryis basedonDempstersruleofcombination to synthesize multiple evidences from various information sources. However, in somecases, counter-intuitive results may be obtained based on Dempsters rule of combination. Lots ofimprovedornewmethodshavebeenproposedtosuppressthecounter-intuitive resultsbasedonaphysicalperspectivethatminimizesthelost ordeviationoforiginalinformation. Inthispaper,inspiredbyevolutionarygametheory, abiological andevolutionaryperspectiveisconsideredtostudy the combination of evidences. An evolutionary combination rule (ECR) is proposed to mimickthe evolution of propositions in a given population and nally nd the biologically most supportedproposition which iscalled asevolutionarily stable proposition (ESP) inthispaper. Our proposedECRprovidesnewinsightforthecombinationofmulti-source information. Experimentalresultsshow that the proposed method is rational and effective.Index TermsDempster-Shafer evidencetheory, Belief function, Evolutionarygametheory, Evolutionarilystable strategy, Replicator equation, Multi-source information fusion.X. DengandY. DengarewiththeSchool of Computer andInformationScience, Southwest University, Chongqing,400715,China.X. Deng and Y. Shyr are with the Center for Quantitative Sciences, Vanderbilt University School of Medicine, Nashville,TN, 37232,USA.D. Han is with the Center for Information Engineering Science Research, School of Electronic and InformationEngineering, Xian Jiaotong University, Xian, 710049,China.J. Dezert is with ONERA The French Aerospace Lab, F-91761 Palaiseau, France.Y. Deng is with the School of Engineering, Vanderbilt University, Nashville, TN, 37235,USA.*Correspondingauthor: Y. Deng, School of Computer andInformationScience, Southwest University, Chongqing,400715,China. E-mail: prof.deng@hotmail.com; ydeng@swu.edu.cn.March 10, 2015 DRAFTI. INTRODUCTIONMulti-source information fusion aims to integrate multiple data and knowledge from mul-tiple information sources into a consistent, comprehensive, and useful estimation for objectsthat beyondtheperformanceofsingleinformationsource. Manyissuesindisciplinescanbe translated to canonical multi-source information fusion problems. Because of that, multi-source information fusion is extensively used in many elds [1][3]. A crucialissue in themulti-source information fusion is that how to represent and dispose the imprecise, fuzzy, am-biguous, even inconsistent and incomplete information. Several branches of theories have beenproposed for the processing of uncertain information. Typically, there are fuzzy set theory [4],rough set theory [5], [6], possibility theory [7], and so forth. Among these theories, Dempster-Shafer evidence theory [8], [9] has attracted increasing interest from scientic communitiesbecause of its inherent advantages in representing and handling uncertain information [10][17].As atool of reasoninginuncertainenvironment, Dempter-Shafer evidencetheoryhasconceived a rounded system for the management of uncertainty. In this theory, the data fromeach information source is represented as a mass function, or called as evidence, Dempstersrule of combination is provided to combine multiple evidences for the fusion of multi-sourceinformation. However, thecombinationruleis controversial. Eventhoughit is of manydesirable characteristics, such as commutativity, associativity, and fast and clear convergencetowardsasolution, but insomecases, especiallywhentheevidencestobecombinedarehighlyconicting, Dempstersruleofcombinationmaybringout counter-intuitiveresults.Typical examples contains Zadehs paradox[18], evidenceshifting[19], [20], dictatorialpower of Dempsters rule [21], [22], and so on.For the reasons of producing counter-intuitive results, there exists many debates, which leadto various alternatives of Dempsters rule or new methods for the combination of evidences,forexample conjunctive rule [23],disjunctive rule[24],cautious conjunctive ruleand bolddisjunctive rule [25], and so on [20], [26][36]. Regarding the debate and improvement aboutDempsters ruleofcombination [37][43],oneshold thatDempsters ruleisinadequatesothat they modify the original rule or propose new rules to synthesize various information, onesdefend Dempsters rule and advocate the modication or preprocess of original evidences [21].Corporately, both of them are expected to bring out an exactly synthetic evidence which canbest represent the system consisting of all original evidences. At present, this is an traditionaland mainstream view. During the process of information fusion, a primary hidden criterion isMarch 10, 2015 DRAFTto minimize the lost or deviation between the information included in the synthetic evidenceand that contained in original evidences. Essentially, this is a typical perspective of physicsor engineering.Instead, inspiredbyevolutionarygametheory, inthis paper we use a biological andevolutionary perspective to study the information fusion. The process of combining evidencesto obtain the most acceptable conclusion is compared to the evolution of species to nd theindividuals with the highest tness who can survive in a population. During this process, evo-lutionary game theory [44], [45] provides a theoretical framework. Specically, the supportedpropositions in evidences are treated as strategies adopted by individuals in a given population.First, by interacting with others, each individual receives payoff which determines its tnessin the population. Then, individuals with high tness have more chances to reproduce so as toincrease their rate in the population. Finally, through the evolution of population, individualswith the highest tness survive, and the strategy adopted by them, i.e. proposition, wins outtobecomethemost supportedoracceptableconclusion. Theproposed methodis calledasevolutionary combination rule (ECR). Here, even it is named as a rule, the ECR actually isnot a traditional combination rule which is expected to obtain a synthetic evidence, but justwant tondthemostsupportedpropositionsinthegivenmulti-evidencesystem. Becauseitcandirectlyndtheobjectwhichhasthehighesttness, potentially, theproposedECRprovides a fast decision-making support for evidence-based multi-source information fusionwithout relying on a transformation function between evidence and probability distribution.In the rest of this paper, it is organized as follows. Section II gives a brief introduction toDempster-Shafer evidence theory and evolutionary game theory. In Section III, the proposedECR is presented, which mainly contains ve steps including evidence weighted averaging,construction of Jaccard matrix game, evolutionary dynamics, evolution of averaging evidence,and two-dimensional measure output. After that, some illustrative examples are given to showtheeffectivenessoftheproposedmethodinSectionIV. Finally, SectionVconcludesthispaper.II. PRELIMINARIESA. Dempster-Shafer evidence theoryDempster-Shaferevidencetheory[8], [9], alsocalledDempster-Shafer theory(DST)orevidence theory, is used to handle uncertain information. This theory needs weaker conditionsthan the Bayesian theory of probability, so it is often regarded as an extension of the Bayesiantheory see discussion in [22]. As a theory of reasoning under the uncertain environment,March 10, 2015 DRAFTDST has an advantage of directly expressing the uncertainty by assigning the probabilityto the subsets of the set composed of multiple objects, rather than to each of the individualobjects.Theprobability assignedtoeachsubset islimited byalowerboundandanupperbound, which respectively measure the total belief and the total plausibility for the objects inthe subset. DST offers a method to combine distinct and cognitively independent bodies ofevidence. Recently, DST has attracted many concern from various elds, such as parameterestimation [46][48], classication and clustering [49][51], decision-making [52][56], etc[57][61]. As an extension of this theory, Dezert-Smarandache Theory (DSmT) of plausibleand paradoxical reasoning [62][64] is also given much attention [36], [65][67].For completeness of theexplanation, afewbasicconcepts onDSTareintroducedasfollows.Let be a set of mutually exclusive and collectively exhaustive events, indicated by = {E1, E2, , Ei, , EN} (1)where set is called a frame of discernment (FOD). The power set of is indicated by2,namely2= {, {E1}, , {EN}, {E1, E2}, , {E1, E2, , Ei}, , } (2)Theelementsof 2orsubset of arecalledpropositions. Forexampleif A2, Aiscalled a proposition.ForaFOD={E1, E2, , EN}, amassfunctionisamappingmfrom2to[0, 1],formally dened by:m : 2 [0, 1] (3)which satises the following condition:m() = 0 and

A2m(A) = 1 (4)InDST, amassfunctionisalsocalledabasicprobabilityassignment (BPA), or abelieffunction, or a piece of evidences. The assigned probability m(A) measures the belief exactlyassigned to A and represents how strongly the evidence supports A. If m(A) > 0, A is calleda focal element, and the union of all focal elements is called the core of the mass function.Associatedwitheachmassfunctionisthebeliefmeasureandplausibility measure, BelfunctionandPl functionrespectively. ForapropositionA, thebelieffunctionBel :2 [0, 1] is dened asBel(A) =

BAm(B) (5)March 10, 2015 DRAFT01Min belief level of A Max belief level of A1Pl(A)Bel(A) The level of ignorance in A 1-Pl(A)Fig. 1. The relationship of belief and plausibility.The plausibility functionPl : 2 [0, 1] is dened asPl(A) = 1 Bel( A) =

BA=m(B) (6)whereA = \ A.Obviously, Pl(A) Bel(A), these functionsBel andPl are the lower limit function andupper limit functionof theprobabilitytowhichpropositionAissupported, respectively.According to Shafers explanation [9], the difference between the belief and the plausibilityof apropositionAexpressestheignoranceof theassessment for thepropositionA. Therelationship of belief and plausibility is shown in Fig. 1.Consider two independent pieces of evidence characterized by their BPAsm1andm2onthe FOD , in DST, m1 and m2 are combined with Dempsters rule of combination, denotedbym = m1 m2, dened as follows:m(A) =___11K

BC=Am1(B)m2(C) , A = ;0 , A = .(7)withK=

BC=m1(B)m2(C) (8)whereKis a normalization constant, called conict coefcient between the two BPAs. Notethat Dempsters rule of combination is only applicable to such two BPAs which satisfy thecondition K< 1. Dempsters rule of combination is the core of DST, it satises commutativeand associative properties, i.e., (i)m1 m2=m2 m1and (ii)(m1 m2) m3=m1 (m2 m3). Thusifthereexist multiplebeliefstructures, thecombinationofthemcanbecarried out in a pairwise way with any order. The vacuous BPA m() = 1 is neutral elementof Dempsters rule.March 10, 2015 DRAFTB. Evolutionary game theoryEvolutionary game theory (EGT) [44], [45] has been developed by John Maynard Smith tostudy the interaction among different players or populations. In recent years, EGT has becomea paradigmatic framework to understand the emergence and evolution of cooperation amongunrelatedindividuals[68][81]. ThemainideaoftheEGTistotrackthechangeofeachstrategys frequency in the population during the evolutionary process. In EGT, evolutionarilystable strategy (ESS) [44] and replicator equation (RE) [82], [83] are the two central and keyconcepts.1) Evolutionarily stable strategy (ESS): In a given environment, an ESS is such a strategy,adopted by a population, that can not be invaded by any other alternative strategy which isinitially rare. The condition required by an ESS can be formulated as [44], [45]:E(S, S) > E(T, S), (9)orE(S, S) = E(T, S), and E(S, T) > E(T, T), (10)for all T=S, wherestrategySis an ESS, Tis an alternative strategy, andE(T, S) is thepayoff of strategyTplaying against strategyS.An evolutionarily stable strategy biologically means that it can not be successfully invadedby any mutant strategies. In game theory, there are two kinds of strategies, pure strategies andmixed strategies. A pure strategy denes an absolutely certain action or move that a playerwill play in every possible attainable situation in a game. In contrast, a mixed strategy is anassignment ofaprobabilitytopurestrategiessoastoallowaplayertorandomlyselectapure strategy. In other words, in a mixed strategy, there are two or more pure strategies thatcan be selected by chance. If the evolutionarily stable strategy (ESS)Sis a pure strategy, Sis called a pure ESS. On the contrary, onceSis a mixed strategy,Sbecomes the so-calledmixed ESS. In [45], it has been proven that a game with two pure strategies always has anESS (pure ESS or mixed ESS).Hawk-Dove game is a classical and paradigmatic example of pure and mixed ESS. Assumethere is a population of animals, in which each individual aggressiveness is different duringtheinteraction with others. Accordingly, their behaviors canbedivided into two types: theaggressive type and the cooperative type. The aggressive type corresponds to strategy Hawk(H), thecooperativetypeisassociatedwithstrategyDove(D). Withineachinteraction,two animals meet and compete for a resourceV(V> 0). When two Hawks meet, they willMarch 10, 2015 DRAFTghtsothatbothofthemhavetheopportunitytoget (V C)/2, whereCisthecost ofinjury in the ght. When two Doves meet, they will sharethe resource,which means eachindividual obtainsV/2. If, however, aHawkmeetsaDove, theformerwill ght andthelatter can only escape. As a result, the Hawk obtains the entire resource without any cost ofinjury, theDove isleftwithnothing. Inthis sense,thepayoffmatrix ofHawk-Dove gameis shown in Eq. (11).A =H DHD___(VC)/2 V0 V /2___(11)Here, E(H, H) =(VC)/2, E(H, D) =V , E(D, H) =0, E(D, D) =V/2, whereE(s1, s2)isthepayoff of s1playingagainst s2. IntheHawk-Dovegame, therearetwopure strategies, namelyHandD, and innumerable mixed strategiesqH + (1 q)Dwhereq(0, 1). AccordingtotheconditionofESS, inthisgame, strategyHisapureESSifV> CbecauseE(H, H) > E(D, H). Conversely, ifV< C, the mixed strategyp= V/Cwhich means that H and D are respectively selected with probability pand 1p, is a mixedESSbecause, p(0, 1)andp=p, E(p, p)=E(p, p)andE(p, p)>E(p, p), whereE(p, p) =V2 (1VC), E(p, p) =V2 (1VC), E(p, p) =V2 (1+VC2p), E(p, p) =V2 (1CV p2).It isnotedthat thepayoffofmixedstrategyp=(p, 1 p)playingagainst mixedpayoffq = (q, 1 q) is calculated by pAq.2) Replicator equation (RE): In EGT, the replicator equation (RE) [82], [83] plays a keyroletodeterminetheevolutionaryprocessofpopulation,whichhasprovidedafrequency-dependent evolutionary dynamics to a well-mixed population.Assume there existsn strategies in a well-mixed population. A game payoff matrixA =[aij] determines the payoff of a player with strategyi if he meets another player who carriesout strategyj. The tness of strategyi is dened by:fi=n

j=1xjaij, i = 1, , n (12)wherexjis the relative frequency of strategyjin the population. The average tness of allstrategies is denoted as, which is dened by: =n

i=1xifi(13)Therelativefrequencyofstrategyi, namelyxi, ischangedwithtimebythisfollowingdifferential equation:dxidt= xi(fi ), i = 1, , n (14)March 10, 2015 DRAFTEq.(14) is the so called replicator equation, which implies that the change ofxidependson the tness of strategy i andxi itself. By solvingdxidt= 0, i = 1, , n, the xed points ofthis evolutionary system, denoted as(x1, , xn), can be found. The stability of each xedpoint is the most focus of concern. Regarding the stability of each xed point(x1, , xn),a theorem is usually used to verify whether the xed point is stable or not, which is givenas below.Theorem 1: [83] Given a set of replicator equationsdxidt= xi(fi), i = 1, , n, thexedpoint p=(x1, , xn)isasymptotically stableifall eigenvaluesassociatedwithpare negative numbers or have negative real parts.For moredetails onTheorem1, pleaserefer to[83]. Sincethereplicator equationisnonlinear, an exact solution is difcult to nd. Hofbauer and Sigmund have proved that eachESS ofA is denitely an asymptotically stable point of the replicator equation [83]. So theESSs are often taken as the asymptotically stable solutions of the replicator equation.III. EVOLUTIONARYCOMBINATIONRULE (ECR)The evolution of species abides by natural selection which provides an excellent mechanismto nd individuals with higher tness. Analogously, information fusion aims to nd the mostsupportedpropositionbyroundlysynthesizinginformationcomingfrommultiplesources.If theinformationis expressedbythemeans of probabilitydistributions, thestatewiththebiggest probabilityis always concernedintensely. InDST, sincetheintroductionofsubjective or epistemic uncertainty, the mass function, also called evidence or basic probabilityassignment (BPA), is employed to represent the uncertain information.Traditional solutionof combiningevidences aims toobtainamass functionthat bestsynthesizes all information. Then, based on the obtained synthetic evidence, a decision canbe made. In this paper, instead of seeking the best synthetic evidence, our purpose is to ndthe best supported proposition (of course, it can also be seenas an evidence who only hasa focal element), analogous to nd the most probable state in probability theory. Obviously,it isnot rational that simplylet thepropositionwiththebiggest belief bethecandidate,because this approach ignores the interaction between propositions. Natural selection and theconcept of tness have inspired us consider this problem from a biological and evolutionarystandpoint.Inthispaper, weproposeanewmethodtocombineevidencescomingfromdifferentinformationsourcesbasedonevolutionary gametheory.Theproposedmethodiscalledasevolutionarycombinationrule(ECR). IntheECR, propositionsintheFODarenaturallyMarch 10, 2015 DRAFT1. Evidence weighted averagingEvidence m1with weight w12. Construction of Jaccard matrix game4. Evolution of averaging evidence3. Evolutionary dynamics5. Two-dimensional measure outputEvidence miwith weight wiEvidence mnwith weight wn inputFrame of discernment (, J)dm/dtAveraging evidence mAveraging evidence mESPmEvolutionary combination rule (ECR)tFig. 2. Framework of the proposed evolutionary combination ruleseenas strategies that canbe adoptedbya populationgivenanenvironment. Throughanevolutionaryprocess of strategies, propositions withthehighest tness canbefoundsoastofacilitatethesubsequent analysisanddecision. Fig. 2showsaframeworkoftheproposed ECR. Mainly, there are ve steps including evidence weighted averaging, construc-tion ofJaccardmatrix game, evolutionary dynamics, evolution ofaveraging evidence, two-dimensional measure output, which will be detailed in the following content of this section,respectively. Most notably, intheproposedECR, therearetwobasicproblems: (i) whatare the interactive relationships between propositions (strategies) ? (ii) how do propositionsevolve in a population? These questions will also be answered in the following content.A. Evidence weighted averagingGiven multiple evidences from different information sources, in accord with the traditionalassumption, these evidences are mutually independent. For these independent evidences, byconsidering the difference of importance among information sources, the weighted averagingMarch 10, 2015 DRAFTapproach is used to integrate multiple evidences.Given aFOD, assume therearenevidencesorBPAsindicatedbym1, , mn,eachevidence has a weighting factor indicated by wi,n

i=1wi = 1. the averaging evidence is denotedasm, which is obtained bym(A) =n

i=1wimi(A), A . (15)B. Jaccard matrix game (JMG)In the ECR, propositions, subsets of the FOD, are treated as strategies. A key problem is todene the interaction rule between these propositions. Given a FOD, suppose propositionA meets proposition B, how many payoff will be obtained for A, and for B as well? To solvethis issue, in this paper the similarity degree between sets is used to express the payoffs ofpropositions since propositions are represented as sets. This idea is based on that an individualcanobtain more benetif heinteracts with individuals being similar with him. Inbiology,thereisaso-calledgreenbeardeffect whichshowsthat cooperativebehaviorsmorelikelyappear between individuals with similar phenotypes [84]. Therefore, each proposition obtainsmore if similar propositions meet, but less if not similar. Specially, in a pair interaction, eachcan obtain 1 if these two propositions are identical, conversely 0 if they are totally different.Mathematically, Jaccardsimilaritycoefcient givesthesimilaritydegreebetweentwosets[85]. Inthispaper, aJaccardmatrixgame(JMG)isproposedtoformalizetheinteractionrelationship between propositions. The denition of the JMG is given as follows.Denition 1: Given a FOD, a Jaccard matrix game (JMG) on is dened as = (, J) (16)where the set of strategies is composed by propositions of the FOD (i.e., the nonempty subsetsof),the payoff matrix isJ=[J(A, B)]A,Bin whichJ(A, B) represents the payoffofpropositionAplayingagainstpropositionB, anditisdenedbytheJaccardsimilaritycoefcient between setsA andB, i.e.,J(A, B) =|AB||AB|.Next, an example is given to show the JMG.Example1: GivenaFODwithtwoelements 1={a, b}, aJMGon1, denotedas(1, J1), can be constructed. In this game, the set of all potential strategies is {a, b, ab} (forMarch 10, 2015 DRAFTthe sake of presentation, set {x, y} is abbreviated asxy), and the payoff matrix is calculatedeasilyJ1=a b ababab__1 0 1/20 1 1/21/2 1/2 1__(17)Similarly, givenaFODwiththreeelements 2={a, b, c}, anewJMG(2, J2) isobtained, where the set of strategies is {a, b, c, ab, ac, bc, abc}, and the payoff matrix is shownas follows.J2=a b c ab ac bc abcabcabacbcabc__1 0 0 1/2 1/2 0 1/30 1 0 1/2 0 1/2 1/30 0 1 0 1/2 1/2 1/31/2 1/2 0 1 1/3 1/3 2/31/2 0 1/2 1/3 1 1/3 2/30 1/2 1/2 1/3 1/3 1 2/31/3 1/3 1/3 2/3 2/3 2/3 1__(18)Again, for each propositionA in DST, it is equivalent to a pure strategy in a JMG.Forexample, given aFOD={a, b},therearethreepropositionsa, bab, eachis apurestrategies in the JMG(, J). All propositions consist of the set of pure strategies adoptedbyapopulation. Moreover, inaJMG, apurestrategycorrespondstoapropositionwhichalso can be seen as a mass function who only has a focal element, and a mixed strategy isassociated with a set of propositions which is a mass function with multiple focal elements.As shown in the above example, a JMG can be constructed if a FOD is given. Formally, thecomplication of a JMG is determined by the size of the given FOD. However, regardlessof the size of, some self-evident corollaries about the JMG are obtained.Corollary 1: A JMG(, J) is symmetric, namelyJ(A, B) = J(B, A).Corollary 2: LetESSbe the set of all ESSs (evolutionarily stable strategies) in a JMG(, J), thenESS= {A|A , A = }Corollary 2 shows a one-to-one correspondence between pure strategies and ESSs in JMGs.Due to the equivalence between propositions and pure strategies in JMGs, in this paper wealso call the evolutionarily stable strategy (ESS)as evolutionarily stable proposition (ESP).Assume the set of ESPs in a JMG is indicated byESP, then we haveESP ESS.March 10, 2015 DRAFTC. Replicator dynamics on the JMGOnce the interactionrelationshipbetweenpropositions has beendetermined, the nextproblemis that howthepropositionsevolve. Inother words, what kindof evolutionarydynamics is adopted in this population? To address this issue, the replicator equation is used tomimick the evolutionary process of population based on two reasons. At rst, it commendablysimulates the evolution of population in a well-mixed environment where individuals couldrandomlyinteract withother membersof thepopulations. Moreimportant, thereplicatorequation is mathematically equivalent to the Lotka-Volterra equation of ecology [83] whichdescribesthedynamicsofspeciesinaninteractingbiological system. TheLotka-Volterraequation is very close to real facts since it does not rely on the rationality assumption whichis fundamental but challengedintheclassical gametheory. So, theequivalent replicatorequation also gets rid of the puzzle caused by the controversial rationality assumption. Thereplicator dynamics on a JMG is dened as follows.Denition2: Givenanevidence mona FOD, thebelief or basicprobabilityforpropositionA, i.e.m(A), evolves in time according to the equationdm(A)dt= m(A) (fA ) , A , A = (19)wherefAis the tness of propositionA, and it is given in terms of a JMG(, J) byfA=

B,B=m(B)J(A, B) (20)whereas the average tness of propositions in the population is =

A

Bm(A)J(A, B)m(B), A, B , and A, B= (21)Regardingthedifferential dynamicsystemshowninEqs.(19)-(21), peopleare mainlyconcerned with the asymptotically stable points (ASPs). Hofbauer and Sigmund [83] provideda theorem to show the relationship between the ASPs of replicator dynamics and ESSs of asymmetric two-player game.Theorem2:[83]Given asymmetric two-playergameG,strategyxisanESSofGifand only ifxis an ASP of the replicator equation evolving onG.Theorem2showsthat thereisanone-to-onecorrespondencebetweenASPsandESSsinasymmetrictwo-playergame. Let usdenotetheset ofASPsofthereplicatorequationinaJMGbyASP, evolutionarilystablepropositions(ESPs) byESP. SinceJMGsaresymmetric, the following corollary is naturally satised.Corollary 3: In JMGs,m ESPif and only ifm ASP.March 10, 2015 DRAFTHere, an example is given to help understand this relationship.Example 2: Given a FOD = {a, b}, a JMG = (, J) is constructed, its payoff matrixJis shown as follows.J=a b ababab__1 0 1/20 1 1/21/2 1/2 1__(22)Nowlet usstudythereplicatordynamicsonthegame. InthisJMG, therearethreepure strategies,a,b, andab. For the sake of presentation, let the basic probabilities of thesepropositions bem(a)=x, m(b)=y, m(ab)=z, wherex + y + z=1. Accordingtothereplicatorequation, wecanwrite(replacing,asusual, thetime derivatives ofx, y, zby x, y, z),___ x = x(fx ) y = y(fy ) z= z(fz )(23)where___fx= x + z/2fy= y + z/2fz= x/2 + y/2 + z(24)and = (x, y, z)J(x, y, z)T= x(x + z/2) + y(y + z/2) + z(x/2 + y/2 + z) (25)According to the stability theory of differential equations, the xed point of the replicatordynamicsystemrepresentedbyEqs. (23)-(25)shouldsatisfy x=0, y=0, z=0, wherex+y+z =1. Soall xedpoints canbe obtained, theyare (x1, y1, z1) =(1, 0, 0),(x2, y2, z2) =(0, 1, 0), (x3, y3, z3) =(0, 0, 1), (x4, y4, z4) =(0, 0.5, 0.5), (x5, y5, z5) =(0.5, 0.5, 0), (x6, y6, z6) =(0.5, 0, 0.5). Thestabilityof eachxedpoint canbecheckedby a Jacobian matrix,JM, as follows.JM=______ xx xy xz yx yy yz zx zy zz______(26)Let us take(x1, y1, z1) as an example. For this xed point,A = JM|(x=x1,y=y1,z=z1)=______1 0 0.50 1 00 0 0.5______. (27)March 10, 2015 DRAFTTABLE IFIXED POINTS AND THEIR STABILITY IN EXAMPLE 2Fixed points (Beliefs of a, b and ab) Associated eigenvalues Stablity(1, 0, 0) -1, -1, -0.5 stable (ASP)(0, 1, 0) -1, -1, -0.5 stable (ASP)(0, 0, 1) -1, -0.5, -0.5 stable (ASP)(0, 0.5, 0.5) -0.75, -0.5, 0.25 unstable(0.5, 0.5, 0) -0.5, 0, 0.5 unstable(0.5, 0, 0.5) -0.75, -0.5, 0.25 unstableThestabilityof xedpoint (x1, y1, z1)isdeterminedbytheeigenvaluesof thefollowingcharacteristic equationdet(AI) = 0 (28)wheredet isthedeterminant of amatrix, andI isanidentitymatrix. So, theeigenvaluecanbereadilycalculated, andtheyare1=1, 2=1, 3=0.5. AccordingtoTheorem 1 [83], the xed points (x1, y1, z1) is an asymptotically stable point (ASP) since alleigenvalues are negative numbers. Meanwhile, ASP(x1, y1, z1) corresponds tom(a) = 1, sowe say propositiona is an evolutionarily stable proposition (ESP).By means of this approach, the stability of other xed points can be found. The results areshown in Tab. I. As shown in Tab. I, there are three ASPs, namelym(a) = 1,m(b) = 1 andm(ab) = 1, which are in one-one corresponding with ESPs, i.e., ESP= ASP. Graphically,a two-dimensional space, called simplex, can clearly represent the evolutionary dynamics ofpropositions a, b and ab, as shown in Fig. 3. In the simplex, every vertex of means that thereonly exists a sole proposition (i.e., strategy) in the population, edges represent that at least aproposition is missing in the population. The interior of the simplex corresponds to the caseofall propositionscoexistence. At eachpoint ofthesimplex, thesumofthebeliefofallpropositions is1.Inaddition, inFig.3,arrowsrepresentthedirections ofevolution, blackcircles indicate ASPs, white circles areunstable xed points. From Fig. 3, it clearly showsthat there arethree ASPs that areprecisely ESPs of, and three unstable xed points thatare located on the midpoints of there edges.March 10, 2015 DRAFT

Fig. 3. Evolutionary dynamics of propositions a, b and ab in Example 2D. Evolution of averaging evidenceAs aforementioned, given an evidence which is usually the average of multiple evidencesin multi-source information fusion, the replicator dynamics determines which proposition thisevidence will evolve to. Here is an example to show the evolutionary process.Example 3: Given an evidencem on a FOD = {a, b, c},m(a) = 0.25,m(b) = 0.25, m(c) = 0.25,m(ab) = 0.05,m(ac) = 0.1,m(bc) = 0.05,m(abc) = 0.05.Wewant todetermine which is themost possible object among objectsa, b, c,in termsofm.Aswesee, thesupport degreesforobjectsa, bandcareverysimilar inm. Facingthis situation, weusetheproposed ECRforanalysis. Let mbetheinitial congurationofthispopulationat t=0, andmt(A)bethemassvalueof m(A)at timet, whereA.The evolutionary process is illustrated bymt+t(A) = mt(A) +dmt(A)dtt (29)In this paper, we simulate the evolutionary process of each proposition by using the fourth-order Runge-Kutta method, as shown in Fig. 4, where the horizontal axis Time indicatestime points intheRunge-Kuttamethod. Accordingto Fig.4,given aninitial congurationMarch 10, 2015 DRAFT0 5 10 15 20 25 30 35 40 45 5000.10.20.30.40.50.60.70.80.91TimeMass of belief m(.) abcabacbcabcFig. 4. The evolutionary process of propositions in Example 3determinedbyevidence m, intheendof evolution, propositionac survives andnallyoccupiesthepopulation, whileotherpropositionsbecomeextinct. So, acistheonlyESPfor evidence m, denotedas ESPm=ac. It impliesthat propositionachas thehighesttness. As a result, it suggests that the most supported object is eithera orc. Moreover, asshown in Fig. 4, by using the Runge-Kutta method, we can track the evolutionary curve ofeachproposition, andobtainthetimerequiredforreachingequilibrium. Inthispaper, weassume that the replicator dynamics equation reaches equilibrium if the maximum incrementor decrement of propositions mass values is less than 103between two adjacent time points.E. Two-dimensional measureByusingourproposedECRmethod, anequilibriumstatecanbeevolvedforanygivenevidences. It couldbeanESPiftheequilibriumstateisstable, asmentionedabove; oth-erwise, itisanunstableequilibriumpoint,aswhilecirclesshowninFig. 3. However, theproblemisnot totallysolved. Consideringthiscasethat twodifferent evidencesevolvetoa same equilibrium state, how can we distinguish them? An effective measure is necessary.Fortunately, since the evolutionary process is dynamic, we nd that, the time evolving to thestable or unstable equilibrium state provides a reasonable measure to reect such difference.March 10, 2015 DRAFT13.0459 16.855300.10.20.30.40.50.60.70.80.91TimeMass of belief m(.) m1: am1: bm1: abm2: am2: bm2: abFig. 5. The evolutionary process of propositions inm1andm2of Example 41) Case of evolving to an ESP: Let us rst consider the case that the population evolvesto a stable equilibrium state, namely an ESP.Example 4: Given two evidences, indicated bym1andm2, on a FOD = {a, b},m1(a) = 0.7,m1(b) = 0.1, m1(ab) = 0.2;m2(a) = 0.5,m2(b) = 0.3, m2(ab) = 0.2.The evolutionary process of propositions in m1 and m2 are shown in Fig. 5. For evidencesm1andm2, the ESPs are botha, namelyESPm1= a, ESPm2= a. But the time evolvingtotheequilibriumstatearedifferent. For m1andm2, thetimeofreachingpropositionaaret1=13.0459 andt2=16.8553, respectively. Fromtheviewofevolution, theseresultsindicate thatm1is more close to ESPa since propositiona is more supported inm1.Example 5: Given a FOD = {a, b, c}, there is an evidencem shown as follows,m(a) = x, m(b) = 0.9 x, m(bc) = 0.05,m(abc) = 0.05.where x [0, 0.9]. Now let us investigate the ESP of m and the time evolving to that ESPdenoted astESPwith the change ofx from 0 to 0.9 where every increment is 0.01.Fig. 6 illustrates the results. From the gure, we can see that the ESP of m is proposition bifx 0.45, and propositiona ifx 0.46. Whenx 0.45,tESPof evolving to propositionbisincreasingwiththedeclineofthebeliefofbduetotheascentof x. Whenx0.46,ESPm=a, andthebeliefof aincreaseswiththeriseof x, asaresult tESPofevolvingMarch 10, 2015 DRAFT0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.968101214161820xtESPESPm = aESPm = bx=0.45x=0.46Fig. 6. Evolutionary results in Example 5toESPadecreases. Thisexampleshowsthat,regardlesstheESPiseitheraorb, tESPischanged with the initial conguration of evidencem. Also, even if two evidences evolve toasameESP, tESPislowerforthisevidencewhichsupportstheassociatedESPmore. Inother words, the higher tness of ESP in an evidence leads to a lowertESPof real evolvingto that ESP.Therefore, we can say that tESPmeasures the temporal cost of an evidence evolving to itsassociated ESP. From an biology perspective, tESPis an evolutionary distance from the givenevidencetoitsassociatedESP. Basedonthisconsideration, weusetESPtofurtherdepictthe relationship between an evidence and its associated ESP. A two-dimensional measure isdened as follows.Denition 3: Assume there are n evidences indicated by m1, , mn, let m be the averageof thesen evidences, ifm evolves to an ESP, the evolutionary output of using the proposedECR is represented as< ESPm, tESP>= fECR(m1, , mn) (30)where ESPm is the ESP of m, andtESPis the time of m evolving to that ESP as a measureof evolutionary distance.Again, in the ECR, we just want to nd the most supported proposition, but not an exactlysynthetical evidence, the time of evolving to an ESP provides a reference for the evolutionaryMarch 10, 2015 DRAFTdistance between the averaging evidence and its associated ESP.2) Caseofevolvingtoanunstable equilibrium state: Itismentioned that, byusing theECR, theaveragingevidencemmaydonot evolvetoaESP, buttoastatewheretwoormore propositions coexist. Mathematically, those states correspond to unstable xed points,graphically illustrated as the white circles in Fig. 3. A very slight disturbance will cause thatmdeviates such unstable equilibrium statesand goes toan ESP. Therefore,the probabilityof evolving to such states is very small in general. Even so, considering the completeness ofmethodology, we also give a denition to formalize the output of ECR in this situation.Denition 4: Givenn evidencesm1, , mn, for their averaging evidencem which willevolves to an unstable equilibrium state, the evolutionary output of using the proposed ECRcan be expressed as< Pm, tES>= fECR(m1, , mn) (31)wherePmisset ofexistent propositionsinthisunstableequilibriumstate, andtESisthetime ofm evolving to that equilibrium state.Exampleswill begiveninthefollowingsection. It isworthynoticedthat, sometimes,ifthepopulationevolvestoanunstableequilibriumstate, it impliesthat therearehighlyconicting information among the original evidences m1, , mn. In this sense, the unstableequilibrium state sometimes can act as an alarm to report the existence of high conict.IV. ILLUSTRATIVEEXAMPLESANDANALYSISA. Combination of highly conicting evidencesConicting evidence combination [86], [87] is a main concern to verify the effectivenessof combination rules in multi-sources information fusion. Here we will give several classicalcases in DST for the verication of the proposed ECR.Example6:Zadehsparadox[18]. Twodoctorsdiagnoseapatient, andtheyagreethatthepatient suffersfromoneofthesethreediseasesincludingmeningitis(M), braintumor(T), andconcussion(C). Hence, aFODisdeterminedas ={M, T, C}. Bothof thesedoctors think that atumor is unlikely, but they hold different opinions for the likely cause.Two diagnosis are given as follows.m1(M) = 0.9, m1(T) = 0.1, m1(C) = 0.0.m2(M) = 0.0, m2(T) = 0.1, m2(C) = 0.9.We can nd that these two evidences are highly conicting. If using the classical Demptersrule of combination to combine them, as shown in Eqs.(7) and (8), the combination result isMarch 10, 2015 DRAFT0 2 4 6 8 1000.10.20.30.40.5TimeMass of belief m(.) MTCFig. 7. Evolutionary result of Zadehs paradoxm(M) = 0,m(T) = 1,m(T) = 0.andtheconictcoefcient K=0.99.Thisisanapparentlycounter-intuitive result.Thepatient most likely does not suffer from tumor in each doctors diagnosis, but the synthesizingresult shows thatthe patient suffers fromtumor with 100%. Soitis counter-intuitive. Thisclassical example is rstgiven byZadeh toshow thedoubts onthe validity ofDempstersrule when information are highly conicting [18].Now, letsusetheproposedECRtointegratethesetwoevidences. Assumetheweightfactor of eachdoctor isidentical, theaveragingevidenceiscalculatedas m(M) =0.45,m(T) = 0.1, m(C) = 0.45. The evolutionary process of each proposition inm is shown inFig. 7. From this gure, we can nd that with the increase of evolutionary time the belief ofTgoes to 0, and nally propositions MandCaveragely share the total belief 1. At the endof evolution, the belief of each proposition becomes m(M) = 0.5, m(T) = 0.0, m(C) = 0.5,which means that Mand C coexist nally. The time of evolving to this unstable equilibriumstableistES=8.7759. Besides,theinstability ofthis equilibrium stableimplies that therearehighlyconictinginformationinmcausedbytheoriginalevidencesm1andm2. TheECR effectively identies this situation.Example7:ModiedZadehsparadox. RegardingZadehsparadox, nowlet usassumethat the two doctors give two outright conicting diagnosis as follows.March 10, 2015 DRAFT0 2 4 6 8 1000.10.20.30.40.5TimeMass of belief m(.) MTCFig. 8. Evolutionary result of modied Zadehs paradoxm1(M) = 1, m1(T) = 0, m1(C) = 0.m2(M) = 0, m2(T) = 0, m2(C) = 1.Nowthereisnot anyagreement betweenm1andm2. AccordingtoDempstersruleofcombination, the conict coefcient K= 1. In this situation, Dempsters rule becomes invalidand can not be used to synthesize them. Instead, let us consider them by using the proposedECR. Theaveragingevidenceism(M)=0.5, m(C)=0.5. Theevolutionaryresultsareshown in Fig. 8. From that gure, we can nd that the belief of each proposition inm doesnot show any change, as theevolutionary time increases.Theaverageevidencemisexactlocatedatanunstableequilibrium point.Therefore, asameasureofevolutionary distance,the time reaching the equilibrium state istES= 0 in this case.Theseabovetwoexamples clearlyshowthat theproposedECRis effectivefor thesecases where Dempsters rule is questionable. Moreover, by using the ECR, Zadehs paradoxistransformedtoasimplemathematicalproblem, andthenexplainedbytheinstabilityofequilibrium point. Even Zadehs paradox is objectively existent, but it is extremely unstable,and a slight numerical change of the inputs could break away from this dilemma.March 10, 2015 DRAFTB. More illustrative examplesIn the following content, we will examine the proposed ECR by using more examples.Example8: In[21], [22], theauthorshavepresentedanemblematicexampletoshowthe inadequate behavior of Dempsters rule of combination. The authors called that behaviorasthe dictatorial power(DP)of Dempsters rule. Specically,in that example the level ofconict can be chosen at any low or high value, which means that the example is not relatedto the level of conict between evidences. A simple version [20] of that example is shownas follows.Given a FOD = {1, 2, 3}, there are four evidences as below:m1(1) = a, m1(12) = 1 a.mi(3) = b, mi() = 1 b, wherei = 2, 3, 4.When using Dempsters rule of combination, one gets:m(1) = a = m1(1), m(12) = 1 a = m1(12).It clearlyshowsthat Dempstersruleisnot respondingtothecombinationofdifferentevidences. It seems that evidence m1 dominates other evidences since the combination resultis alwaysm1, which does not accord with peoples expectation or intuition.Now let us reconsider this example by using the proposed ECR. Moreover, for the sake ofcomparison, several improved methods for the combination of evidences, including Murphyssimple average [28], Dengs weighted average [29], Hans sequential weighted combination[20], proportional conict redistribution(PCR6)rule[63], havealsobeenusedtotest theresults. Because PCR6 is not associative, to get optimal results, the PCR6 rule is implementedin this paper by combining all evidences altogether at the same time. Here, assumea = 0.7,b = 0.6. The results are listed in Tab. II. As illustrated in that table, the combination resultsare always the same as m1 if using Dempsters rule of combination, which is counter-intuitive.However, the counter-intuitive results have been eliminated in the results of Murphys simpleaverage, Dengs weighted average, Hans sequential weighted combination, and PCR6rule.Similarly,intheproposedECR, thecounter-intuitive behaviorsarealsosuppressed,whichshows the advantageoftheproposed method. Inevery caseofcombination, the most sup-portedpropositionobtainedbytheECRtotallyaccordswiththeresultsobtainedbyotherreasonable methods. Moreover, the decrease oftESPindicates that the evolutionary distancetotheESPisreducingwiththeaccumulationofevidences, whichfurthercoincideswithpeoples expectation.Example9: In[20], theauthorsstudiedtheproblemof target recognitioninamulti-sensor system. In a multisensor-based automatic target recognition system, assume the FODMarch 10, 2015 DRAFTTABLE IIRESULTS FOR EXAMPLE 8Evidences m1, m2 m1, m2, m3 m1, m2, m3, m4Dempsters rule of combinationm(1) = 0.7000, m(1) = 0.7000, m(1) = 0.7000,m(12) = 0.3000. m(12) = 0.3000. m(12) = 0.3000.Murphys simple averagem(1) = 0.5250, m(1) = 0.3379, m(1) = 0.1794,m(12) = 0.1179, m(12) = 0.0615, m(12) = 0.0292,m(3) = 0.3000, m(3) = 0.5622, m(3) = 0.7711,m() = 0.0571. m() = 0.0384. m() = 0.0203.Dengs weighted averagem(1) = 0.5250, m(1) = 0.1032, m(1) = 0.1032,m(12) = 0.1179, m(12) = 0.0290, m(12) = 0.0093,m(3) = 0.3000, m(3) = 0.8122, m(3) = 0.9344,m() = 0.0571. m() = 0.0555. m() = 0.0246.Hans sequential weighted combinationm(1) = 0.5250, m(1) = 0.2362, m(1) = 0.0676,m(12) = 0.1179, m(12) = 0.0363, m(12) = 0.0089,m(3) = 0.3000, m(3) = 0.6369, m(3) = 0.8298,m() = 0.0571. m() = 0.0906. m() = 0.0937.PCR6 rulem(1) = 0.5062, m(1) = 0.3432, m(1) = 0.2464,m(12) = 0.1800, m(12) = 0.1028, m(12) = 0.0642,m(3) = 0.3138. m(3) = 0.4306, m(3) = 0.4921,m() = 0.1234. m() = 0.1973.The proposed ECR

is={1, 2, 3}. Foranunknowntarget, thesystemhascollectedveevidencesshownas follows.m1(1) = 0.60, m1(2) = 0.10, m1(2, 3) = 0.30.m2(1) = 0.65, m2(2) = 0.10, m2(3) = 0.25.m3(1) = 0.00, m3(2) = 0.90, m3(2, 3) = 0.10.m4(1) = 0.55, m4(2) = 0.10, m4(2, 3) = 0.35.m5(1) = 0.55, m5(2) = 0.10, m5(2, 3) = 0.35.Byusingdifferent methods, thecombinationresultsarederived, asshowninTab. III.From Tab. III,in the combination results based on Dempters rule, m(1)always equals toMarch 10, 2015 DRAFTTABLE IIIRESULTS FOR EXAMPLE 9Evidences m1, m2 m1, m2, m3 m1, m2, m3, m4 m1, m2, m3, m4, m5Dempsters rule of m(1) = 0.7723, m(1) = 0.0000, m(1) = 0.0000, m(1) = 0.0000,combinationm(2) = 0.0792, m(2) = 0.8421, m(2) = 0.8727, m(2) = 0.8981,m(3) = 0.1485. m(3) = 0.1579. m(3) = 0.1273. m(3) = 0.1019.Murphys simple averagem(1) = 0.7716, m(1) = 0.3526, m(1) = 0.5167, m(1) = 0.6706,m(2) = 0.0790, m(2) = 0.5978, m(2) = 0.4573, m(2) = 0.3169,m(3) = 0.1049, m(3) = 0.0380, m(3) = 0.0189, m(3) = 0.0088,m(23) = 0.0444. m(23) = 0.0116. m(23) = 0.0071. m(23) = 0.0037.Dengs weighted averagem(1) = 0.7716, m(1) = 0.6013, m(1) = 0.7987, m(1) = 0.8975,m(2) = 0.0790, m(2) = 0.3302, m(2) = 0.1698, m(2) = 0.0897,m(3) = 0.1049, m(3) = 0.0532, m(3) = 0.0227, m(3) = 0.0088,m(23) = 0.0444. m(23) = 0.0153. m(23) = 0.0088. m(23) = 0.0040.m(1) = 0.7716, m(1) = 0.5591, m(1) = 0.6781, m(1) = 0.8103,Hans sequential m(2) = 0.0790, m(2) = 0.4021, m(2) = 0.3054, m(2) = 0.1797,weighted combination m(3) = 0.1049, m(3) = 0.0297, m(3) = 0.0071, m(3) = 0.0014,m(23) = 0.0444. m(23) = 0.0091. m(23) = 0.0093. m(23) = 0.0086.PCR6 rulem(1) = 0.7371, m(1) = 0.4224, m(1) = 0.4755, m(1) = 0.5111,m(2) = 0.0644, m(2) = 0.4729, m(2) = 0.3849, m(2) = 0.3244,m(3) = 0.1370, m(3) = 0.0483, m(3) = 0.0351, m(3) = 0.0276,m(23) = 0.0615. m(23) = 0.0564. m(23) = 0.1045. m(23) = 0.1369.The proposed ECR

0 after combiningm3, while regardless of the support for1inm4andm5. Evidently, thisis counter-intuitive. In contrast, as we can see in Tab. III, based on the other four methods,the counter-intuitive results are suppressed. In Dengs weighted average and Hans sequentialweighted combination, the most supported is always1for any case, and the support for1is increasing after the arrival ofm4andm5. For Murphys simple average, PCR6 rule, andthe proposed ECR, even though the most supported is2in the case of combiningm1, m2andm3,but it promptly changesto1assoon asthearrival ofm4,which also overcomesthecounter-intuitive behavior.Moreover,intheproposedECR, theincreaseofthesupportMarch 10, 2015 DRAFTTABLE IVRESULTS FOR EXAMPLE 10The number of evidencesl l = 10 l = 25 l = 50Dempsters rule of combinationm(1) = 0.4614, m(1) = 0.7871, m(1) = 0.9547,m() = 0.5386. m() = 0.2129. m() = 0.0453.Murphys simple averagem(1) = 0.4614, m(1) = 0.7871, m(1) = 0.9547,m() = 0.5386. m() = 0.2129. m() = 0.0453.Dengs weighted averagem(1) = 0.4614, m(1) = 0.7871, m(1) = 0.9547,m() = 0.5386. m() = 0.2129. m() = 0.0453.Hans sequentialweighted combinationm(1) = 0.3195, m(1) = 0.3684, m(1) = 0.3924,m() = 0.6805. m() = 0.6316. m() = 0.6076.PCR6 rulem(1) = 0.4614, m(1) = 0.7871, m(1) = 0.9547,m() = 0.5386. m() = 0.2129. m() = 0.0453.The proposed ECR

for1is expressed by means of the decline oftESPwhich implies the evolutionary distanceto1is reduced. Therefore, our proposed ECR is still effective in this example.Example 10: Evidence shiftingparadox[19], [20] describes another counter-intuitivebehavior inDemspter-Shafer evidencetheory. Let us consider this scenethat atarget isevaluatedbyl different expertswiththesameimportance. TheFODis ={1, 2, 3}.Each expert gives an identical assessment as below.mi(1) = 0.06,mi() = 0.94, wherei = 1, , l.By using Dempsters rule, the combination result is shown as follows.m(1) = 1 0.94l, m() = 0.94l.Ifl is a big number, for example 100, m(1) = 1 0.94100= 0.9979 which is very largealthough foreachevidencetobecombinedmi(1)=0.06whichisverysmall.Theresultshows that theaggregation ofthewisdom ofcrowdsmay generatecounter-intuitive resultsby using Dempsters rule of combination.Now, lets studythis paradoxbyusingtheproposedECRas well as other improvedmethods. The results derived based on different methods are listed in Tab. IV. As illustratedinTab. IV, withtherise of thenumber of evidences tobecombined, Dempsters ruleof combination, Murphys simpleaverage, Dengs weightedaverage, andPCR6ruleallMarch 10, 2015 DRAFTgeneratecounter-intuitiveresults. AlthoughHanssequential weightedcombinationbringsout reasonable results, but in that methodm(1) has a rising trend asl increases. A counter-intuitiveresult that m(1)>0.5wouldalsoproducewhenl becomeslargeenough. Onlythe proposed ECR brings out the most reasonable result that the most supported propositionisalways, andtheevolutionarytimealsodoesnotchangewiththeincreaseof l. Here,the ECR presents a similarly idempotent character which helps to settle the evidence shiftingparadox.V. CONCLUSIONIn this paper, we have proposed an evolutionary combination rule (ECR) for the evidence-based multi-source information fusion from an evolutionary game theory perspective. WithintheproposedframeworkofECR, original evidencesareaveragedbytheirweights, andaJaccard matrix game is presented to formalize the interaction relationship between proposi-tions, then we use the replicator dynamics equation to mimick the evolution ofpopulation,nally a proposition with the highest tness is identied as the biologically most supportedor possibleconclusion. Experimental resultsshowthat theproposedECRhassuppressedthecounter-intuitive resultscausedbytheclassicalDempstersruleofcombination, whichdemonstrates the rationality and effectiveness of the proposed method.In this work, we import a biological and evolutionary idea into DST, which is not presentedin previous studies and contributes a new insight for multi-source information fusion. Basedon our proposed ECR, the most supported proposition, in the biological sense, can be foundfor decision-making. Of course, the ECR is still of some problems to be solved in the futureresearch. We summarized several noticeable issues as follows.Firstly, the ECRis not associative, all evidences must be combined together at the sametime. Two main reasons, average mechanism and replicator equation, lead to the non-associativity.Inthe initial stageof ECR, multi-source information is synthesized by averagemechanismwhichisalsousedinmanyexistingevidencecombinationapproaches, suchasMurphyssimple average [28], Dengs weighted average [29], Hans sequential weighted combination[20]. As a result, all these approaches are not associative. In addition, the replicator equationruling the evolution of mass value of each proposition does not meet associativity as well.Secondly, theECRdoesnot preservetheneutralityforavacuousevidencemv()=1whereistheFOD, namelym ECRmv=m. Intheframeworkof ECR, assameasotherpropositionA, theFODisregardedasastrategywhichcanbeadoptedbyindividualsinanassumedpopulation. Thevacuousevidencemv()=1doesnot meansMarch 10, 2015 DRAFTtotallyunknown, but meansthat everyindividualadoptsstrategyinthepopulation. So,the ECR does not meet the neutrality.Thirdly, some counter-intuitive results are still hard to interpret in the framework of ECRat present. Forexample, in[37]theauthorpresentedanexamplethat combinesm1(A)=m1(B, C)=0.5andm2(C)=m2(A, B)=0.5. Dempstersrulegivesm(A)=m(B)=m(C) = 1/3. As reported by Voorbraak [37], this result is counter-intuitive, since intuitivelyB seems to share twice a probability mass of 0.5, while both A and C only have to share once0.5 withBand areonceassigned 0.5 individually. Sointuitively,Bis less conrmed thanA andC, but they are equally conrmed by Dempsters rule. Byusing the proposed ECR,we obtainm(AB) = 0.5,m(BC) = 0.5 which corresponds to an unstable equilibrium state.The results seem to also be counter-intuitive, but the ECR is unable to interpret currently.In summary, although there are some drawbacks in the proposed ECR at present, this workisstillmeaningful andinnovative becauseoftheexploitingofbiologicalandevolutionarystandpoint for evidence combination. In many cases, it is effective and useful. In the futureresearch, we will continue to improve and perfect the framework of ECR.ACKNOWLEDGMENTTheworkis partiallysupportedbyChina ScholarshipCouncil, National Natural Sci-enceFoundationofChina(Grant No. 61174022),SpecializedResearchFundfortheDoc-toralProgramofHigherEducation(GrantNo. 20131102130002), R&DProgramofChina(2012BAH07B01), National High Technology Research and Development Program of China(863 Program) (Grant No. 2013AA013801), the open funding project of State Key Laboratoryof Virtual Reality Technology and Systems, Beihang University (Grant No.BUAA-VR-14KF-02), Fundamental Research Funds for the Central Universities (Grant No. XDJK2014D034).REFERENCES[1] R. P. Mahler, Statistical Multisource-MultitargetInformation Fusion. Artech House, Inc., 2007.[2] N. XiongandP. Svensson, Multi-sensormanagement forinformationfusion: issuesandapproaches,InformationFusion, vol. 3, no. 2, pp. 163186,2002.[3] B. Khaleghi, A. Khamis, F. O. Karray, and S. N. Razavi, Multisensor data fusion: A review of the state-of-the-art,Information Fusion, vol. 14, no. 1, pp. 2844, 2013.[4] L. A. Zadeh, Fuzzy sets, Information and Control, vol. 8, no. 3, pp. 338353,1965.[5] Z. Pawlak, Rough set theory and its applications to data analysis, Cybernetics & Systems, vol. 29, no. 7, pp. 661688,1998.[6] Z. Pawlak, L. Polkowski, and A. Skowron, Rough set theory, Wiley Encyclopedia of Computer Science andEngineering,2008.March 10, 2015 DRAFT[7] D. Dubois and H. Prade, Possibility Theory. Springer, 1988.[8] A. Dempster, Upper and lower probabilities induced by a multivalued mapping, Annals of Mathematics and Statistics,vol. 38, no. 2, pp. 325339,1967.[9] G. Shafer, A Mathematical Theory of Evidence. Princeton: Princeton University Press, 1976.[10] J. Liu, D. A. Maluf, andM. C. Desmarais, Anewuncertaintymeasurefor belief networkswithapplicationstooptimal evidential inferencing, IEEE Transactions on Knowledge and Data Engineering, vol. 13, no. 3, pp. 416425,2001.[11] S. Xia andJ.Liu, A belief-basedmodelfor characterizingthe spreadof awarenessandits impacts on individualsvaccination decisions, Journalof The Royal Society Interface, vol. 11, no. 94, p. 20140013, 2014.[12] G. Shafer, A betting interpretation for probabilities and Dempster-Shafer degrees of belief, International Journal ofApproximate Reasoning, vol. 52, no. 2, pp. 127136,2011.[13] G. Shafer, P. R. Gillett, andR. B. Scherl, Anewunderstandingofsubjectiveprobabilityanditsgeneralizationtolower and upper prevision, InternationalJournal of Approximate Reasoning,vol. 33, no. 1, pp. 149,2003.[14] X. Deng, Y. Hu, Y. Deng, and S. Mahadevan, Supplier selection using AHP methodology extended by D numbers,Expert Systems with Applications, vol. 41, no. 1, pp. 156167,2014.[15] J.Abell anandA. Masegosa, Requirementsfor totaluncertaintymeasuresin Dempster-Shafer theoryof evidence,InternationalJournalof General Systems, vol. 37, no. 6, pp. 733747,2008.[16] B. Kang, Y. Deng, R. Sadiq, and S. Mahadevan, Evidential cognitive maps, Knowledge-Based Systems, vol. 35, pp.7786,2012.[17] M. Shoyaib, M. Abdullah-Al-Wadud, and O. Chae, A skin detection approach based on the Dempster-Shafer theoryof evidence, InternationalJournal of Approximate Reasoning,vol. 53, no. 4, pp. 636659, 2012.[18] L. A. Zadeh, Review of a mathematical theory of evidence, AI magazine,vol. 5, no. 3, pp. 8183, 1984.[19] F. YangandX. Wang, CombinationMethodof ConictiveEvidencesinD-SEvidenceTheory. Beijing: NationalDefence Industry Press, 2010.[20] D. Han, Y. Deng, and C. Han, Sequential weighted combination for unreliable evidence based on evidence variance,Decision SupportSystems, vol. 56, pp. 387393, 2013.[21] J. Dezert, P. Wang, andA. Tchamova, Onthe validityof Dempster-Shafer theory, inThe 15thInternationalConference on Information Fusion (FUSION). IEEE, 2012,pp. 655660.[22] J. DezertandA. Tchamova, OnthevalidityofDempstersfusionruleandits interpretationasageneralizationofBayesian fusion rule, InternationalJournalof Intelligent Systems, vol. 29, no. 3, pp. 223252, 2014.[23] P. Smets and R. Kennes, The transferable belief model, Articial Intelligence, vol. 66, no. 2, pp. 191234,1994.[24] D.DuboisandH. Prade,A settheoretic viewof belief functions:Logicaloperationsandapproximationsbyfuzzysets, InternationalJournal of General System, vol. 12, no. 3, pp. 193226,1986.[25] T. Denoeux, Conjunctive and disjunctive combination of belief functions induced by nondistinct bodies of evidence,Articial Intelligence, vol. 172, no. 2, pp. 234264,2008.[26] R. R. Yager, On the Dempster-Shafer framework and new combination rules, Information Sciences,vol. 41, no. 2,pp. 93137,1987.[27] J.-B. Yang and M. G. Singh, An evidential reasoning approach for multiple-attribute decision making with uncertainty,IEEE Transactions on Systems, Man and Cybernetics, vol. 24, no. 1, pp. 118, 1994.[28] C. K.Murphy,Combiningbelief functionswhen evidenceconicts, Decision Support Systems, vol.29,no.1, pp.19, 2000.[29] Y. Deng, W. Shi, Z. Zhu, and Q. Liu, Combining belief functions based on distance of evidence, Decision SupportSystems, vol. 38, no. 3, pp. 489493, 2004.March 10, 2015 DRAFT[30] M. C. Florea, A.-L. Jousselme,E. Boss e, and D. Grenier, Robust combination rules for evidence theory, InformationFusion, vol. 10, no. 2, pp. 183197, 2009.[31] J. Ma, W. Liu, D. Dubois, and H. Prade, Revision rules in the theory of evidence, in 2010 22nd IEEE InternationalConference on Tools with Articial Intelligence (ICTAI), vol. 1, 2010, pp. 295302.[32] J.-B. YangandD.-L. Xu,Evidentialreasoningrule forevidencecombination,ArticialIntelligence, vol. 205, pp.129,2013.[33] X. Deng, Y. Deng, and F. T. Chan, An improved operator of combination with adapted conict, Annals of OperationsResearch, vol. 223, no. 1, pp. 451459, 2014.[34] R. R. Yager, Combining various types of belief structures, Information Sciences,vol. 303, pp. 83100,2015.[35] L. Zhang, Representation, independence,and combination of evidence in the Dempster-Shafer theory, in Advancesin the Dempster-Shafer theory of evidence. John Wiley & Sons, Inc., 1994,pp. 5169.[36] F. Smarandache and J. Dezert, On the consistency of PCR6 with the averaging rule and its application to probabilityestimation, in The 16th International Conference on Information Fusion (FUSION). IEEE, 2013, pp. 11191126.[37] F. Voorbraak, Onthejusticationof Dempstersruleof combination,Articial Intelligence, vol. 48, no. 2, pp.171197, 1991.[38] E. Lefevre, O. Colot,and P. Vannoorenberghe, Belief function combinationand conictmanagement,InformationFusion, vol. 3, no. 2, pp. 149162,2002.[39] R. Haenni, ArealternativestoDempstersruleofcombinationreal alternatives?: CommentsonAbout thebelieffunction combination and the conict managementproblem-Lefevre et al, Information Fusion, vol. 3, no. 3, pp.237239, 2002.[40] F. Cuzzolin, Geometryof Dempsters rule of combination,IEEE Transactionson Systems, Man,andCybernetics,Part B: Cybernetics, vol. 34, no. 2, pp. 961977, 2004.[41] W. Liu and J. Hong, Reinvestigating Dempsters idea on evidence combination, Knowledge and Information Systems,vol. 2, no. 2, pp. 223241,2000.[42] F. Cuzzolin, A geometric approach to the theory of evidence, IEEE Transactions on Systems, Man, and Cybernetics,Part C: Applications and Reviews, vol. 38, no. 4, pp. 522534, 2008.[43] E. Lef` evre and Z. Elouedi, How to preserve the conict as an alarm in the combination of belief functions? DecisionSupportSystems, vol. 56, pp. 326333, 2013.[44] J. Maynard Smith and G. Price, The logic of animal conict, Nature, vol. 246, pp. 1518,1973.[45] J. Maynard Smith, Evolution and the Theory of Games. Cambridge University Press, 1982.[46] T. Denoeux, Maximum likelihood estimation from uncertain data in the belief function framework, IEEE Transactionson Knowledge and Data Engineering, vol. 25, no. 1, pp. 119130,2013.[47] Z.-g. Su, Y.-f. Wang, andP.-h. Wang, Parametricregressionanalysisofimpreciseanduncertaindatainthefuzzybelief function framework, International Journalof Approximate Reasoning, vol. 54, no. 8, pp. 12171242, 2013.[48] X. Deng, Y. Hu, F. T. Chan, S. Mahadevan, andY. Deng, Parameter estimationbasedoninterval-valuedbeliefstructures, European Journalof Operational Research, vol. 241, no. 2, pp. 579582, 2015.[49] Y. Bi, J. Guan, and D. Bell, The combination of multiple classiers using an evidential reasoning approach, ArticialIntelligence, vol. 172, no. 15, pp. 17311751, 2008.[50] T.DenuxandM.-H.Masson, EVCLUS: evidentialclusteringofproximitydata, IEEE TransactionsonSystems,Man, and Cybernetics, Part B: Cybernetics, vol. 34, no. 1, pp. 95109,2004.[51] Z.-g. Liu, Q. Pan, J. Dezert, and G. Mercier, Credal classication rule for uncertain data based on belief functions,Pattern Recognition, vol. 47, no. 7, pp. 25322541, 2014.[52] M. J. Beynon, Understanding local ignorance and non-specicity within the DS/AHP method of multi-criteria decisionmaking, European Journal of Operational Research, vol. 163, no. 2, pp. 403417,2005.March 10, 2015 DRAFT[53] M. Fontani, T. Bianchi, A. DeRosa, A. Piva, andM. Barni, AframeworkfordecisionfusioninimageforensicsbasedonDempster-Shafer theoryof evidence,IEEETransactionsonInformationForensicsandSecurity, vol. 8,no. 4, pp. 593607,2013.[54] J. M. Merig o and M. Casanovas, Induced aggregation operators in decision making with the Dempster-Shafer beliefstructure, International Journalof Intelligent Systems, vol. 24, no. 8, pp. 934954,2009.[55] R. P. Srivastava, An introduction to evidential reasoning for decision making under uncertainty: Bayesian and belieffunction perspectives, International Journalof Accounting Information Systems, vol. 12, no. 2, pp. 126135,2011.[56] R. R. Yager and N. Alajlan, Decision making with ordinal payoffs under Dempster-Shafer type uncertainty,InternationalJournalof Intelligent Systems, vol. 28, no. 11, pp. 10391053, 2013.[57] T. Denoeux, Likelihood-basedbelieffunction:justicationandsomeextensionstolow-qualitydata,InternationalJournalof Approximate Reasoning, vol. 55, no. 7, pp. 15351547, 2014.[58] E. Ramasso and T. Denoeux, Making use of partial knowledge about hidden states in HMMs: an approach based onbelief functions, IEEE Transactionson Fuzzy Systems, vol. 22, no. 2, pp. 395405,2014.[59] J.-B. Yang, J. Liu, J. Wang, H.-S. Sii, and H.-W. Wang, Belief rule-base inference methodology using the evidentialreasoningapproach-RIMER,IEEETransactionsonSystems, ManandCybernetics, Part A:SystemsandHumans,vol. 36, no. 2, pp. 266285,2006.[60] F.Cuzzolin,Twonewbayesianapproximationsofbelief functionsbasedonconvexgeometry,IEEE Transactionson Systems, Man, and Cybernetics, Part B: Cybernetics, vol. 37, no. 4, pp. 9931008,2007.[61] A. P. Dempster, TheDempster-Shafercalculusforstatisticians,International Journal of ApproximateReasoning,vol. 48, no. 2, pp. 365377,2008.[62] J. Dezert,Foundationsforanewtheoryofplausibleandparadoxical reasoning,InformationandSecurity, vol. 9,pp. 1357, 2002.[63] F. Smarandache and J. E. Dezert, Eds., Advances and applications of DSmT for information fusion. Am. Res. Press,Rehoboth,NM, USA, 20042015, vol. 14. [Online]. Available: http://fs.gallup.unm.edu//DSmT.htm[64] F. SmarandacheandJ. Dezert, Informationfusionbasedonnewproportional conict redistributionrules,inThe7th International Conference on Information Fusion (FUSION). IEEE, 2004, pp. 907914.[65] X. Li, J. Dezert, F. Smarandache, andX. Huang, Evidence supportingmeasure of similarityfor reducingthecomplexity in information fusion, Information Sciences, vol. 181, no. 10, pp. 18181835, 2011.[66] J. Zhou, D. Jianmin, and Y. Guangzu, Occupancy grid mapping based on DSmT for dynamic environment perception,InternationalJournalof Robotics and Automation, vol. 2, no. 4, pp. 129139,2013.[67] Z.-g. Liu, J. Dezert, G. Mercier,and Q. Pan, Dynamic evidentialreasoningfor changedetection in remote sensingimages, IEEE Transactions on Geoscience and Remote Sensing, vol. 50, no. 5, pp. 19551967, 2012.[68] M. A. NowakandR. M. May, Evolutionarygamesandspatial chaos,Nature, vol. 359, no. 6398, pp. 826829,1992.[69] Z. Wang, A. Szolnoki, andM. Perc, Evolutionofpubliccooperationoninterdependent networks: Theimpact ofbiased utility functions, EPL (Europhysics Letters), vol. 97, no. 4, p. 48001,2012.[70] G. Szab o and G. Fath, Evolutionary games on graphs, Physics Reports, vol. 446, no. 4, pp. 97216,2007.[71] R. M. Axelrod, The Evolution of Cooperation. Basic books,2006.[72] Z.Wang,X. Zhu, and J.J.Arenzon, Cooperationandage structurein spatialgames, PhysicalReview E,vol. 85,no. 1, p. 011149,2012.[73] R. Chiong and M. Kirley, Effects of iterated interactions in multiplayer spatial evolutionary games, IEEE Transactionson Evolutionary Computation, vol. 16, no. 4, pp. 537555, 2012.[74] Z. Wang, S. Kokubo, J. Tanimoto, E. Fukuda, and K. Shigaki, Insight into the so-called spatial reciprocity, PhysicalReview E, vol. 88, no. 4, p. 042145,2013.March 10, 2015 DRAFT[75] F. C. Santos and J. M. Pacheco, Scale-free networks provide a unifying framework for the emergence of cooperation,Physical Review Letters, vol. 95, no. 9, p. 098104,2005.[76] S. Boccaletti, G. Bianconi,R. Criado, C. Del Genio,J. G omez-Garde nes, M. Romance, I. Sendi na-Nadal,Z. Wang,andM. Zanin, Thestructureanddynamicsofmultilayernetworks,PhysicsReports, vol. 544, no. 1, pp. 1122,2014.[77] Z. Wang,A. Szolnoki,and M.Perc, Interdependent networkreciprocity in evolutionarygames, Scientic Reports,vol. 3, p. 1183, 2013.[78] M. A. Nowak, Evolutionary Dynamics. Harvard University Press, 2006.[79] X. Deng, Q. Liu, R. Sadiq, and Y. Deng,Impact of roles assignationon heterogeneouspopulations in evolutionarydictator game, Scientic Reports, vol. 4, p. 6937, 2014.[80] X. Deng, Z. Wang, Q. Liu, Y. Deng, andS. Mahadevan, Abelief-basedevolutionarilystablestrategy,Journal ofTheoretical Biology, vol. 361, pp. 8186,2014.[81] K.Shigaki, J.Tanimoto,Z. Wang,S.Kokubo, A.Hagishima,andN. Ikegaya, Referringto the socialperformancepromotes cooperation in spatial prisoners dilemma games, Physical Review E, vol. 86, no. 3, p. 031141,2012.[82] P. D. Taylor and L. B. Jonker, Evolutionary stable strategies and game dynamics, Mathematical Biosciences, vol. 40,no. 1, pp. 145156,1978.[83] J. Hofbauer and K. Sigmund, Evolutionary Games and Population Dynamics. Cambridge University Press, 1998.[84] A. Gardner and S. A. West, Greenbeards, Evolution, vol. 64, no. 1, pp. 2538,2010.[85] P. Jaccard, Etude comparative de la distribution orale dans une portion des alpes et du jura, Bulletin de la Soci et evaudoise des sciences naturelles, vol. 37, pp. 547579,1901.[86] J. Schubert, Conict management inDempster-Shafertheoryusingthedegreeoffalsity,International Journal ofApproximate Reasoning, vol. 52, no. 3, pp. 449460,2011.[87] W. Liu, Analyzing the degree of conict among belief functions, Articial Intelligence, vol. 170, no. 11, pp. 909924,2006.March 10, 2015 DRAFT