Neutrosophic Sets and Systems, Vol. 2, 2013 Surapati Pramanik, Tapan Kumar Roy, Neutrosophic Game Theoretic Approach to Indo-Pak Conflict over J ammu-KashmirNeutrosophic Game Theoretic Approach to Indo-Pak Conflict over J ammu-Kashmir Surapati Pramanik1, Tapan Kumar Roy2 1 Nandalal Ghosh B.T. College, Panpur, P.O.-Narayanpur, Dist-North 24 Parganas, PIN Code-743126, West Bengal, India. E-mail: sura_pati@yahoo.co.in 2 Bengal Engineering and Science University,P.O.-B. Garden, District Howrah-711103, West Bengal, India. E-mail: roy _t_k@yahoo.co.in Abstract.Thestudydealswiththeenduringconflictbetween IndiaandPakistanoverJammuandKashmirsince1947.The ongoing conflict is analyzed as an enduring rivalry; characterized bythreemajorwars(1947-48),1965,1971,lowintensity militaryconflict(Siachen),miniwaratKargil(1999),internal insurgency, cross border terrorism. We examine the progress and thestatusofthedispute,aswellasthedynamicsoftheIndia PakistanrelationshipbyconsideringtheinfluenceofUSAand Chinaincrisisdynamics.Wediscussthepossiblesolutions offeredbythevariousstudygroupsandpersons.Mostofthe studiesweredoneincrispenvironment.PramanikandRoy(S. PramanikandT.K.Roy,Gametheoreticmodeltothe Jammu-KashmirconflictbetweenIndiaandPakistan. InternationalJournalofMathematicalArchive(IJMA), 4(8) (2013), 162-170.) studied game theoretic model toJammu and Kashmir conflict in crisp environment. In the present study wehaveextendedtheconceptofthegametheoricmodelofthe JammuandKashmirconflictinneutrosophicenvirorment.We haveexploredthepossibilitiesanddevelopedargumentsforan applicationofprincipleofneutrosophicgametheoryto understand properly of the Jammu and Kashmir conflict in terms of goals and strategy of either side. Standard 22 zero-sum game theoretic model used to identify an optimal solution. Keywords: Conflict resolution, game theory, Jammu and Kashmir conflict, neutrosophic membership function, optimal solution sad-dle point, zero-sum game.1 Introduction The purpose of this study is to develop neutrosophic game theoreticmodeltoIndia-Pakistan(Indo-Pak)crisis dynamicsandcontributetotheneutrosophicanalysisof conflictsandtheirneutrosophicresolution.M.Intriligator [1]reviewedmathematicalapproachestothestudyof conflictresolutionsincrispenvironment.Heprepareda listofprimarymethodologicalthrustsasdifferential equations,decisionandcontroltheory,gameand bargainingtheory,uncertaintyanalysis,stabilitytheory, action-reactionmodelsandorganizationtheory. AnandalingamandApprey[2]proposedmultilevel mathematicalprogrammingmodelinordertodevelopa conflictresolutiontheorybasedontheintegrationofnon-cooperativegamewithinamathematicalparadigm.They postulatedconflictproblemasaStackelberg[3] optimizationwithleadersandfollowers.However,the modelissuitableonlyforthenormalversionof information distribution [4] when the strategy of all lower-levelplayersiscompletelyknowntotheleader.Yakir Plessner [5] employed the game theoretic model to resolve the conflict between Israel and the Palestinians.Pramanik andRoy[6]studiedgametheoreticmodeltotheJ&K conflictbetweenIndiaandPakistanincrispenvironment. ButthesituationandrelationbetweenIndiaandPakistan arenotstaticbutdynamicandneutrosophicinnature.So new approach is required to deal with the conflict.Ourcontributiontotheliteratureistodiscussbrieflythe genesis of the conflict and apply neutrosophic game theory for conflict resolution. Restofthepaperisorganizedinthefollowingway. Section2presentssomebasicsofneutrosophyand neutrosophic sets and their operations. Section 3 describes abriefhistoryandthegenesisofJammuandKashmir conflict.Section4isdevotedtoformulationneutrosophic gametheoreticmodeltoJammuandKashmirconflict between India and Pakistan. Section 5 presents concluding remarks. 2. Basics of neutrosophy and neutrosophic setsInthissection,wepresentsomebasicdefinitionsof neutrosophy,andneutrosophicsetsandtheiroperations due to Smrandache [7] and Wang et al.[8]. Definition1.Neutrosophy:Anewbranchofphilosophy, introducedbyFlorentinSmarandachethatpresentsthe origin,nature,andscopeofneutralities,aswellastheir interactions with different ideational spectra. Neutrosophy is the basis ofneutrosophic set, neutrosophic probability, and neutrosophic statistics. Definition2.Infinitesimalnumber: issaidtobe infinitesimal number if and only if for all positive integers n, < 1/nDefinition3.Hyper-realnumber:Let >0bean infinitesimalnumber.Thehyper-realnumbersetisan 82 Neutrosophic Sets and Systems, Vol. 2, 2013 Surapati Pramanik, Tapan Kumar Roy, Neutrosophic Game Theoretic Approach to Indo-Pak Conflict over J ammu-Kashmirextension of the real number set, which includes classes of infinite numbers and classes of infinitesimal numbers.Definition4.Non-standardfinitenumber:1+=1+ , where1isitsstandardpartand its -non-standard part Definition5.Non-standardfinitenumber: -0=0- ,and 0 is standard part and is its non-standard part. Definition 6. A non-standard unit interval: A non-standard unitintervalcanbedefinedas||--0,1+-||.Here -0isnon-standardnumberinfinitelysmallbutlessthan0and1+is non-standard number infinitely small but greater than 1. MainPrinciple:Betweenanidea< >anditsopposite ,thereisacontinuum-powerspectrumofneu-tralities .Fundamental Thesis: Any idea is T% true, I% inde-terminate, and F% false, where T, I, F belong to subset of nonstandardunitinterval||-0,1+||andtheirsumisnot restricted to 100%. Definition7.LetXbeaspaceofpoints(objects)with generic element in X denoted by x. Then a neutrosophic set A in X is characterized by a truth membership function TA, anindeterminacymembershipfunctionIAandafalsity membershipfunctionFA.ThefunctionsTA,IAandFAare real standard or non-standard subsets of] 0-, 1+[ i.e. TA : X ]0-, 1+[ ; IA : X ]0-, 1+[;FA : X]0-, 1+[ It should be noted that there is no restriction on the sum of TA(x), IA(x), FA(x) i.e.0-TA(x) + IA(x) +FA(x) 3+.Definition8.ThecomplementofaneutrosophicsetAis denoted by cA and is defined by= ) x ( TcA) x ( T } 1 {A+;) x ( I } 1 { ) x ( IA cA =+; ) x ( F } 1 { ) x ( FA cA =+Definition 9. A neutrosophic set A is contained in the other neutrosophic set B, A B if and only if the following result holds. ) x ( T inf ) x ( T infB A ,) x ( T sup ) x ( T supB A (1) ) x ( I inf ) x ( I infB A ,) x ( I sup ) x ( I supB A (2) ) x ( F inf ) x ( F infB A ,) x ( F sup ) x ( F supB A (3) for all x in X. Definition 10. Single-valued neutrosophic set (SVNS): Let Xbeauniversalspaceofpoints(objects)withageneric elementofXdenotedbyx.Asingle-valuedneutrosophic setX ~N is characterized by a true membership function) x ( T~N,afalsitymembershipfunction) x ( F~Nandan indeterminacyfunction) x ( I ~Nwith ), x ( T~N), x ( I ~N ) x ( F~N [0, 1] for all x in X.When X is continuous a SVNSs,N~can be written as =x~ ~ ~ , x ) x ( F ), x ( I ), x ( T~N N NN . X x and when X is discrete aSVNSsN~can be written as x / ) x ( F ), x ( I ), x ( T~m1 i~ ~ ~==N N NN ,. X x SVNS is an instance of neutrosophic set that can be used in reallife situations like decisionmaking,scientific anden-gineering applications. In case of SVNS, the degree of the truthmembership ), x ( T~Ntheindeterminacymembership ) x ( I ~N and the falsity membership) x ( F~N values belong to [0, 1]. It should be noted that for a SVNS , N~3 ) x ( F sup ) x ( I sup ) x ( T sup 0 ~ ~ ~N N N+ + , . X x (4) and for a neutrosophic set, the following relation holds , 3 ) x ( F sup + ) x ( I sup + ) x ( T sup 0+~ ~ ~-N N N. X x (5) Definition 11. The complement of a neutrosophic setN~ is denoted by c~N and is defined by = ) x ( T~cN) x ( F~N; = ) x ( I~cN 1 ) x ( I ~N ;= ) x ( F~cN) x ( T~NDefinition 12.A SVNS A~Nis contained in the other SVNS B~N , denoted as A~NB~N , if and only if ) x ( T ) x ( TB~A~N N; ) x ( I ) x ( IB~A~N N ; ) x ( F ) x ( FB~A~N N. X x Definition 13. Two SVNSs A~Nand B~Nare equal, i.e. A~N= B~N , if and only if A~NB~Nand A~N B~N . Definition 14. Union: The union of two SVNSsA~Nand B~Nis a SVNSC~N , written as C~N = A~NB~N . Its truth membership, indeterminacy-membership and falsity mem-bership functions are relatedas follows: )) x ( T ), x ( T ( max = ) x ( TB~A~C~N N N;)) x ( I ), x ( I ( max = ) x ( IB~A~C~N N N; )) x ( F ), x ( F ( min = ) x ( FB~A~C~N N Nfor all x in X. Definition 15. Intersection: The intersection of two SVNSs A~Nand B~Nis a SVNSC~N , written as C~N = A~N B~N , whose truth membership, indeterminacy-membership and falsity membership functions are relatedas follows: )) x ( T ), x ( T ( min = ) x ( TB~A~C~N N N; )) x ( I ), x ( I ( min = ) x ( IB~A~C~N N N; )) x ( F ), x ( F ( max = ) x ( FB~A~C~N N Nfor all x in X. 3. BriefhistoryandthegenesisofJammuandKashmir conflict ItissaidthatKashmirismorebeautifulthantheheaven, and the benefactor of the supreme blessing and happiness. The account of Kashmir is found in the oldest extant book- NilamatPurana.Kalhan,Kashmirsgreatesthistorian scholarlydepictedthehistoryofKashmirstartingjust 83 Neutrosophic Sets and Systems, Vol. 2, 2013 Surapati Pramanik, Tapan Kumar Roy, Neutrosophic Game Theoretic Approach to Indo-Pak Conflict over J ammu-Kashmir beforethegreatMahabharataWar.AccordingtoKalhan, theMauryanemperorAshokaannexedKashmirin250 B.C.HeembracedBuddhismaftertheKalingawar.He madeitastatereligion.HebuiltmanyBihars,temples speciallyShivatemple.AccordingtoChinesetraveler, HuenTsangoverfivethousandBuddhistMonkssettled down in Kashmir during the reign of Ashoka. After the fall of Maurya dynasty, it is believed that Kashmir for over two hundredyearswasruledbyIndo-GreekKingsbeforethe startof"Turushka"(Kushan)ruleinthestate.Thus,the peopleofKashmircameincontactwiththeGreeks.The reflectionofwhichisfoundonthebeautifularchitectural andsculpturalstyleofoldKashmiritemples,andthe coinage of the later Kashmiri kings.ThezenithofBuddhistpowerinKashmirwasreachedin thereignofkingKanishka.InfluencedbyIndianculture, Kanishka adopted Buddhism and made it the state religion. Duringhisreign,itisbelievedthattheforthBuddhist CouncilwasheldatKundalavanainKashmir.Itwas enthusiasticallyattendedbyalargenumberofscholars, theoreticians,andcommentators..Duringhisreign, BuddhismpropagatedinTibet,ChinaandCentralAsia. However,Buddhismwasfollowedbyarevivalof Hinduism and Hindu rulers ruled Kashmir up to 1320.Rinchan(1320-1323)ascendedthethroneon6thOctober 1320. He was the first converted Islam ruler in the history of Kashmir. Shah Mir ascended the throne with the title of SultanShamsuddin(1339-1342)in1339A.D.andShah-Mirdynasty(1339-1561)ruledthestatefor222years. ShahMirdynastyisoneofthemostimportantinthe annalsofKashmir,inasmuchasIslamwasfirmly establishedhere.DuringChakrule(1561-1586)Sunnni Muslims and Hindus alike were persecuted. Akbar,theMughalEmperorannexedKashmirin1586.It isimportanttonotethatundertheMughalreign(1586-1752),peoplegotslightreliefandlivedhonorably. However,theMughalusedforcedlaborintheirvisitsto Kashmirintermsofahugeretinueofunpaidlaborersto carry their goods and other supplies for the journey. Afghanrule(1752-1819)succeededinmaintainingtheir suzeraintyoverKashmirforaspanofsixty-sevenyears. The Afghans were highly unscrupulous in the employment offorcedlabor.ThecommonKashmirianpeoplewere tiredoftheirferocity,barbarityandpersecution.Itistrue historyofKashmirthatallsectionsofpeoplesuffered during Afghan rule but the principal victims of these cruel werethepeasants.Duringthiseraallcruelandinhuman measuresofAfghanrulerscouldnotputanendthebasic tradition of Kashmiri. The reign of Sikh Power (1819-1846) in Kashmir lasted for only27years.ItistobenotedthattheSikhscontinued withthepracticeofforcedlaborinordertotransportof goodsandmaterials.AccordingtoLawrence[9],"toall classesinKashmirtoseethedownfalloftheevilruleof Pathan,andtononewasthereliefgreaterthantothe peasantswhohadbeencruellyfleecedbytherapacious sardarsofKabul.IdonotmeantosuggestthattheSikh rulewasbenignorgood,butitwasatanyratebetterthat that of the Pathans. 3.1 Dogra rule (1846-1947) Dogradynestyplayedaninimportantroleindeveloping Jammu and Kahmir State. 3.1.1 Gulab Singh (1846-1857) TheStateofJammuwasconferredonGulabSinghwith thetitleofRajabyMaharajaRanjitSinghofPunjabin 1820. He annexed Ladakh in September 1842. Some parts ofGilgitandBaltistanwereinvadedbefore1846.The StateofJammuandKashmir(J&K)isfoundedthrough Amritsartreatyin1846betweentheEastIndiaCompany and Raja Gulab Singh who buys Kashmir Valley from the EastIndiaCompanyforRs.7.5millionandannexesitto Jammu and Ladakh already under his rule. Thus the Dogra dynestyestablishesin1846.GulabSinghconquered Muzaffarabad in 1854.3.1.2 Ranbir Singh (1857-1885) RanbirSingh(1857-1885)ascendedthethroneafterhis fatherdeathin1857A.D.,whoruledfrom1857to1885 A.D.LordNorthbooksGovernmentrecommendedfora politicalofficertoresidepermanentlyattheMaharajas CourtinSeptember26,1873A.D.ABritishResident remainedpermanentlyatthecourtof Maharajarelatingto the external relations of British India from 1873.3.1.3Maharaja Pratap Singh (1885-1925) MaharajaPratapSingh(1885-1925)ascendedthethrone afterhisfatherdeathin1885.Duringhisrule,British powerwasdeeplyinterestedinKashmirandthrough BritishResidentMaharajaPratapSinghwaskeptunder pressure.InSeptember1885duringtheinitialstageof PratapSinghsrule,theBritishGovernmentchangedthe status of the British officer Special Duty in Kashmir to that ofapoliticalResident.PratapSingh'sAddressinDurbar October19,1885revealedthatthepositionofpolitical officerinKashmirhasbeenplacedonthesamefooting with that of Residents in other Indian States in subordinate alliance with the Government. British Government of India disposedMaharajain1889.Maharajawasofferedan allowance, which was ungenerously described as sufficient for dignity but not for extravagance, would be made to him. No specific period was set for this arrangement to come to anend.ColonelNisbet,ResidentofKashmirbecamethe virtualrulerbecausealthoughtheCouncilofminister wouldhavefullpowersofadministration,theywouldbe expectedtoexercisethosepowersundertheguidanceof the British Resident. Without consulting with him, Council wouldnottakeanyimportantdecisionandtheCouncil would follow Residents advice whenever it was offered.In1889,theBritishGovernmentinstitutedGilgitAgency underthedirectruleofBritishpoliticalagent.Colonel AlgeronDurand[10],thefirstBritishAgentinGilgit 84 Neutrosophic Sets and Systems, Vol. 2, 2013 Surapati Pramanik, Tapan Kumar Roy, Neutrosophic Game Theoretic Approach to Indo-Pak Conflict over J ammu-Kashmirrecords the Russian influence for creation of Gilgit Agency in his Book, The Making of a Frontier. He remarked in a statementWhyithasbeenaskedshoulditbeworthour while to interfere there whatever happened? The answer is ofcourseRussiaExpensiveastheGilgitgamemight havebeen,itwasworththeCandle.ViceroyLord Curzon reinstated Maharaja Pratap Singh in power in 1905 A.D. The State Council is abolished in May 1906 A.D. 3.1.4 Hari Singh (1925-1947) HariSingh(1925-1947)ascendedthethroneafterhis grandfather, Pratap Singhs death in 1925.During his rule theagitationagainsttheDograrulestartedmainlyagainst themisrule,corruptadministration,autocraticrule, repressiononthesubjectsattheslightestexcuseandlack ofadministrativeefficiency.MaharajaHariruthlessly crushedamassuprisingin1931.HariSinghconstituted GrievancesEnquiryCommissionheadedbyB.J.Glancy on12November1931foraprobeintothecomplaintsof thepeopleofKashmir.InApril1932,thecommission recommendeditssuggestions.Amongthese recommendations,theimportantonewasthesteptobe takenforpropagatingeducationforKashmiriMuslims. TheCommissionrecommendedtogivepayment[11] Kashmiri people for Government work. Inthe order dated 31May1932,MaharajaHariSinghacceptedthe recommendationofthePresidentoftheKashmir ConstitutionalReformsCommittee,B.J.Glancyand appointedaFranchiseCommitteetoputforwardtentative suggestionregardingtheimportantquestionofthe franchiseandthecompositionoftheassembly.Inthis backgroundAllJammuandKashmirMuslimConference (AJKMC)wasformedundertheleadershipofSheikh Abdullahin1932inOctoberinSrinagar.Theconference held from 15 to 17 October 1932. In1934,theMuslimConferencedemonstrateditssecular viewwhenitforwardedmemorandumdraftedbyGhulam Abbus to the Maharaja demanding early implementation of thereportofGlancyCommissionandspecificallyurged forasystemofjointelectoratesintheState.In1934, Maharaja introduced a Legislative Assembly. However 35 of its 75 members were to be nominated. 8 per cent of the populationwasallowedtocastvote.Tobecomeavoter literacyandpropertyqualificationswerespecified.The Assemblyenjoyedonlyconsultativepowers.Maharaja furtherreformedmakingtheprovisionforCouncilof Ministersandajudicialandlegislativebranchofpublic administration in 1939.However, Maharaja enjoyed most of the decision powers under the new reforms. On 26 March 1938 Sheik Abdullah iterated two important points: i) to put an end communalism by ceasing to think in termsofMuslims-non-Muslimswhendiscussingpolitical problems.ii)Universalsuffrageonthebasisofjoint electorate.It is to be noted that the national demand issued in August 1938 was signed among others by Pandit Jia Lal Kilam, Pandit Lal Saraf, Pandit Kasyap Bandhu. Under the leadership of Sheik Abdullah AJKMC felt the necessity of common platform to struggle against the rule of Maharaja. Afterseriesofdiscussionsanddebates,theworking committeeofAJKMCtookthehistoricdecisionofre-christeningtoJammuandKashmirNationalConference (orsimplyNationalConference)on24June1938.On27 April1939,NationalConferencecameintobeing.Its secular credentials set a new pace for the politics of Jammu Kashmir.NationalConference[12]consistedofmany leadersofminoritycommunitieslikeHindu,Sikhetc during 1940s. In the history of India subcontinent, the Pakistan resolution demanding the creation of an independent state comprised of all regions in which Muslims are the majority is passed atIqbalPark,LahoreonMarch23,1940byMuslim League. ThesecularizationofKashmirpoliticsandredefinitionof the goal helped immensely National Conference to come in closecontactwiththeIndianNationalCongress.In1942 NewKashmirmanifestowasformulatedunderthe leadership of Dr. N. N. Raina by a brilliant group of young communistoperatingwithintheNationalConferencewho weremostlyresponsibleforintroducingthenationalist movementtotheconceptofsocialistpatternofsociety based on equality, democracy and free from exploitation. It consists of two parts: a) the constitution of the state; b) the NationalEconomicPlan.Underthesoundleadershipof Abdullah,NationalConferenceledapowerfulmass movementinordertofindanewpoliticalandeconomic orderinKashmirandotherpartsofJammuregion.The NationalConferencestartedagitationagainsttheDogra rule in 1945.In the grave political situation, offering him allcharges,RamChandraKakwasappointedasPrime Ministerinordertobringtheagitationincontrol.InMay 1946NationalConferencelaunchedQuitKashmir movementfollowingtheQuitIndiaMovementin1942 led by the Indian National Congress. Mohamod Ali Jinnah wasnotinterestedintheQuitKashmirMovement[13] ratherblamedthemovementasactofGundas.InMarch 1946 Crisps Mission came to visit India. Sheikh Abdullah sentatelegrambydemandingfreedomofpeopleof Kashmir on withdrawal of British power from India. PrimeMinisterofJ&KRamChandraKakdeclared emergencytocrackdownthemovement.Abdullahwas arrested on 20 May 1946. The State Government employed a wave of arrests and a policy of repression throughout the State.Thepeopleprotestedstronglyandseveralagitated Kashmiri people were killed and injured due to clash with armedforcesofMaharaja.TheIndianNationalCongress andtheAllIndiaStatespeoplesConferencesupported NationalConferencestrongly.SheikAbdullahwas imprisonedforthreeyearsforantinationalactivities. NationalConferencewasbanned.InJanuary1947, NationalConferenceboycottedelectionsbecauseof 85 Neutrosophic Sets and Systems, Vol. 2, 2013 Surapati Pramanik, Tapan Kumar Roy, Neutrosophic Game Theoretic Approach to Indo-Pak Conflict over J ammu-Kashmir repression.MuslimConferencegrabbedtheopportunity and won 16 out of 21 Muslim seats.3.2 Muslim ConferenceMuslimConferencedidnotsupporttheQuitKashmir agitation.MuslimConferencediscouragedthepeopleof Kashmirfromjoiningtheagitationinthesametuneof MuslimLeague.On30May1946,ChaudhuryGulam AbbasthePresidentofMuslimConferencestatedthatthe agitation had started at Congress leaders behest in order to restorethelostprestigeoftheNationalist.TheMuslim ConferenceadoptstheAzadKashmirResolutionon26 July 1946 calling for the end of autocratic Dogra rule in the region and claiming the right to elect their own constituent assembly. He said that the primary task [14]was to restore the unity of the Muslim nation and there be no other place foranhonestandself-respectingMuslimbutinhisown organization.On25October1946,StateGovernment arrestedanddetainedfourtopleadersofMuslim Conferences. 3.3 British Cabinet Mission InMarch1946,TheBritishCabinetMissionheld conference about a week at Simla with four representatives, two each of the Congress and the Muslim Leagues and the conference broke down on the issue of Pakistan and parity in the proposed interim government.On 16 May 1946, the CabinetMissionannouncedtheirownproposals,the essenceofwhichwasthecreationofaConstituent Assembly to frame the Constitution of India, which was to bebasedontheprinciplethattheCenterwouldcontrol onlythreesubjects,viz.,Defense,ForeignAffairsand Communicationsandthecreationofthreegroupof provinces-two of the areas claimed by Muslim League for Pakistan in the east and the west and the third of the rest of the subcontinent [15]. 3.4 Interim Government announced On25June1946,theCongressWorkingCommittee announcedtheirrejectionoftheplanofInterim Government.OnJune26,1946,LordWavellannounced thathewouldsetupatemporarycaretakerGovernment of officials to carry on in the interim period. In July 1946,theMuslimLeague withdrewitsacceptance oftheCabinetMissionsplanandresolvedthatnowthe timehascomefortheMuslimnationtoresorttodirect actiontoachievePakistan,toasserttheirjustrights,to vindicatetheirhonorandtogetridofthepresentBritish slaveryandthecontemplatedfuturecaste-Hindu domination at a meeting in Bombay.Accepting the invitationfromtheViceroytoconstitutean interim Government, on 6 August 1946, Jawaharlal Nehru formedit,whichconsistedofsixHindus,includingone DepressedClassmember,threeMuslimsofwhomtwo belongedneithertotheCongressnortotheLeague,one Sikh,oneChristian-andoneParsee.Itstartedfunctioning on2ndSeptember1946.TheLeaguejoinedtheInterim Government in the last week of October 1946 but was not prepared to join the Constituent Assembly, which led every dayamoreandmoredifficultanddelicateonaccountof thedifferencesbetweenthecabinetministersofCongress and the Muslim League.0n 26 November 1946, Mr. Atlee invitedLordWavellandrepresentativesoftheCongress andtheMuslimLeaguetomeetinLondontoattemptto resolve the deadlock The discussions were held from 3 to 6 December1946butdidnotyieldanyagreedsettlement. The first meeting of the Constituent Assembly of India was heldinon11December1946.TheMuslimLeague boycotteditanditdevelopedastakeinsabotagingthe Assemblys work.On20February1947,PrimeMinisterAtleedeclaredthat Britain would transfer power by June 1948, by which time the Congress and the Muslim League were supposed to re-solvetheirdifferences.On24MarchMountbattenwas sworninasViceroyandGovernorGeneralofIndiain place of Wavell. After negotiations with the leaders of dif-ferentpoliticalparties,Viceroy,LordMountbattenan-nounced that long before June 1948, the Dominions of In-dia and Pakistan would be created and that the question of Indian states would be dealt with in the light of the Cabinet Mission'smemorandum[16]of12May1946.Toapprove the Mountbatten plan accepted by British Cabinet, a confe-rencebetweenMountbattenandrepresentativesofthe Congress and the Muslim League was held on 2 June 1947. On 3 June 1947 a White paper was issued which stated the detailprocedureofthepartitionofIndia.Regardingthe PrincelyStatesitdeclaredthatBritishpolicytowardsIn-dianStatescontainedintheCabinetMissionsmemoran-dum [17] of 12th may, 1946 remained unchanged. 3.5 Partition Plan accepted by CongressOn14June1947,inahistoricsessionofAllIndia Congress Committee (AICC) in New Delhi, Pandit Ballabh PantmovedtheresolutiondealingwiththeMountbatten planforpartitionBritishIndia.MahatmaGandhi intervenedinthedebateintheseconddayandexpressed thathewasalwaysagainstthepartitionbutsituationhad changedandappealedtosupporttheresolution.On15 June 1947, the resolution was passed with 29 votes in favor and 15 against.Mr. Jinnah clearly expressed Muslim League view [18] on the question of Princely States on 17 June 1947 by saying "ConstitutionallyandlegallytheIndianstateswillbe independentsovereignstatesontheterminationof paramountcy and they will be free to decide for themselves andadoptanycoursetheylike;itisopentothemtojoin the Hindustan Constitutional Assembly or decide to remain independent. In case they opt for independence they would enter into such agreements or relationships with Hindustan or Pakistan as they may choose".3.6 Partition and riots 86 Neutrosophic Sets and Systems, Vol. 2, 2013 Surapati Pramanik, Tapan Kumar Roy, Neutrosophic Game Theoretic Approach to Indo-Pak Conflict over J ammu-Kashmir Calcutta,capitalofBengalwitnessedabeginningof holocaustonanunprecedentedscaleon16August1946, whichwasdeclaredapublicholidaybytheMuslim LeagueGovernmentofBengal.Itwasestimatedthat Jinnahs direct action [18] caused death of more than 5000 lives, and over 15000 people were injured, besides 100000 beingrenderedhomeless.Afterafortnight560people werekilledinBombay.AfterCalcutta,onOctober1946, seriousanti-HinduriotseruptedinNoakhaliinEast BengalfollowedbymassacredofMuslimsin BiharThe chainreactionofriotsstartedinthePunjabcausinglarge scalekillingsofHindus,Sikhs,andMuslimsshortly afterwards.3.7 Development in Jammu and Kashmir Basedontwo-nationtheory,Indiawaspartitionedinto Pakistan and India in August 14, 1947. The princely states wereofferedtherightundertheIndianIndependentAct 1947andGovernmentofIndianAct1935[19]to accede either to India or Pakistan or remain independent. It seemedthatHariSingh,thethenMaharajaofJammuand KashmirhopedtocreateindependentKingdomor autonomyfromIndiaandPakistan.Hedidnotaccedeto either of two successor dominions at the time of accession. AllJammuandKashmirRajyaHinduSabhapasseda resolution [20] expressing its faith in Maharaja Hari Singh andextendeditssupporttowhateverhewasdoingor mightdoontheissueofaccessionin1947.On15June 1947,animportantresolution[21]regardingtheprincely states saying the lapse of paramountcy does not lead to the independence of the princelystateswas adoptedbyAICC unanimously.Contrarytothis,Mr.Jinnahclearly expressed the view [18] of Muslim League on the question ofPrincelyStateson17June1947bysaying "ConstitutionallyandlegallytheIndianstateswillbe independentsovereignstatesontheterminationof paramountcy and they will be free to decide for themselves andadoptanycoursetheylike;itisopentothemtojoin theHindustanConstitutionalAssemblyorthePakistan Constituent Assembly, or decide to remain independent. In thelastcase,theyenterintosuchagreementsor relationshipwithHindusthanorPakistanastheymay choose ". On19July1947,theworkingcommitteeofMuslim Conferencepassedamodifiedresolution[22]infavorof independence, which respectfully and fervently appealed to the Maharaja to declare internal autonomy of the state and accedetoPakistanregardingtodefense,communication andexternalaffairs.KhurshidAhmad,Jinnahspersonal SecretaryduringhisstayinKashmironthecrucialdays for the question of accession gave Maharaja assurance [23] thatPakistanwouldnottouchahairofhisheadortake awayaiotaofhispower.BeforepartitionBritish Government restored the Gilgit area, an important strategic region,hithertoadministeredbyaBritishagency,toJ&K without taking the verdict of the local people. 3.8 Standstill AgreementPakistan became independent on 14 august 1947. India and fewprincelystates,whichdidnotjoineitherofIndiaor Pakistan,becameindependenton15August1947.Inthis way J&K attained the statusof independenton15 August 1947.On15AugustpostofficesinJ&Khoistedthe Pakistaniflags.MaharajaHaiSinghsignedastandstill agreement with Pakistan on 16 August 1947 with regard to Statespostalservices,railways,andcommunicationsand hopedtosignsimilaragreementwithIndiawithregardto externalaffairs,controlofstateforces,defenseetc.India [23]didnotshowanyinterestintheacceptanceofthe offerofstandstillagreement..On18August1947a controversycameintolightwhenSirCyrilRadcliffe awarded a portion of Muslim majority Gurudaspur District toIndiacausingfundamentaldifferencesinJ&Ks geopoliticalsituation.Thesubcontinentexperienced communalriotsduringthesedays.Bythistime,Muslim majorityPoonchestatewithintheJammuregion experiencedserioustroubleswithregardtosomelocal demandsliketherehabilitationof60,000demobilized soldiersoftheBritisharmybelongingtothearea.The agitationfinallytransformedintocommunalformhaving mixed with other issues. The state army refused to fire on the demonstrators with whom they had religious and ethnic ties.Theagitationturnedtotheformofarmedrevolt because of mass desertion from army. The supply of arms andammunitionandotherassistancefromoutsidethe bordermagnifiedtherevolt.TheKashmirSocialistparty passed a resolution on 18 September 1947 to join Pakistan andnotIndia.ThepartyimpressedonMaharajathat withoutanyfurtherunnecessarydelayheshouldmakean announcementaccordingly.Itistobenotedherethata convention of Muslim Conference workers formally asked for accession to Pakistan on 22 September 1947.MaharajaHariSinghreleasedSheikhAbdullahfrom prison along with some other National Conference workers on29September1947buthedidnotreleasetheworkers ofMuslimConferenceduetogravesituationofthestate. PakistantermedAbdullahsreleaseasaconspiracy becauseworkersofMuslimConferencewerenot simultaneouslyreleased.ByOctober,communalriots spreadalloverJ&K.Themassinfiltrationbakedby Pakistaniarmyjeopardizedtheenvironmentofthestate.Pakistanviolatedthestandstillagreementbystopping regularsupplyoffood,salt,petrolandessential commoditiesfromPakistan.Thecommunicationsystem controlledby PakistaniGovernmentdid notrenderproper service.On21October1947,Pakistandecidedtosettlethefuture of Kashmir with the power of gun suspecting that Mahara-ja was likely to accede to India. Jinnah, the Governor Gen-87 Neutrosophic Sets and Systems, Vol. 2, 2013 Surapati Pramanik, Tapan Kumar Roy, Neutrosophic Game Theoretic Approach to Indo-Pak Conflict over J ammu-Kashmir eral of Pakistan personally authorized a plan [25] to launch aclandestineinvasionbyaforcecomprisedofPathan (Afghan) tribesmen, ex-servicemen and soldiers on leave. Itwaswitnessedthatchargesandcounterchargeswere beingmadebyboththegovernmentofJ&KandPakistan duringthemonthofOctoberandfinallyOn22October 1947,2000tribesmenfromNorthwestFrontierProvince (NWFP)ofPakistanandotherPakistaninationalsfully armedwithmodernarms,underthecommandoftrained generals,startedinvasiontocapturing thestatesterritory. TheMuslimsintheWesternpartofKashmirestablished their own independent (Azad) Kashmir Government on 24 October 1947.The State forces were wiped out in fighting. The tribesmen resorted to indiscriminate slaughter of both HindusandMuslims[26].Theyreachedwithin15miles fromcapitalSrinagar.Underthisgreatemergencyofthe situation, Maharaja sought Indian military assistance in his letter dated 26 October 1947 along with the Instrument of Accession[27]toMountbatten,theGovernorGeneralof India.ThereaftertheMaharajasignedtheinstrumentof accession,whichtheGovernorGeneralMountbatten acceptedon27October1947byaddingthatthequestion of accession [28] should be settled by a referendum. Indian forces[29]airliftedatSrinagaralmostatthecrucial moment,for,afewminuteslatertheairfieldmightwell havebeeninenemyhands.MembersoftheNational ConferenceprovidedlogisticalsupportfortheIndian forces.InfuriatedbyIndianintervention,on27October 1947, Pakistani Governor General, Mohammed Ali Jinnah orderedLt.GeneralSirDouglasGracey,Chiefofthe PakistaniArmy,tosendPakistaniregulartroopsto Kashmir,butFieldMarshallAuchinleck,theSupreme Commanderofthetransitionperiodsucceededin persuadinghimtowithdrawhisorders.Amessage[30] wassenttotheGovernorGeneralandthePrimeMinister of India to go to Lahore for discussion regarding Kashmir.3.9 Indo-Pak talks On 1November 1947,at ameetingofGovernorsGeneral of India and Pakistan at Lahore, Mountbatten offered to re-solve the J&K issue by holding referendum. Rejecting the Mountbatten formula, M.A. Jinnah remarked that a plebis-citewasredundantandundesirable.H.V.Hodson[31] has recorded in his book, The Great Divide, that M.A. Jin-nahobjectedthatwithIndiantroopspresentandSheikh Abdullah in power the people would be frightened to vote forPakistan.Jinnahproposedasimultaneouswithdrawal ofallforces-theIndiantroopsandtheinvadingforces. Hereitisinterestingtonotethatwhenhewasaskedhow anyone could guarantee that the latter would also be with-drawn,Jinnah[30]repliedIfyoudothisIwillcallthe whole thing off. In connection with the steps to ascertain the wishes of the people of J&K, Mountbatten was in favor of a plebiscite under the auspices of United Nations while M.L.JinnahproposedthatheandMountbattenshould have plenary power to control and supervise the plebiscite.Ultimately, the first direct bilateral talks broke down. On1January1948,basedontheadviceofMountbatten, IndialodgedacomplaintwiththeSecurityCouncil invokingarticles35ofChapterVIoftheUNCharterto recommendappropriateproceduresormethodsof adjustmentforthepacificsettlementofdisputesandnot for action with respect to acts of aggression as provided for in Chapter VII of the Charter [32]. India reiterated her pledge of her conditional commitment to a plebiscite under internationalauspicesoncetheaggressorwasevicted. PakistancontradictedthevalidityoftheMaharajas accessiontoIndia[33],and urgedthe SecurityCouncilto appointacommissionforsecuringacease-fireand ensuringwithdrawalofoutsideforces,andconductinga plebiscite in order to determine the future of J&K. 3.10RoleoftheUnitedNationSecurityCouncil (UNSC)Both India and Pakistan denied implementing the UN reso-lutions [34-36] for a free and impartial plebiscite in order to put an end to the situation for the accession of J&K. Having taken note of the developments in J&K, the United Nations Commission for India and Pakistan UNCIP sub-mitted a draft resolution [36] consisting of three parts to the council on 13 August 1948. Part I of the resolution comprised of instruction for a cea-sefire.PartIIoftheresolutiondealtwiththeprincipleofatruce agreementwhichcalledforPakistantowithdrawtribes-men, Pakistani nationals not normally resident therein who had entered the State of J&K for the purpose of fighting, to evacuatetheterritoryoccupiedbyPakistanandafterthe noticeoftheimplementationoftheabovestipulationby the UNCIP India was to withdraw the bulk of her forces in stagesfromJ&Kleavingminimumstrengthwiththeap-proval of the commission in order to ensure law, order and peace in the State. PartIIIoftheresolutionappearedtobeimportantasit clearlyexpressedthatboththeGovernmentofIndiaand theGovernmentofPakistanreaffirmtheirwishthatthe futurestatusoftheStateofJ&Kshallbedeterminedin accordance with the will of the people.The second resolution [37] specified the basic principle of plebiscitewasformallyadoptedon5January1949after acceptanceofIndiaandPakistanon23and25December 1948 respectively. AnimportantdevelopmentoccurredwhenbothIndiaand Pakistan agreed to the cease-fire line in 1949. This enabled the UN to finally send a Military observer Group to super-vise the line [38]. The ceasefire came into effect on 1 Jan-uary 1949. The most important long- term outcome of the first Indo-Pak war was the creation of ceasefire line. Thus UNCIPsucceededinimplementingtheimportantprovi-sionofPartIoftheresolution.Inordertomonitortothe 88 Neutrosophic Sets and Systems, Vol. 2, 2013 Surapati Pramanik, Tapan Kumar Roy, Neutrosophic Game Theoretic Approach to Indo-Pak Conflict over J ammu-Kashmir ceasefire line (CFL), the UNCIP sent a Monitoring Group forIndiaandPakistan(UNMGIP)toJ&Kon24January 1949 relying on its resolution of 13 august 1948. In Kara-chion27July1949,themilitaryrepresentativesofIndia and Pakistan, duly authorized, approved CFL and thus ap-proved the presence of UNMGIP [39]. In March 1949, the conflicting attitudes came into light as IndiaandPakistanexpressedtheirviewpointsbeforethe trucesubcommitteeoftheUNCIP.On15April1949, UNCIP transmitted to the governments of India and Pakis-tan its own proposals [40], which were: i)tocreateacease-fireline,eliminatingallno manslandsandbasedonthefactualpositionofthe troopsin January 1949. ii)todrawaphasedprogramofwithdrawalofPa-kistanitroopstobecompletedinsevenweeks, and the withdrawal of all Pakistani nationals. iii)toaskIndianforcesalsotowithdrawinaccor-dance with a phased program after the withdrawn oftribesmenandPakistaninationalandafterthe declaration of UNCIPs satisfaction regarding the troops withdrawal of Pakistan. iv)to release all prisoners of war within one month. v)to repeal all emergency laws.vi) to release all political prisoners.BothIndiaandPakistan[41] couldnotaccept the propos-als because of their own interest. TheUNCIPproposedarbitrationontheissuesregarding thepartIIoftheresolutioninalettertothetwoGovern-ments on 26 August 1949 and named Fleet Admiral Ches-ter Nimitz as the Arbitrator.Pakistan accepted the propos-al on 7 September 1949 but India rejected this proposal of arbitration.TheCzechoslovakrepresentativeoftheUN-CIP, Dr. Oldrich Chyle (Chyle took the post after resigna-tion of Korbel) criticized the UNCIPs work [42]. Accord-ing to him, the arbitration move was a pre-planned attempt on the part of the USA and UK to intervene in the dispute. On17December1949,theUNSCnameditspresident GeneralA.G.L.McNaughtonofCanadaastheInformal Mediator [43], instead of commission to negotiate a demi-litarization plan in consultation with India and Pakistan. He submitted his proposal on 22 December 1949. Pakistan ac-cepted the proposals, suggesting minor amendments while Indiasuggestedmajoramendments:onecallingforthe disbandinganddisarmingofAzadforces,andtheother dealingwiththereturnoftheNorthernAreastoIndiafor purposesofdefenseandadministrationofJ&K.Pakistan was unable to accept Indian amendments [44] as a clear re-jectionoftheproposals.Pakistanagreedtosimultaneous demilitarization but Indian rejected it on the grounds of the legal and moral aspects of the plan. TheUNSCadoptedanotherresolutionintroducedbyC. BlancoofCubaonbehalfoffourpowersCuba,Norway, UKandUSAon14March1950,whichcalleduponthe twonations,withoutprejudicetotheirrightsorclaimsto prepareandexecutewithinthestipulatedperiodoffive months for the demilitarization of J&K based on proposals of McNaughton and for self determination [45] through an impartialplebiscite.TheresolutionterminatedtheUNCIP andtransferredtheirpowersandresponsibilitiestoaUN representative.3.10.1 Dixon mediation SirOwenDixon,UNRepresentativesubmittedhisrec-ommendationstotheUNon15September1950.Hesug-gestingaunique proposal [46] limiting theplebiscite only to the Kashmir Valley claimed by both by Pakistan due to itsMuslimmajorityandthewatersofJhelum.Indiaand Pakistan rejected the plan. UN representatives worked to negotiate for free and impar-tialplebisciteinJ&Kuntil1953buttheireffortsbrought no fruit. The UN continued its efforts for a plebiscite [47], but all attempts of UN failed due to the conflicting and di-vergentattitude ofthe GovernmentsofIndiaand Pakistan towards the dispute and the cold war [18].The fifth report of Dr. Frank P. Graham [48] suggested direct negotiations between India and Pakistan.Thus the UN attempts at solv-ingtheproblemofJ&Kcametoendwhichreflectedthe limitations of the UN. ThearmiesofIndiaandPakistanwagedaninconclusive war(1947-48)foroverayearinJ&K.TheIndianarmy occupiedalmosttwothirdofJ&Kremaining1/3portion wasunderthecontrolofPakistanwhichiscalledAzad Kashmir or Pakistan occupied Kashmir (POK).3.11 Indo-Pak negotiation (1962-1963) India experienced a huge defeat in 1962 war against China. The J&K dispute became the subject of Indo-Pak negotia-tion in late 1962 but no agreement could be signed for res-olutionofJ&Kquestiondespitesixroundtalksbetween an Indian delegation headed by Swaran Singh and a Pakis-tanidelegationheadedbyZ.A.Bhuttofrom27December 1962 and 16 May 1963. 3.12 Sino-Pak border agreement 1963 On2March1963,theSino-PakistanBorderAgreement was signed in Peking and they had agreed that after the set-tlementoftheKashmirdisputebetweenPakistanandIn-dia, the sovereign authority concerned would reopen nego-tiationswithChinaontheboundaryasdescribedinAr-ticle.BythisagreementPakistan[49]succeededinstabi-lizing Pakistans position regarding Kashmir in the eyes of ChineseGovernmentandcompellinghertorejectune-quivocally the contention that Kashmir belonged to India. 3.13 The Kutch conflict- a low intensity war In19April1965,Pakistanipermanentrepresentative[50] inUNmadeclaimsabout8960squarekilometersareaof Rann. Pakistaniclaimtothe Rann ofKutch wasbasedon 89 Neutrosophic Sets and Systems, Vol. 2, 2013 Surapati Pramanik, Tapan Kumar Roy, Neutrosophic Game Theoretic Approach to Indo-Pak Conflict over J ammu-KashmirthefactthattheRannwasalakeandaccordingto international law [51], the boundary line between India and PakistanmustbedrawnthroughthemiddleoftheRann. OnotherhandIndiaarguedthat theRannofKutchwasa marshlandratherthanalake.IndiaaskedPakistanto restore the status quo ante. Tikka Khan, in command of the 18InfantryDivision,didpainstakinglyprepareforthe operationsandsucceededinadvancinginsideIndian territoriesinstrength,causingtothefalloftheIndian forwardposthastilypositionedthere.IndiaandPakistan foughtalowintensitywar.Theimportantaspectofthe conflict lies in the historic fact that both India and Pakistan accepted a ceasefire and arbitration on British intervention. On the other hand, India captured some Pakistani Posts in theKargilareaofLadakh.TheKutchdispute[52]was referredtoatribunalcomprisingofthreemembers,one nominatedbyIndia,anotherbyPakistanandaChairman chosenbytheUNSecretaryGeneral.Afteralong deliberationthetribunalawardedPakistan317square milesoutof3500squaremilesclaimedbyher.Indialeft the occupied posts of Pakistan in Kargil. 3.14 Indo-Pak war in 1965ThePakistaniGovernmentwasgreatlyemboldenedby presumably military success in the Rann of Kutch in 1965.InAugust1965infiltrationhadstartedinJammuand Kashmir to wage what Zulfikar Ali Bhutto called a war of liberation. On 10 August 1965, Z. A. Bhutto [53] publicly declaredhiscountrysfullsupporttothepeopleofKash-mirbutdeniedhiscountrysinvolvementintheKashmir trouble.On1September,1965Indianforcescrossedthe international border and sealed the borders of Kashmir. On 4September,Malaysiamovedaresolutionco-sponsored byBolivia,theIvoryCost,Jordan,theNetherlands,and UruguayproposinganimmediateceasefireinKashmir without calling Pakistan as an aggressor in the UNSC [54]. But it did not succeed in stopping the fighting. Ayub Khan backedtheinfiltrationwithafull-fledgedattackinthe Chhambsectorbycrossingtheinternationalborder,lead-ing to effective progress to reach Jaurian. On 5 September 1965,Indianforceslaunchedthree-prongedthrustinof West Pakistan in Lahore Sector and in Sialkot sector a day later.Followingthisdevelopment,Malaysianrepresenta-tivesubmittedanotherresolution[55]supportedbyBoli-via,theIvoryCost,Jordan,theNetherlands,andUruguay callinguponboththecountriestoceasehostilitiesand withdrawtheirtroopstothepositionsheldby5August 1965,whichwaspassedunanimouslyon5September 1965.ThegoodwillmissiontoIndiaandPakistanbythe U.N.SecretaryGeneral,UThantdidnotsucceed.Both countries were requested by U Thant to stop fighting with-outimposinganyconditiononeachother[56].Indiaac-ceptedunconditionalceasefirebutPresidentAyubKhan [57]imposedcertainpre-conditions:(i)Withdrawalofall forcesofbothIndiaandPakistan(ii)Inductionofforeign forces,preferablyAfro-AsianunderUNauspicious,(iii) Holding a Plebiscite in Kashmir within three months of the cease fire. Armiesofboththecountriesengagedinlarge-scale combat in a series of sharp and intense actions along theceasefirelineinJ&Kandtheinternationalborderin Punjab,Rajasthan,andGujaratbyemployingimport weaponrysystembutoutmodedwarstretegies.They reachedtothepointofexhaustion,battlefatigue.The representativeoftheNetherlandsmovedthedraft resolution[58],whichwasaccepted,bybothIndiaand PakistanintheUNSCon20September1965.Itwas adopted by ten votes to nil, with Jordan abstaining. On 20 September1965,thesuperpowerUSAconcurredwith USSRintheSecurityCounciloncallingceasefirewithin 48hours.PakistanandIndiaacceptedthecall[59]on21 and22September1965respectively.Theceasefire,the UNenforcedbecameeffectiveat03:30hoursof23 September 1965. Both India and Pakistan lost nearly 3000 peopleeachinthewar.Economyofboththecountries suffered a setback. AlthoughfightingendedinconclusivebothIndiaand Pakistanclaimedvictories,.ChinaidentifiedIndiaasan aggressorandsupportedtheKashmirisrightofself-determination.3.15 Tashkent agreement 1966 TheTashkentDeclarationwassignedbetweenIndian PrimeMinisterL.B.ShastriandPakistaniPresidentafter sixdaysofhardbargainingon10January1966.They agreed that all armed personnel of the two countries should bedrawnnotlaterthan25February1966totheposition theyheldpriorto5August1965,andbothsidesshould observetheceasefiretermsontheceasefireline.They affirmed to employ peaceful means to solve their conflicts. Neithersidewasallowedtoenjoythegainsofwar. Pakistanwas notevenmentionedas theaggressornor did it admit having engineered the infiltration in J&K. 3.16 Indo-Pak war in 1971InthegeneralelectionheldinPakistanin7December 1970, the Awami League led by Mujibur Rehman secured majority in the national assembly by winning 158 seats out of300seats.HedemandedcompleteautonomyforEast Pakistan.TheEastPakistanisformedMuktiBahini (LiberationForce)andcivilwareruptedinEastPakistan. India supported the Movement. Pakistan used armed forces tocurvethemovement.Thefightingforced10million EastPakistanistofleeinIndianterritories.Indiaaccused the Government of Pakistan of committing brutal genocide in the East Pakistan. India asked Pakistan to negotiate with Rehamanforapoliticalsettlement.On3December1971, PakistanlaunchedattackonIndianairfieldsalongthe frontierof Punjab,Rajasthan,andJ&K[60].Ontheother hand,PakistanallegedthatIndianforcesattackedon21 November1971inthesouth-easternsectorofEast Pakistan. India is the first country who recognized formally 90 Neutrosophic Sets and Systems, Vol. 2, 2013 Surapati Pramanik, Tapan Kumar Roy, Neutrosophic Game Theoretic Approach to Indo-Pak Conflict over J ammu-Kashmir thebirthofBangladesh[61]on6December1971.The IndianArmyalongwiththeMuktiBahini(Liberation Army)foughtthePakistaniarmedforces.Thenewsof sending a naval task force from the US Seven Fleet [62] to theBayofBengalfromtheIndo-Chinatheatrecaught muchattention.ButtheUSSR[63]confirmedIndiathat theSovietpowerfulnavalfleetwouldfollowtheSeven Fleet.On15December1971theIndianarmyreachedthe outskirts of Dacca. On 16 December 1971, 9000 Pakistani forcesalongwiththeircommanderGeneralNiazi surrenderedtotheJointCommandofIndiaand Bangladesh.Indiadeclaredaunilateralceasefire[64] effectivefrom20:00hourson17December1971and YahyaKhanacceptedit.YahyaKhanhadtoresign because of huge defeat in East Pakistan. He handed power toZ.A.Bhutto.AlthoughIndiaandPakistanfoughta thirdwaroverEastPakistan,J&Kdisputewasonlya peripheralissuebutvitaloneinthecaseofJ&K.Attime ofceasefire,Indiaoccupied204.7sqkmsofterritoryof PakistanadministeredKashmir,957.31sqkmofPunjab and12198.84sqkmsofKutchwhilePakistanoccupied 134.58sqkmsofterritoryofIndianadministeredJ&Kin theChhambsector,175.87sqkmsinPunjaband1.48sq kms in Rajasthan [65].3.17 Role of UNTheUNintervenedtoarrangecease-firesduringthewar 1971.USSRexercisedhervetopowerseveraltimesin favor of India. The Secretary General [66] was authorized toappointment,ifnecessary,aspecialrepresentativeto help in the solution of humanitarian problem. The issue of Indo-Pakconflictcametoanendon25December1971 with the appointment by U Thant, the Secretary General, of V.W.Guicciardi,asSecretaryGeneralsspecial representativeforhumanitarianproblemsinIndiaand Pakistan. 3.18 Simla agreement 1972 The Prime Minister of India and President of Pakistan had talks in Simla from 28 June 1972 to 2 July 1972 and signed the Simla Agreement [67] on 2 July, 1972.By signing the agreement,both IndiaandPakistan committedthemselves tosettlingtheirdifferencesthroughbilateralnegotiations orbyanyotherpeacefulmeansmutuallyagreedupon between them.Hopefully, they also agreed that in Jammu and Kashmir, the line of control (LOC) resulting from the cease-fireofDecember17,1971,shallberespectedby bothsideswithoutprejudicetotherecognizedpositionof either side.TheSimlaAgreementwasratifiedbybothcountries[68] and it came into force on 5 August 1972. To delineate the lineofcontrolGeneralBhagatandGeneralHamidKhan hadtoholdtenmeetingsbetween10Augustto7 December1972.On11December1972,they[69]metat Suchetgarh and jointly signed 19 maps delineating the line ofcontrolfromChhambtoTurtuk,coveringabout800 kilometers.BoththeGovernmentsapprovedthe delineation[70]almostnextday.Oncompletionof adjustmentinthelineofcontrol,IndiaandPakistan withdrewtroopsfromtheoccupiedterritoriesinorderto restore the status quo ante on the international border on 20 December1972.Pakistan[71]hasrecognizedBangladesh inFebruary1974.Theissueregardingprisonerofwars [72]closedwiththerepatriationofthelastgroupalong with Gen. Niazi at Wagah on 29 April 1974. East Pakistan crisisreflectedthatthetwo-nationtheoryfailedmiserably in the subcontinent. 3.19 The conflict at Siachen (1984 onwards) The conflict between India & Pakistan over Siachen origi-natedduetothenon-demarcationsonthewesternsideof the map beyond a grid point known as NJ 9842.The CFL, which was established because of the first Indo-Pak war of 1947-48 and the intervention of the UN, runs along the in-ternationalIndo-Pakborderandthennorthandnortheast until map grid-point NJ 9842, located near the Shyok River at the base of the Saltoro mountain range. Unfortunately, it wasnotdelineatedbeyondthegridpointknownasNJ 9842 as far as the Chinese border but both countries agreed vaguelythattheCFLextendstotheterminalpointNJ 9842, and "thence north to the Glaciers". After second In-do-Pak war in 1965, obeying the Tashkent Agreement both countries withdrew forces along the 1949 CFL. After third Indo-Pakwar1971,theSimla Agreementof1972created a new LOC based on December 1971 cease-fire. However, the Siachen Glacier region was left un-delineated where no hostilitiesoccurred.Theauthoritiesofbothcountries showednointeresttoclarifythepositionoftheLOC beyond NJ 9842. Due to lack of strategic viewpoint and se-riousnesstheLOCwaspoorlydescribedasrunningfrom Nerlin (inclusive to India), Brilman (inclusive to Pakistan), uptoChorbatLaintheTurtoksector.InApril1984,In-dianarmyoccupiedkeymountainPassesandestablished permanentpostsattheSiachenheights.Indiantroops broughtcontrolovertwooutofthreepassesontheSia-chen,SiaLaandBilfondLa,whilethethirdpass,Gyong LaremainedunderPakistan'scontrol.TheIndianforces arepermanentlydeployedallalongthe110-kmlongAc-tualGroundPositionLine(AGPL).ArmiesofbothIndia andPakistanfightinlethal10-22000footaltitudeinSia-chen Glacier.Pakistan retaliates in the worlds highest war zone.3.20KashmirinsurgencyinIndianadministered KashmirTheassemblyelectionsinJ&Kon23March1987were partlymanipulatedandriggedwhichtheNational Conference-Congresscoalitionwonalandslidevictory. TheoppositionpartyMuslimUnitedFront(MUF)called thevictoryasblatantlyfraudulentandrigged.Alarge numberyoungpeopleofKashmirwerealienatedbythis perception.StateGovernmentofJ&Kwitnessedvarious 91 Neutrosophic Sets and Systems, Vol. 2, 2013 Surapati Pramanik, Tapan Kumar Roy, Neutrosophic Game Theoretic Approach to Indo-Pak Conflict over J ammu-Kashmirdemonstrationandagitationbetweenmid-1987andmid-1989basedmanifestationofanaccumulatedanger comprised of many components such as administrative (the curtailnumberofOfficesthatmovetothewintercapital Jammu), the regional autonomy, economic policy (increase ofpowertariffs),religioussentiments,civilliberties (custodiandeath),andanti-Indiademonstrationof14and 15 August, 26 October (accession day) and 26 January. On 8 December 1989, the militants kidnapped Rubaiye Sayeed, daughterofIndianHomeMinsterMuftiMohammed Sayeed.TheprestigeofFarooqAbdullahledState Governmentsufferedserioussetbacksforrepressionofany form of protest Farooq Abdullahs resignation with the appointment of Jagmohan as Governor for the second time on19January1990,broughtCentralGovernmentinto directconfrontationwiththe variousrebelgroups.At5a. m. on 20 January 1990, Indian paramilitary forces crackeddown on a part of Srinagar city and began the most intense house-to-housesearchandroundedupoverthreehundred people.Mostofthem,however,werelaterreleasedand arrested persons complained to be beaten up or dragged out of their houses. People got frightened first, but discovering thecourageofdesperation,thepeoplestartedpouringout intothestreetdefyingthecurfew,toprotestagainstthe allegedexcessiveuseofforceinsearchoperationinnext day. The administration got completely unnerved and gave orders to fire at when most of the groups of demonstrators converged at Gau Kadal. The number of deaths [73, 74] is disputed;however,thepressreported35dead.Thenthe implicitsupportfortheseparatistsforindependence transformedintoexplicitduetomainlythehigh-handed searchesorderedbyJagmohan,theGovernorofJ&K.On 19February1990GovernordissolvedtheStateassembly andGovernorrulewasimposed.TheJagmohanregime [13] witnessed sadly the exodus of almost the entire small KashmiriPanditcommunityfromthevalleyand20000 thousandMuslimhadbeenforcedtomigrate.TheState assemblyelectionof1990resultedinAbdullahdownfall followingtheoutbreakofaMuslimuprising.Duringthe 1990s,severalnewmilitantgroupswereformed,having radical Islamic views. The large numbers of Islamic Jihadis, who had fought in Afghanistan against the Soviet Union in the 1980s, joined the movementManyumbrella groups were responsible for the uprising in J&K.Amongthem,thefirstumbrellagroupistiedtothe JammuandKashmirLiberationFront(JKLF).They demanedindependentKashmir.Thesecondgroup comprisedofMuslimfundamentalistsandhaslinkswith thefundamentalistPakistanparty,Jammait-I-Islam.No doubtthegrouphasapro-PakistanOrientation.Thethird groupisJammuandKashmirPeople`sLeaguethathasa pro-Pakistanorientation.Thegroupsdemandedplebiscite sothatpeopleofJ&Kcouldexercisetheirrightofself determination.Indiaadoptedamultipleprolonged approach to deal with the insurgency in J&K. In 1990, the then Governor Jagmohan announced the implementation of ArmedForcesSpecialPowersActof1958(AFSPA)for J&KandJ&KDisturbedAreasActtoputdownthe militancy.Indiansecurityforcesallegedlycommitteda seriesofhumanrightabuses[75]inJ&K.Itisobserved thattheencounterbetweenIndiansecurityforcesandthe militantscausedmorethan50,000deaths[75]including manyhundredsofinnocentcivilians.Kashmirimilitants have been also accused of killing moderate Muslim leaders, Hindus,bombingpassengerbussesandrailwaybridges andpublicestablishments.InSeptember1996,National Conference had won a landslide victory in J&K Assembly election,althoughthe30-disparatepartycoalition,known asAll-PartyHurriyatConference(APHC)didboycottthe election.IndianauthoritiesformedseveralMuslim counterinsurgencygroupstocombattheinsurgencyalong withIndiansecurityforces.Duetotheacutefailureof Indian authorities to address the socio-economicproblems and ambition of autonomy to some extent of the people of J&KandPakistansactiveroleinfosteringcrossborder terrorism, the situation in J&K becomes more complex and volatile and neutrosophic in nature.3.21 Nuclear rivalry between India and Pakistan Indiahadconductedherfirstnucleardevicein1974.In May1998,Indiaconductedseveralnucleartestsinthe desertofRajasthan.Pakistangottheopportunitytocon-ductnucleartest,andhopefullygrabbedtheopportunity and conducted six tests in Baluchistan in order to balancing nuclear power with India. The arm race between Indo-Pak caught international attention. The UNSC condemned both thecountriesforconductingnucleartestsandurgedthem tostopallnuclearweaponsprogram.On23September 1998, new development occurred following at UN General Assembly session. Both India and Pakistan agreed to try to resolvetheKashmirquestionpeacefullyandtofocuson tradeandpeopletopeoplecontact.Pakistansenther cricketteaminIndiaasgoodwillgestureonNovember 1998afteradecadeabsence.Ontheotherhand,India agreedtobuysugarandpowderfromPakistan.InFebru-ary1999,busservicebetweenNewDelhitoLahore started.AcceptinganinvitationfromSharif,Vajpayeevi-sited Lahore by bus. His visit to Pakistan is known as bus diplomacy. Itdrewmuchattentionand atend ofthesum-mittheyissuedLahoreDeclarationthatwasbackedby Memorandum of Understanding (MOU) [76]. 3.22 Kargil conflict in 1999 In May 1999, Pakistan backed militants together with Pa-kistaniregularforcescrossedtheLOCand infiltratedinto India-heldKargilregionofNorthKashmir.Themilitants occupiedcovertlymorethanthirtywell-fortifiedpositions themostinhospitablefrigidlycoldridgesatanaltitudeof 16000-18000feet,intheGreatHimalayanrangefacing Dras, Kargil, Batalik and the Mushko Valley sectors. India retaliated by launching air attacks known as Operation Vi-jay (victory) on 26 May 1999. India identified Pakistan as anaggressorthatviolatedtheLOC.Asthebattleturned 92 Neutrosophic Sets and Systems, Vol. 2, 2013 Surapati Pramanik, Tapan Kumar Roy, Neutrosophic Game Theoretic Approach to Indo-Pak Conflict over J ammu-Kashmirmore intense, the Clinton administration intervened to help defuse the conflict. It was witnessed that on 15 June Clin-tonmadeseparatetelephonicconversationswithboththe PrimeMinistersofIndia andPakistan.Heasked Sharifto withdrawinfiltratorsfromacrosstheLOC.Hecordially appreciated Vajpayee for his display of restraint in the con-flict. Pakistan was isolated from world community regard-ingtheKargil-issue,onlySaudiArabiaandUnitedEmi-rates supported Pakistan. On 4 July 1999, Sharif and Clin-ton held a three-hour meeting and issued a joint communi-qu in which Sharif agreed to withdraw the intruders.On 11July1999,inaccordancewiththeagreementtheinfil-tratorsstartedretreatingfromKargilasIndiaset16July, 1999 as the dead line for the total withdrawal. On 12 July 1999, Sharif defended his 4 July agreement with Mr. Clin-tonanddefendedhisKargilpolicythatdesignedtodraw theinternationalattentionoftheinternationalcommunity to Kashmir issue.In the war [77], India lost more than 400 forces.670intrudersand30Pakistaniregularforceswere also killed excluding the injured. 3.23 Agra summit (14-16 July 2001) Agra Summit was held between the Indian Prime Minister Atal Behari Vajpayee and Pakistan's President Pervez Mu-sharraf in Agra, from July 14 to 16, 2001. The summit be-gan amid high hopes of resolving various disputes between the two countries including complex J&K issue. Both sides remained inflexible on the core issue of J&K and ultimate-lythebilateralsummitfailedtoproduceanyformal agreement.3.24 The threat of war between India and Pakistan and the role of Bush administration On13December2001,fivemilitantsattackedIndianna-tional parliament house causing the deaths of 13 people in-cluding five terrorists.India held Pakistan responsible for the incidence. India immediately reacted by deploying a si-zeableforcealongtheLOCinJ&K.Pakistanfollowed suit,untilbothnationshadalignedavastarrayoftroops and weapons against one another. Armies of both countries frequently exchanged of artillery fires. The mobilization of troopssparkedworldwidefearsofadeadlymilitarycon-flict between India and Pakistan. InordertodefusethegrowingtensionsBushAdministra-tion took initiatives and succeeded in getting both sides to makeconciliatorymove.On12January2002,inhisad-dress to his nation, Musharraf committed Pakistans polit-ical,diplomaticandmoralsupporttothestruggleof peopleofKashmir.HewentontocriticizethePakistani militant Islamic groups for i) creating violent activities, ii) aggravatinginternalinstability,iii)harboringsectarianism inPakistanpoliticsiv)creatingwarlikesituationagainst India.Hebannedtwomilitantgroups,theLasker-e-Toiba and Jais-e- Mohammed. In the following weeks, 2,000 ac-tivists of the banned militant groups were arrested in Pakis-tan.Musharrafregimecloseddownsomeofmilitants groupsofficesandrecruitingcenters.Indiawelcomed thesemeasurescautiouslyandthetensionwassomewhat defused. On May 14, 2002, three militants disguised in In-dian Army uniforms shot passengers indiscriminately on a publicbusandthenkilled40people(mostlywivesand children of army personnel) including eight bus passengers at Kaluchak of Jammu before militants were gunned down. India reacted by threatening to strike at the terrorist camps situatedinPOKandtooktoughstanddeclaringsome measures: i) expelled the Ambassador of Pakistan to India, ii) withdrewherdiplomaticpersonnelfromPakistan,iii)imposedbanonPakistanicommercialairflightsfromIn-dian air space, iv) mobilized 100000 Indian forces close to LOC. On22May2002,onhisvisittothefrontlinesinJ&K, Vajpayee called for a decisive battle.Pakistani authority declaredthatitwoulddefendPakistaniadministered Kashmirbyanycost.Musharrafmobilized50,000troops totheborders.On27May2002,Musharraf[78]warned Indiabydeclaring,ifwaristhrustuponus,wewillre-spond with full might. EvenPakistanthreatenedtousenuclearweaponsagainst India.ThisthreatdrewpointedattentiontotheUSAand UK.TheBritishForeignSecretary,JackStrawandUS Deputy of Secretary, Richard Armitage and Defense Secre-tary, Donald Ramsfeld rushed to both India and Pakistan in May-June 2002 in order to defuse tension. They were suc-cessful in their mission to defuse tension and succeeded in getting promise from Musharraf to stop cross-border infil-trationintoJ&K.Afterwitnessingaslowdownininfiltra-tion,IndiatriedtoimproveherrelationwithPakistanby reestablishingdiplomaticties,recallinghernavalshipsto theirBombaybase,andopeningherairspacetoPakistani commercial flights.3.25 Musharrafs proposals for J&K resolution On25October2004,whileaddressinganIftarparty, PresidentMusharrafannouncedanimportantdeclaration regardingsettlementoftheJ&KacceptabletoPakistan, IndiaandpeopleofJ&K.Heremarkedthatthesolution would have to be met thee steps: i) Bothsidesshouldidentifytheregionsonbothsides of LOC,ii) Demilitarize these regions,iii) Determinetheirstatusthroughindependenceorjoint control or UN mandate.HeopinionedthatPakistanisdemandforaplebiscitewas impracticalwhileIndiasofferformakingLOCaperma-nentborderwasunacceptable.TheMusharrafsan-nouncements drew much attention but Indian Prime Minis-terM.Singhrefusedtocommentdescribingthemasof the cut remarks.President Musharraf [79] proposed four point solutions re-garding the resolution of J&K disputes as follows:93 Neutrosophic Sets and Systems, Vol. 2, 2013 Surapati Pramanik, Tapan Kumar Roy, Neutrosophic Game Theoretic Approach to Indo-Pak Conflict over J ammu-Kashmir i)troopswillbewithdrawnfromtheregionina staggered mannerii)theborderwillremainunchanged,howeverpeoplewillbeallowedtomovefreelyinthere-gion iii)selfgovernanceorautonomybutnotindepen-denceiv)ajointmanagementmechanismwillbecreated withIndia,PakistanandKashmiriRepresenta-tives On5December2006,duringaninterviewwithNDTV President Musharraf opinioned that Pakistan is prepared to give up her claim on Kashmir, also ready to give up her old demand for a plebiscite and forget all UN resolution if In-dia accepts the four-point resolution of Kashmir dispute of-feredbyhim.HeremarkedthatPakistanisabsolutely against the independence of Kashmir so is India.He stated that for compromise solution both countries would have to give up their positions and step back. On 31 December 2006, G. N.Azad, the Chief Minister of J&Kstatedthatjointmechanismispossibleinthefield of trade, water, tourism and culture, and this could lead the way for a resolution to the longstanding Kashmir problem.On8January2007,hefurtherstatedthatthelatestfour point-solutionofferedbyPresidentMusharrafshouldnot be put aside without discussing positively. On19January2007,followingthemeetingwithIndian ExternalAffairsMinister,ChiefMinister,G.N.Azadof J&KsaidThedateforthecompositedialoguehasbeen fixedfor 13-14March 2007andIamsureall outstanding issues and proposals floated from time to time by President Musharrafwillbediscussed.OnthesamedayAPHC leaderMirwaizUmarFarooqtoldtheBBCthatthenext three months would be crucial. On2February2007,PresidentMusharrafsaid:Wecan-not take people on board who believe in confrontation and whothinkthatonlymilitancysolvestheproblem.On3 February 2007, Indian Prime Minister M. Singh welcomed President Musharrafs statement that militancy or violence cannot resolve the Kashmir issue. On 4 February 2007, In-dianExternalAffairsMinister,PranabMukherjeecom-mentedonMusharrafsproposals:Itisgood.Everybody shouldadviseterroriststogiveupviolenceandjointhe process of dialogue.The idea of four point resolution was purelypersonalthatdidnothavethemandate.However, Musharraf had to resign from his post for internal problem. His endeavor to resolve the J&K problem failed due to no response from India. 3.26 Terror attack at Mumbai On 26 November 2008, Mumbai, the capital of Maharastra andfinancialcapitalofIndiawitnesseddeadlyterrorat-tacks.Indiaadoptedatougher-thanusualstandagainst PakistaninthewakeoftheMumbaiterrorattackandde-manded to hand over 20 terrorists including Pakistan-based underworld Dawood Ibrahim and Jaish-e-Mohammad chief MaulanaMasoodAzharstayinginPakistan..Todefuse the tension between India and Pakistan, Secretary of State oftheUSA,Condoleeza RicevisitedIndiansubcontinent. Ultimately USA succeeded in defusing the tension. 3.27 Is China a third party in J&K conflict? Indianstandonthequestioniscontradictory,ambiguous and unclear and neutrosophic in nature. India strongly ob-jectedtheborderagreementbetweenPakistanandChina signedin1963bywhichChinagained2700squaremiles ofthePakistanoccupiedKashmir.BasedontheSimla Agreement 1972 between India and Pakistan, India strong-ly is of the opinion that J&K problem is a bilateral dispute. ThirdpartyinvolvementisnotwelcomedbyIndiatore-solve the issue. However, Pakistan wants that China would playadefiniteroletoresolvetheJ&Kconflict.China adopts a neutral role as seen in the Kargil conflict in 1999.So,dependinguponSimlaAgreement,Indianpointof view and present status, J&K conflict is considered as a bi-lateral problem between India and Pakistan. 3.28 Internal development in Indian administered J&K Special status wasconferredonJ&KunderArticle370of IndianConstitution[80].ConstituentAssemblywas electedbyJ&Kon31October,1951.Theaccessionof J&KtoIndiawasratifiedbytheStatesConstituentAs-sembly in 1954. The Constituent Assembly also ratified the Maharajasaccessionof1947in17November1956.In 1956,thecategoryofpartBwasabolishedandJ&Kwas includedasoneoftheStatesofIndiaunderarticle1.On January26,1957ConstituentAssemblydissolvesitself.On30March1965,article249ofIndianConstitutionex-tendedtoJ&KwherebytheCentralGovernmentatNew Delhi could legislate on any matter enumerated in state list. DesignationlikePrimeMinisterandSarder-i-Riyasatare replacedbyChiefMinsterandGovernorrespectively.In 1964, decision to extend Articles 356 and 357 of the Union Constitution of India to J&K announced.On 12 February 1975,ChiefMinisterAbdullahsignedKashmirAccord that affirmed its status as a constituent unit of the India and the State J&K will be governed by Article 370 of the Con-stitution of India.3.29Devel opmentofPakistanadministered Jammu and KashmirAzadKashmirwasestablishedon24October1947.The UNCIPresolutiondepictsthestatusofAzadKashmirasneitherasovereignstatenoraprovinceofPakistan,but rathera"localauthority"withresponsibilityoverthearea assignedtoitundertheceasefireagreement.The"local authority" or the provisional government ofAzad Kashmirhad handed over matters related to defense, foreign affairs, negotiations with the UNCIP and coordination of all affairs relatingtoGilgitandBaltistantoPakistanunderthe Karachi Agreement [81] of April 28, 1949.94 Neutrosophic Sets and Systems, Vol. 2, 2013 Surapati Pramanik, Tapan Kumar Roy, Neutrosophic Game Theoretic Approach to Indo-Pak Conflict over J ammu-Kashmir ZulfiqarAliBhuttosgovernmentvirtuallyannexedthe POKbypromulgatingtheAzadJammuandKashmirAct of1974.AzadKashmiradoptsIslamasthestatereligion videArticle3.Theconstitutionprohibitsactivities prejudicialordetrimentaltotheideologyoftheState's accessiontoPakistan(Article7).Itdisqualifiesnon-Muslims from election to the Presidency and prescribed in the oath of office the pledge 'to remain loyal to the country andthecauseofaccessionoftheStateofJammuand KashmirtoPakistan'.Itprovidestwoexecutiveforums, namelytheAzadKashmirgovernmentinMuzaffarabad and the Azad Kashmir council in Islamabad. The Pakistan governmentcandismissanyelectedgovernmentinAzad Kasmir irrespective of the support it may enjoy in the Azad Kasmir Legislative Assembly [82] by applying the Section 56 of the constitution,. The Northern Areas have the status of a Federally Administered Area.3.30 The Indo-Pak conflict over Jammu and Kashmir Theconflictwasbasedontheneutrosophicexplanation andunderstandingoftheneutrosophicsituationinIndia andPakistan.FromPakistanipointofview,shehoped J&K was going to accede to Pakistan as the majority of the population being Muslims. If Junagadh, despite its Muslim rulers accession to Pakistan, belonged to India because of its Hindu majority, then Kashmir surely belonged to Pakis-tan.PrincelystateHydrabadbecameIndependenton15 August1947likeJ&K.ButIndiainvadeditbecauseof Hindumajority.PakistanregardedtheAccessionofJ&K as a coerced attempt to force the hand of Maharaja. Popu-lar outbursts took place but J&K had acceded to India, be-causetherulerwasaHindu.FromIndianpointofview, J&Khadaccededtoher.ArmedPakistaniraidershaving Pakistanicomplicityandsupportinvadedsomeportionof J&K. Both the countries failed to implement the UN reso-lutions.Theplebiscitehasneverbeenheld.Indiaviewed thatthatthetimehaschanged.Indiastronglyarguesthat legislative measures subsequently legitimized the question ofaccession.AfterSimlaAgreement,J&Kproblembe-camebilateralissueanditssolutionshouldbebasedon Simla Agreement 1972.4 Neutrosophic game theoretic model formulation of the Indo-Pak conflict over Jammu and Kashmir FollowingtheabovediscussionandbasedonSimla Agreement 1972 , game theoretic modelis formulated. Theproblemismodeledasastandardtwoperson(22) zero-sum game. Table 1: Payoff matrixPakistan III I 1 -1II 0 -1IndiaIII0-1 IV 1 0 Pakistan strategy vector: i)Full compliance with Simla Agreement 1972,ii) Partial or non-compliance with Simla agreement 1972 Indias strategy vector: i) Make territorial concessions, ii) Accept the third party mediation ( USA),iii)Apply the strategy of all-out military campaign,iv)Continue fencing along the LOC (see Fig. 1). Fig.1 Photograph of fence along LOC Theabovepayoffmatrixhasbeenconstructedwith referencetotherowplayeri.e.India.Intheprocessof formulatingthepayoffmatrix,itisassumedthatthecombination (I, I) will hopefully resolve the conflict while thecombination(IV,II)willbasicallyimplyastatusquo with continuing conflict. If Pakistan can get India to either make territorial concessions (Muslims dominated Kashmir valley or other important stragic areas of J&K such Kargil) or accept the third party mediationlike USA without fully complyingwiththeSimlaAgreementi.e.strategy combinations(I,II)and(II,II),thenitreflectsthat Pakistan will be benefited but India will be loser.If India acceptsthethirdpartymediationandPakistanagreesto complyfullywiththeSimlaAgreement,thenthoughit potentiallyendstheconflict,thereshouldbeapolitical jeopardyinIndiaasaresultoflackofstrategicand political concensusamong thepoliticalpartiesandsothe strategycombination(II,I)isnotafavorablepayofffor India.IfIndiaemploysanall-outmilitarycampaign,an devastatingwarseemstooccurasboththecountriesare capableofusingnuclearpowersi.e.strategycombination (III, I) would not producea positive payoff for either side.Ifthereoccursaunexpectedandsuddenchangeofmind setupforJ&KwithinthePakistanileaderships(from insideoroutsidepressure)andPakistanchoosestofully abidebytheSimlaAgreement1972consideringLOCas thepermanentinternationalborderi.e.strategy combination(IV,I)willbringapotentialendtothe conflict as both countries may think that they will be loser95 Neutrosophic Sets and Systems, Vol. 2, 2013 Surapati Pramanik, Tapan Kumar Roy, Neutrosophic Game Theoretic Approach to Indo-Pak Conflict over J ammu-Kashmir and winner at the same time (neutrosophically true) in the sensethattheywillnotgetthewholeJ&KbutPakistan cancansolehersayingthatshehasgainedonethirdof J&K while India may think she gained two third of J&K.Inthepayoffmatrix(seeTable1),alltheelementsofthe first,secondandthirdrowsarelessthanorequaltothe correspondingelementsoftheforthrow,thereforefrom thegametheory[83]pointofview,forthrowdominates thefirstthreerows.Ontheotherhand,everyelementof the first column is greater than the corresponding elements of the second column, therefore, first column is dominated bysecondcolumn.Itshowsthattheabovegamehasa saddlepointhavingthestrategycombination(IV,II), which reflects that in their very attempt to out-bargain each other both countries actually end up continuing the conflict indefinitely!Thusthemodelmodeloffersanequilibrium solution.Inthesubcontinentpoliticalarena,Pakistanileaderships bestinterestwastotransformtheconflictmorecomplex and keep theconflictmorealivewithfullstrenght to gain politicalsupportfrominsideandoutsideandultimately compelledIndiatomaketerritorialorotherconcessions. However,forinternationalpressuremainlyfromUSA, Pakistan had to state some overt declaration that negotiated settlementoverJ&KbasedonSimlaAgreement1972is possible. However, Pakistan covertly continues her sincere helptoseparatistgroupsbymeansofmonetary,logical, psychologicalandmilitaryequipments.Bydoingso Pakistanisnowindeeptroubalwithvariousmilitant groups and Jihadi groups. She has to deal internal security problescausedbyPakistaniTalibangroups.Undersuch volatilecircumstances,itwouldbequiteimpossiblefor Pakistan to chalk out a distinct governing strategy to meet with counter strategy. Boththecountries,ingeneral,playedtheirgamesunder internationalpressure.AlthoughPakistansignedLahore declarationwithIndiabythethenPrimeMinisterNawaj Sarif,PakistanimilitarybossMr.Musharrafoccupied some heights of Kargil in 1999 to derail the peace process anddrawinternationalattentiontotheJ&Kconflict.An important lesson of the Kargil conflict seems to be that no militaryexpeditioncouldbeasuccessifitispursued withoutpayingtoseriousattentiontothetotalityofthe scenarioshavingdomestic,political,economic, geographical,internationalopinionsandsensitivities.AnotherimportantofKargilconflictseemsthatnational communitydoesnotwanttomilitarysolutionrelatingto J&K problem.However, one positive aspect of the Lahore declarationreflectsthatboththecountriesarecapableof transformingthegamescenarioanopenoneinthesense thattheconflictingcountriesarecapableofdynamically constructingandformulatingobjectivesandstrategiesin thecourseoftheirpeaceful,mutualinteractionwithina formally defined socio- political set up.During the Agra summit in 2001, when probably President Musharrafwasthinkingtomaketheconflictveryalive while offering the impression to the other side (India) that theywerewholeheartedlyseekingstrategiestoputanend theconflict.PresidentMusharrafplayedverycleverand diplomaticrolebyusingthemediaverycautiouslyand cleverlytomaketheinternationalcommunityandhis countryunderstandthathetrieshislevelbesttoreacha meaningful, desicive and effective agreement but fails due to Indian rigid position regarding J&K issue. He left Agra and thereby tried to obtain his acceptance to his own nation and international community. This immediately shows why such a negotiation would break down at early stage. ThegovernmentofIndiaandPakistanaredealingwith militancyandterrorisminownland.Butmainissue remains the J&K conflict.4.1 Case for applying neutrosophic game theory It is experienced that none of the strategy vectors available toeithersidewillremaintemporarilystationarybecause crucial events come into light on the global political arena, ingeneral,andthesouthAsiansubcontinentinparticular. Moreover, there is a broad variety of ambiguities about the motivesbehindPakistanauthoritysprimarygoalsabout the driving forcebehind Pakistan authoritys primary goal andthestrategiesitadoptstoachievethatgoal.PakistansprincipalallyUSAisalsoagreatfactor.The influence of USA has a great impact on forming strategies.TheterroristactivitiesbyPakistanbakedterroristgroups are sometimes monitored by the wishes of USA. Although Pakistanhasnothundredpercentcontroloverforeign mercenariescomingfromdifferentpartofworldnamely, SaudiArabia,Afghanistan,Chechnia,Sudanetc.Pakistan isconstantlytryingtobringIndiaunderpressureby harboringterroristattacksonIndianruledKashmirand destabilizingthenormalcyinthestateinorderto understandtheinternationalcommunitythatinternational interventionisrequierdtoresolvetheJ&Kconflict.Itis alsodifficult tostate apart a truebargainingstrategyfrom onejustmeanttobeapoliticaldecay.Inthehorizonof continuousconflict,webelieveandadvocatean applicationoftheconceptualframeworkofthe neutrosophicgametheoryasageneralizationofthe dynamic fuzzy game paradigm. Itgeneralizedterms,awell-specifieddynamicgameata particular time t is a particular interaction between decision makerswithwell-definedrulesandregulationsandroles for the decision makers, which remain in place at time t but are free to change over time. However, it is likely that the decision makers may suffer from the role of ambiguity i.e. atypicalsituationwherenoneofthedecisionmakersare exactlysurewhattoexpectfromothersorwhattheother decision makers expect from them. In the context of indo-Pakcontinuingconflict,forexample,Pakistanleadership wouldprobablynothavebeensureofitsrolewhenMr. MusharrafmetwithIndianprimeministerAtalBehari VajpayeeattheAgrasummittochalkoutapeace agreement.Mr.Musharuffwenttothatsummitunderthe international pressure or to prove himself to be an efficient 96 Neutrosophic Sets and Systems, Vol. 2, 2013 Surapati Pramanik, Tapan Kumar Roy, Neutrosophic Game Theoretic Approach to Indo-Pak Conflict over J ammu-KashmirleaderofPakistan oragainsthisfreewillandwouldhave liked to avoid Agra if he could because he did not want to signanyfinalagreementonJ&Kwhichcanbeagainst commonfeelingofpeopleofPakistan.Mushauff demanded for declaration of J&K to be a disputed territory atleast.Havingnosuchcapitulationforthcomingfrom India,MusharuffleftAgrawithoutsigninganyjoint statement. In this political context, Pakistani leaderships best interest was to keep the conflict alive with full strength ultimately compelledIndiatomaketerritorialorotherconcession. However,forinternationalpressuremainlyfromUSA, Pakistanhadtooffersomeovertdeclarationthat negotiatedsettlementoverJammuandKashmirbasedon SimlaAgreement1972ispossible.Pakistancovertly continueshersincerehelpseparatistgroupsbymeansof monetary,logical,psychologicalandmilitaryequipments. Undersuchvolatilecircumstances,itwouldbequite impossibletochalkoutadistinctgoverningstrategyto meet with counter strategy. However,boththecountries,ingeneral,playedtheir gamesunderinternationalpressure.AlthoughPakistan signedLahoredeclarationwithIndiabythethenPrime MinisterNawajSarif,PakistanimilitarybossMr. MusharrafoccupiedsomeheightsofKargilin1999to derail the peace process. An important lesson of the Kargil conflict seems to be that no military expedition could be a success if it is pursued without paying to serious attention tothetotalityofthescenariohavingdomestic,political, economic,geographical,internationalopinionsand sensitivities.AnotherimportantofKargilconflictseems that national community does not want to military solution relatingtoKashmirproblem.However,onepositive aspectaboutLahoredeclarationorIndo-Pakjoint declarationindicatestheytransformthegamescenarioan openoneinthesensethattheconflictingpartiesare capableofdynamicallyconstructingandformulating objectivesandstrategiesinthecourseoftheirpeaceful, mutualinteractionwithinaformallydefinedsocio- political set up. Thus, the negotiation space may be represented as: NPakistan NIndia . AccordingtotheopinionofBurnsandRowzkowska[84] eachplayerspersonalviewsconstituteadeal.Thefuzzy semantic space comprises of such deals i.e. personal views. Personalvaluejudgments,acquiredexperiencesand expectationsaboutthepossiblebestorworstoutcomes from a negotiation are crucial to constitute such views. The fuzzysemanticspace is adynamicalsystemand is freeto modify according to the need and desire of the players and practical situations in the light of new information. Thesemanticspacehoweverremainsfuzzyowingto vaguenessabouttheexactobjectivesandlackofprecise understandingoftheexactstand,whichtheopponent parties have from their viewpoints. That is why none of the conflictingpartiescaneffectivelyreadandprecisely understand each others nature of expectations.ThiswasreflectedinAgrasummitwhenprobably Musharraf (Pakistan) was thinking in terms of keeping the conflictalivewhileofferingtheimpressiontotheother side(India)thattheywerewholeheartedlyseeking strategies to put an end the conflict. Mr. Musharraf played verydiplomaticrolebyusingthemediaverycautiously andcleverlytomaketheinternationalcommunityandhis country understand he tried his level best to reach a fruitful agreement but failed and left at midnight and thereby tried to obtain his acceptance to his own nation and international community.Thisimmediatelycomestolightwhysucha negotiation would break down at early stage.IftheIndo-PakconflictoverJammuandKashmiris constitutedasfuzzydynamicfuzzybargaininggame,the playersfuzzysetjudgmentfunctionsoverexpected outcomescanbeformulatedasfollowsaccordingtothe rules of well-developed fuzzy set theory due to Zadeh [85] ForPakistan,thefuzzyevaluativejudgmentfunctionat timet, ( ) t P, J(P,t)willberepresentedbyfuzzyset membership function as follows: ( )( ) < for x, 0 = for x , 5 . 0 < x < for, 1 , 5 . 0 = WorstWorstBest Worstt , JHere, the symbol Best indicates the best possible outcome thatPakistanwouldexpect;whichwouldprobablythe annexation of Jammu-Kashmir to Pakistan according to the two-nationtheoryofMuslimLeague.Similarly, WorstindicatesprobablytheconversionofLOCasthe permanent international borderline. On the other hand, for India the fuzzy evaluative judgment functionat timet,J(I,t)willberepresented by thefuzzy set membership function( ) t , Jas follows:( )( )WorstWorstBest WorstBestt , J yfor, 0 = y for, 5 . 0 < y < for, 1 , 5 . 0 y for, 1= Here Worst indicatestheworstpossiblenegotiation outcomeIndiacouldexpect,whichwouldbecoincidence with the best-expected outcome for Pakistan.ItistobenotedthatsemanticspaceNPakistanNIndia is moregenerallyframedasaneutrosophicsemanticspace, whichconsidersathree-waygeneralizationofthefuzzy semanticspace.Sinceneutrosophicsemanticspace comprisesofneutralpossibility,itcannotbedefuzzified into two crisp zero-one states owing to the incorporation of aninterveningstateofindeterminacy.Such 97 Neutrosophic Sets and Systems, Vol. 2, 2013 Surapati Pramanik, Tapan Kumar Roy, Neutrosophic Game Theoretic Approach to Indo-Pak Conflict over J ammu-Kashmir indeterminacy would be practically encountered due to the fact any mediated, two-way negotiation process is likely to becomeovercatalystbythesubjectiveutilitypreferences of the mediator. LetT,I,Frepresentrealsubsetsoftherealstandardunit interval[0,1].Statically,T,I,Faresubsetswhile dynamically,inthecontextofadynamicgame,theymay beconsideredasset-valuedvectorfunctions.Ifalogical proposition is said to bet % true in T, i % indeterminate in Iandf%falseinF,thenT,I,Fareconsideredasthe neutrosophiccomponents.AccordingtoSmarandache [7] neutrosophicprobabilityisusefultoeventsthatare shrouded in a veil of indeterminacy like the actual implied volatile of long-term options.Bhattacharya et al.y applied theconceptofneutrosophicprobabilityinordertoformulateneutrosophicgamethereticmodel[86]toIsrael Palestineconflict.Itisworthyofmentionthatthe proposedapproachusesasubset-approximationfortruth-values,indeterminacyandfalsity-values.Itiscapableof providing a better approximation than classical probability to uncertain events. Therefore,theneutrosophicevaluativefunctionfor Pakistanattimet,JN(P,t)willberepresentedbythe neutrosophicsetmembershipfunction ( )( ) x t , NJ as follows: ( )( ) x t , NJ= Tx , O < x for , 0Tx , O = x for , 5 .Tx , O < x < O for ), 1 , 5 (.WorstWorstBest Worst Ontheotherhand,theneutrosophicevaluativejudgment function for India at time t,JN(I, t) will be represented by theneutrosophicsetmembershipfunction ( )( ) y t , NJas follows:( )( ) y t , NJ= ( )Fy , yfor, 0Fy , = y for, 5 . 0Fy , < y < for, 1 , 5 . 0 Fy , y for, 1WorstWorstBest WorstBest Relyingonthree-wayclassificationofneutrosophic semanticspace,itist%trueinsub-spacethatbilateral negotiationwillproduceafavourableoutcomewithinthe evaluativejudgmentspaceofthePakistanwhileitisf% false in the sub-space F that the outcome will be favorable withintheevaluativejudgmentspaceofthePakistan. However,thereisani%indeterminacyinsub-spaceI wherebythenatureoftheoutcomemaybeneither favorablenorunfavorablewithintheevaluativejudgment space of either competitor.5 ConclusionWehavediscussedthecrisisdynamicsofthecontinuing Indo-Pak conflict over J&K. we have briefly examined the effortsmadebyvariousstudygroupsandpersonsand India and Pakistan in resolving the conflict and the reasons fortheirfailure.WehavelookedtheIndo-Pakrelations and recent developments and their views on J&K situations. Alternative possible solutions are also considered. Most of the solutions are either impractical or unacceptable to India, Pakistan, and or the various militant groups. Pro Pakistani militantgroupsandPakistanwouldoptforfreeand impartialplebiscite.Evensomemilitantgroupswould oppose the plebiscite because the UNSC resolutions do not offer them the very option of independence. In the process offormulatingthepayoffmatrixforgametheorymodel, the combination (I, I) will hopefully resolve the conflict i.e. maintainingthestatusquoalongtheLOCwithsome borderadjustmentsfavorabletoPakistan.Otherwise,the the proposed model offers the solution wich state that both thecountrieswillcontinuetheconflictindefinitely.The application of game theoretic method to the ongoing Indo-PakconflictoverJ&Kisbasedonidentifyingand evaluating the best options that each side has and is trying toachievechosenoptions.TheSimlaAgreement1972is usedasaninstancewithPakistanbeinglefttochoose between two mutually exclusive options. ThesolutionreflectstherealfactsthatPakistandoesnot want to ever agree to have full compliance with the Simla Agreement1972,asshewillseealwaysherselfworseoff thatway.RealizingthatPakistanwillneveractually complywithSimlaAgreement1972,Indiawillfindher bestinteresttocontinuethestatusquowithanongoing conflict, as she will see herself ending up on the worse end of the bargaining if she chooses to apply any other strategy.It is experienced that none of the strategy vectors available toeithersidewillremaintemporallystationarydueto actionandreactionofmilitantgroupsandIndainsecurty forcesinJ&K.Moreover,thereexistsabroadvarietyof ambiguities concerning primary goals of the two countries and the strategies they adopt to achieve those goals.Due to the impact of globalization, people have to interact withpeoplefromothercountries.Inthedaysofcross-border strategic alliances and emphasis on various groups, itwouldreallybetragicifothernationsremainstandstill orignoretheconflict.ThegovernmentofIndiafailsto solveitsownproblemsofnortheaststatessuchas Nagaland,Assam,Mizoram,andArunachalPradesh.On theotherhand,Pakistanisconstantlyfacingwiththe myriad problems of democracy and absence of it. Pakistan failstocurveinsurgenciesinBalochistan,thelargest provinceofPakistanhavingresourceofnaturalgasand mineralandsparselypopulated.Wehavediscussedthe development of the Jammu and Kashmir conflict. We have examinedthevariouseffortsbyIndo-Pakandother countriestoresolvethegraveconflict.Theeffectof nuclear power acquisition by both the countries,9/11 terror attack on USA and USA and its allies invasion in Iraq. The mostbeneficiarypartyofJammuandKashmirconflictis therepublicofChina.Itinvadedabout38000square kilometers territory from Indian ruled Jammu and Kashmir in1962warandlaterPakistanceded5180square 98 Neutrosophic Sets and Systems, Vol. 2, 2013 Surapati Pramanik, Tapan Kumar Roy, Neutrosophic Game Theoretic Approach to Indo-Pak Conflict over J ammu-Kashmirkilometers areas to Beijing under a 1963 pact. The ongoing conflict between and Pakistan reflects the fact that both the countriesareincapableofsolvingJ&Kconflictandother coreissues.Attheinternationallevel,thirdparty arrangementscanbeestablishedwiththeparticipationof aninterveningstate,groupofstates,orinternational organizationsinordertoplayacrucialroleinhelpingto overcomeastrategicimpasse.Thequestionremains whetherIndiaandPakistanwouldacceptthirdparty mediation. Limits on international direct intervention in the J&K grave conflict have historically had to do with Indias insistencethatthe 1972 SimlaAgreementbetween India and Pakistan is sufficient to deal with the issue bilaterally. Itistonotedthatwhenwellconceivedan