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Electronic copy available at: https://ssrn.com/abstract=2876315 The Decline of Violent Conflicts: What Do The Data Really Say? Pasquale Cirillo and Nassim Nicholas Taleb 1 Nobel Foundation Symposium 161: The Causes of Peace Summary: We propose a methodology to look at violence in particular, and other aspects of quantitative historiography in general, in a way compatible with statistical inference, which needs to accommodate the fat-tailedness of the data and the unreliability of the reports of conflicts. We investigate the theses of “long peace” and drop in violence and find that these are statistically invalid and resulting from flawed and naive methodologies, incompatible with fat tails and non-robust to minor changes in data formatting and methodologies. There is no statistical basis to claim that “times are different” owing to the long inter-arrival times between conflicts; there is no basis to discuss any “trend”, and no scientific basis for narratives about change in risk. We describe naive empiricism under fat tails. We also establish that violence has a “true mean” that is underestimated in the track record. This is a historiographical adaptation of the results in Cirillo and Taleb (2016). Preamble The first theory of “long peace” is as follows. In 1858, one H.T. Buckle wrote: That this barbarous pursuit is, in the progress of society, steadily declining, must be evident, even to the most hasty reader of European history. If we compare one country with another, we shall find that for a very long period wars have been becoming less frequent; and now so clearly is the movement marked, that, until the late commencement of hostilities, we had remained at peace for nearly forty years: a circumstance unparalleled (...) The question arises, as to what share our moral feelings have had in bringing about this great improvement. (Buckle, 1858) . 1 Delft Institute of Technology and Tandon School of Engineering, New York University. The contributions of the authors to this paper and the general statistical studies associated with it, are equal. We thank Captain Marc Weisenborn for his diligent and indefatigable work in collecting, checking, and comparing data.

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Electronic copy available at: https://ssrn.com/abstract=2876315

TheDeclineofViolentConflicts:WhatDoTheDataReallySay?PasqualeCirilloandNassimNicholasTaleb1

NobelFoundationSymposium161:TheCausesofPeace

Summary:Weproposeamethodologytolookatviolenceinparticular,andotheraspectsofquantitativehistoriographyingeneral,inawaycompatiblewithstatisticalinference,whichneeds toaccommodate the fat-tailednessof thedataand theunreliabilityof the reportsofconflicts.Weinvestigatethethesesof“longpeace”anddropinviolenceandfindthattheseare statistically invalid and resulting from flawed and naive methodologies, incompatiblewithfattailsandnon-robusttominorchangesindataformattingandmethodologies.Thereisnostatisticalbasistoclaimthat“timesaredifferent”owingtothelonginter-arrivaltimesbetween conflicts; there is no basis to discuss any “trend”, and no scientific basis fornarratives about change in risk. We describe naive empiricism under fat tails. We alsoestablishthatviolencehasa“truemean”thatisunderestimatedinthetrackrecord.ThisisahistoriographicaladaptationoftheresultsinCirilloandTaleb(2016).

Preamble

Thefirsttheoryof“longpeace”isasfollows.In1858,oneH.T.Bucklewrote:

Thatthisbarbarouspursuitis,intheprogressofsociety,steadilydeclining,mustbeevident, even to the most hasty reader of European history. If we compare onecountry with another, we shall find that for a very long period wars have beenbecominglessfrequent;andnowsoclearly isthemovementmarked,that,until thelatecommencementofhostilities,wehadremainedatpeacefornearlyfortyyears:acircumstance unparalleled (...) The question arises, as to what share our moralfeelingshavehadinbringingaboutthisgreatimprovement.(Buckle,1858).

1DelftInstituteofTechnologyandTandonSchoolofEngineering,NewYorkUniversity.Thecontributionsoftheauthorstothispaperandthegeneralstatisticalstudiesassociatedwithit,areequal.WethankCaptainMarcWeisenbornforhisdiligentandindefatigableworkincollecting,checking,andcomparingdata.

Electronic copy available at: https://ssrn.com/abstract=2876315

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Bucklewasperhaps“right”(withminorhick-ups)foranotherfivedecades,butmoralfeelingsornot,thecenturyfollowingMr.Buckle’sproseturnedouttobethemostmurderousinhumanhistory.

Thefirst–obvious–problemisthatBucklemadeaseverelyflawedriskassessment.Thesecondisthathefeltobligatedtomountanarrativeentailing“moralfeelings”forwhatheperceivedwerethechangesintheenvironment.NotethatBucklewasplacinghimselfinthetraditionof“scientific”socialscienceofAugusteComte.

Anotherevent.In2004,BenBernanke,thenamemberoftheboardoftheFederalReserveBankoftheUnitedStatesproclaimedthateconomiclifewasundergoinga“greatmoderation”,onthebasisofanunprecedentedstabilityofeconomicvariables(Taleb,2007,2010).LikeBuckle,hefoundthereasonsforthat.Thetheorybecamethenormuntilthecrisisof2007whenweexperiencedasimilarrevisionofbelief.

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Thisarticleisorganizedasfollows.Firstwepresenttheproblemsassociatedwithhistoricalanalyses of violence. Second, we discuss the quantitative approaches since Richardson(1948) and present the statistical flaws and methodological errors in the widely heldtheoriessuchasthoseinPinker(2011).Third,wediscussourapproach.Fourth,weprovideaslightlymoretechnicalbackup.Fifth,wegiveourverdict.

SomeProblemsinQuantitativeHistoriography

Studying thehistoryof violence todetect trendsand changesover a timeperiod is anon-trivial taskforascientistconstrainedbyrigor. Welist fiveproblemsthatareparticulartoviolence,butmaybeuniversaltoanyformofquantitativeandstatisticalhistoriography.

Problem 1: Fat tails. First, we are dealing with a “fat-tailed” phenomenon. We defineviolenceseenquantitativelyaseither“fatalitiesoveraspecifictimeperiod”or“fatalitiesperspecific event” and both are fat tailed variables. What characterizes fat tailed variables?Thesehave theirproperties (suchas theaverage)dominatedbyextremeevents, those "inthe tails".Themostpopularlyknownversion is the"Pareto80/20"(80%of thepeople inItaly,Paretonoticed,own20%oftheland,andviceversa,whichbyrecursionleadsto1%ofthepeopleowning53%oftheland).

Thesetoolsarenotjust“changingthecolorofthedress”,buttheyrequireanewstatisticalframework and a different way of thinking, going from the tail to the body (standing theusualstatisticallogic,whichconsistsofgoingfromthebodytothetail,onitshead)–andthegreatmajorityofresearcherswhoaretrainedinstatisticsarenotfamiliarwiththebranchesof thedisciplineandtheoremsneededfor fat tails(seeTaleb2007/2010,2016).Toaddtothe problem, our examination shows thatwar turned out to be themother of fat tails, farworse than the popular 80/20 rule: there are fewphenomena such as fluid turbulence orthermalspikesonthesurfaceofthesunthatcanrivalthefat-tailednessofviolence.Further,historical data are temporal (spread out over time) and statistical analyses of time series(suchasfinancialdata)requirefarmoresophisticationthansimplestatisticaltestsfoundinempirical scientific papers. For instance one cannot blindly use the same methods tocomputethestatisticalpropertiesofcitysizeandthetimeseriesofwar–sinceinthelattercase theobservedpropertiesdependonour survival (saya1960nuclearwarwouldhavepreventedusfromhavingthisdiscussion),hencerestrictionapplytowhatcanorcannotbe

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inferred (there is a difference between ensemble probability and time probability, thoughnotalways,andtheeffectofthebiasneedstobeestablished).

Problem2:Boundedness.Notonlyarewedealingwithextremefattails,buttheeffect isboundedquantitatively,withanalmostpreciselyknownupper limit–nowarcankillmorethan thepopulationof theplanet. Thisbrings inanadditionalmathematical complication,since all techniques for fat tails requires an infinite support for the variable. Theboundednessrequiressomeformulaicadjutmenttothestatisticsofviolence–which,asweshowinsomedetail,hashadsofaramathematicallyandstatisticallynaiveliterature.Butitisnotallbadnewssincetherewillbeastatistical“mean”,which,carefullyinterpreted,canhelp in the analysis –while naive statistical analysis produces an “infinite” or “undefined”mean.

Problem 3: Reliability of historical data. The analysis needs to incorporate theunreliabilityofhistoricaldata–thereisnowaytogobackandfact-checkthecasualtiesinthePeloponesianwarandwerarelyonlyhavemorethanonesidetothestory.Estimatesofwarcasualties are often anecdotal, spreading via citations, and based on vague computations,almost impossible to verify using period sources. Even more recent events, such as theAlgerianwarofthe1960shavetwopolarizedsidestothestory;and,insomecasessuchastheArmenian-Syriacgenocideof1915-1917,thenumbersdonotconverge,andhaveactuallybeendivergingoveracentury.Thisunreliabilityrequirestakingintoaccountanotherlayerofuncertainty.Inaddition,theanalysismustconsiderthatmanywarswentunreported–wejustdonotknowhowmany,andsuchanumberisitselfarandomvariable.Thereexistsevenathirdlayerofuncertainty:thenumberofgapsbetweenwarscanbetreatedasarandomvariable,anditseffectmustbetakenintoconsiderationintheinterpretationoftheresults.

Problem4:Thedefinitionofanevent.Ahurdlecanbetheprecisedefinitionofanevent,whichappearstobeafunctionofthesophisticationofthehistorianandhisorherclosenesstoonesideinvolvedinit.Wementionedearliertheunderlyingstatisticalvariabledefinedas“fatalitiesperspecificevent”,but theverydefinitionofaneventmatters–and theanalysisshouldnotbefragiletothespecification.Forinstance,Pinker(2011)treatsasasingleeventthe“Mongolianinvasions”whichlastedmorethanacenturyandaquarter.Thisswelledthenumbersper event over theMiddle Ages and contributed to the illusion that violence hasdroppedsince, given that subsequent “events”hadshorterdurations. Effectively themainsourcessuchasPhilipsandAxelrod’s3-volumeEncyclopediaofWarlistnumerousconflictsin place of "Mongol Invasions" –themore sophisticated the historians in a given area, themore likely they are to break conflicts into different "named" events and, depending onhistorians,Mongolianwars range between 12 and 55 conflicts. If someMongolian-centricPinker’s counterpartwrote aboutEuropeanwars, hewouldhavebundled theperiod fromtheFranco-PrussianwartoWWIIas"NorthernEuropeanorWesternwars".

Not only are "Named" conflicts – part of what Richardson called “deadly quarrels” –anarbitrary designation that, often, does notmake sense statistically, but a conflict can havetwo or more names; two or more conflicts can have the same name, and we found nosatisfactoryclear-cuthierarchybetweenwarandconflict.Oursolutionis1)totreateventsastheshorterofeventoritsdisaggregationintounitswithamaximumdurationof25yearseach,whichcorrespondstoageneration,and2)performastudyofstatisticalrobustness(onwhich much, further down) to assess whether other time windows and geographicredefinitionsproducedifferentresults.

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Likewise,thedatamakesithardtoassesswhetherthenumbersincludepeoplewhodiedofside effects ofwars –say for example itmakes a differencewhether the victims of faminefromthesiegeofJerusalemareincludedornotinthehistoricalfigures.

Problem5:UnitsfortheAnalysis.Shouldoneconsider,inatrendanalysis,raworrelativenumbers,thatisactualnumberofpeoplekilledortheirratiotothetotalpopulation?Giventhat thepopulationof the earthhasbeen increasingover time, a constant rateof violencewouldgivetheillusionofriseincasualties.

OurpaperCirillo andTaleb (2016) set to construct themost statistically robustpictureofhistoricalviolencefor“namedconflicts”wecouldmakeundertheconstraintsgivenbythesefive problems. To deal with the first problem (Fat tails) we made use of a branch calledExtremeValueTheory.Forproblem2wehadtodevelopatechniquetoallowboundednessintheanalysisandpublishasstandaloneasitappliestoothersimilarproblems(Taleb,2016,TalebandCirillo,2015). Todealwithproblems3through5wereliedonvariousmethodsfromthebranchofrobuststatistics.

FatTailsandTheoriesofViolence

ThefirstmainattempttomodelviolenceusingpowerlawswasdonebythepolymathLewisFry Richardson, in 1948,when he fit a visual power law using Zipf Plots to data between1820and1945.Zipfplotsareavisual techniquenamedafterGeorgeZipfwhopopularizedPareto’s law by applying it to phenomena such as linguistics (word frequency) ,demographics(sizeofcities),andothers.Richardsonhimselfcamefromdiscoveriesofself-similarity and scaling in nature, particularly coastlines and some turbulent phenomena(Mandelbrot1983).

Intuitively,wecanexplainpowerlawsasfollows:#!"!#$% !"##"$% !"#$ !!!" !""!#!"!#$% !"##"$% !"#$ !!!" !"!

approximately

equal to#!"!#$% !"##"$% !"#$ !!!" !""!#!"!#$% !"##"$% !"#$ !!!" !""!

. This “scalability” is crucial as it makes the law bothmoreintuitiveandtractable.Morepreciselythereisan“exponent”inthetails,suchthat,foralargedeviationK,

𝑃𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝑜𝑓 𝑒𝑥𝑐𝑒𝑒𝑑𝑖𝑛𝑔 𝐾 = 𝑝𝑟𝑜𝑝𝑜𝑟𝑡𝑖𝑜𝑛𝑎𝑙 𝑡𝑜 𝐾!!

where “alpha” is the tail exponent and the major determinant to the shape of thedistribution.Thelowerthealpha,thefatterthetails. Notehereahelpfulpropertythatthealphadoesnotchangeifonetakeshalfthedataanddoublesit.Itmeansthealphaisrobusttomanymistakesindata.Alsonotethattheproportionalityholdsinthetailsi.e.Kverylarge,butnotnecessarilyforsmallervalues.

Many research papers subsequently confirmed the power law or “Paretianity” of the data(although Cirillo 2013 shows thanmany phenomena identified as power laws are in factlognormal,thoughstillfat-tailedbecauseoftheirhighvariance)–butthepracticeinthatfieldwas to find a mechanism that justifies a statistical law prior to adopting it. Mechanismsabound, and after the works of Mandelbrot (1983) linking the phenomenon to fractalgeometry(asRichardsondid),manybranchesof“complexitytheory”wereborn.Cederman(2003)lookedatabroadersetand“justified”theprocessthankstoaclassofmodelscalledagent-basedthathavebeenknowntoproducepowerlaws.Cellularautomataandmodelsofinteractionbetweenagentshavenow flourishedprovidingus a large computational-based

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modeling apparatus; it suffices to specify the conditions and the mathematical andcomputational framework in place allows us to checkwhether power laws emerge or not(seeWolfram,2002andMathematica’sprogramming language;note that thesemodelsarenotanalytic-functionalbutalgorithmic requiringcomputationalmethods). It isnotpartofthisanalysistodiscussthesemodels–ourdomainisstatisticalinferencenotmodelbuilding;wefocusonthestatisticalbackupthatallowstherejectionor“acceptance”ofsomemodels.

We note that all “fat tails” are not power laws, there are distributions that produceconcentration –it is just that power laws aremorenatural becauseof their scalability andappealtousersastheyareverytractableanalytically.Butthereareclaimsoneshouldnevercasuallymake in thepresence of fat tails, power lawsor not. Let us nowput together theproblemsofstatisticalinferenceunderfattails.

FatTails,LongPeace,andtheFoundationalPrinciplesofStatisticalInference

Pinker (2011, 2011b) started the promotion of an idea that violence has dropped, withsimilartoBuckle–eerilysimilar—aninvocationtothevariousmoralvaluescausingwhathecalls “theobsolescenceofmajorwars” (Pinker2011b). Hewrites,Buckle-like, “Themostpromising explanation, I believe, is that the components of the human mind that inhibitviolence—whatAbrahamLincolncalled“thebetterangelsofournature”—havebecomeincreasinglyengaged.”

It is important todiscuss thatbookbecause ithasbeencitedas “evidence” for thedrop inviolence across political science. Pinker deals with the phenomenon of violence, and itsmanifestationsatdifferentscales, fromhomicidesandrapes, toriotsandwars, fromdeathpenalty and torture issues, to civil rights violations anddenial. To explain and sustainhisvision about the general decline in violence, Pinker develops the metaphor of a constantbattle,withinhumanity,amongsome``InnerDemons”,likeforexamplerevenge,sadismandideology, and some ``BetterAngels”,such as empathy, self-control and reason (even if it isnotcompletelyclear,at least tous,what it is thatPinkercallsreason).Demonsaremainlyexpressionsofatavicfeelingsandcompulsions,whicharerelatedtotheoriginalbeastinus.Angelsarearesultofcivilevolutionandreasondevelopment.Andsincecivilizationseemstobeanunstoppableprocess,Angelsareboundtowinthebattle.

Usingasortofmeta-analysis,relyingonothers’results,Pinkercollectsabunchoffigurestosupporttheideaofadeclineinviolenceinthehistoryofhumanity.

Toarmedconflicts,inalltheirpossibleexpressions,hedevotestwochapters:thefifth,``TheLong Peace”, and the sixth, ``The New Peace”. In the specific case of wars, he relies onpreviousanalysesby,forexample,Cederman(2003)andRichardson(1960).Butthewayinwhichhereadsandinterpretstheresultsofscholars likeRichardsonrevealsanattemptofbendingempiricalevidencetohisowntheory,e.g.whenhedealswiththePoissonnatureofthenumberofarmedconflictsovertime.Aswealsofindoutinourdataanalysis,consistentwith Richardson (1960), there is no sufficient evidence to reject the null hypothesis of ahomogenous Poisson process,which denies the presence of any trend in the belligerence ofhumanity. Nevertheless, Pinker refers to some yet-unspecified mathematical model thatcouldalsosupportsuchadeclineinviolence,whathecallsa“nonstationary”process,evenifdata look theway they look. It is on the basis of this and other apodictic statements thatPinkerbuildshisnarrativeaboutviolence.

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Pinker,inadditionbuildsatheorythatisnotatallstatisticallyrobusttoproblems3and4oftheprevioussection–suchasaschangesintheAnLushanestimatesorthegranularityoftheMongoliannamedconflict.

But at the core, Pinker’s severe mistake is one of standard naive empiricism –basicallymistaking data (actually absence of data) for evidence and building his theory of whyviolencehasdroppedwithout even ascertaintingwhetherviolencedid indeeddrop. This isnot to say that Pinker’s socio-psychological theories can’t be right: they are just notsufficientlyconnectedtodatatostartremotelylookinglikescience.Fundamentally,statisticsis about ensuring people do not build scientific theories from hot air, that is withoutsignificantdeparturefromrandom.Otherwise,itispatently"fooledbyrandomness".Andwehaveaveryclearideawhatdeparturefromrandommeans.

Forfattailedvariables,theconventionalmechanismofthelawoflargenumbers(onwhichstatisticalinferencereposes)isconsiderablyslowerandsignificancerequiresmoredataandlonger periods. Taleb (2016) shows that the Pareto 80/20 takes 1013more data than acorrespondingNormaldistributionthatisubiquitousintextbooksifonelooksatthesampleaverage.Simply,thesampleaverageisnotagoodestimatorofthe“true”mean;ithaswhatiscalled a small sample bias when data is one-tailed (i.e. can only take either positive ornegativevalues,asisthecasewithviolence). Inotherwords,notonlydoweneedalotofdatatoknowwhat’sgoingon,but,asinthecaseofviolence,weshouldexpectthatthemeanviolenceasmeasuredinsampletobelowerthanthetruemean.Thestatisticianwouldnevermeasure themean in-samplebutuseandwewill resort to “back-door”methodsandmorerigorousmaximum-likelihoodtechniques.

Ironically,thereareclaimsthatcanbedoneonlittledata:inferenceisasymmetricunderfat-taileddomains.Werequiremoredatatoassert that therearenoblackswansthantoassertthatthereareblackswanshencewewouldneedmuchmoredatatoclaimadropinviolencethantoclaimariseinit.

Finally,statementsthatarenotdeemedstatisticallysignificant–andshowntobeso–shouldnever be used to construct scientific theories. Descriptive statistics, though deemedunscientificandanecdotal,canbeusefulforexploratorydiscussions,butnotwithfattailedprocesseswhentherandomvariableentailsexposuresratherthanbinaryoutcomes.

Thesefoundationalprinciplesareoftenmissedbecause,typically,socialscientists’statisticaltraining is limited tomechanistic tools fromthin taileddomains. Inphysics,onecanoftenclaim evidence from small data sets, bypassing standard statistical methodologies, simplybecausethevarianceforthesevariablesislow(ortheprocesshasastrongtheoryverifiedona high signal to noise ratio). The higher the variance, the more data one needs to makestatisticalclaims.Forfat-tails,thevarianceistypicallyhighandunderestimatedinpastdata.And,asweshowed itdropsveryslowlyunderaveraging(the lawof largenumbersmeansthesampleaveragebecomeslessandlessvolatileasoneincreasesdata,typicallyattherateofthesquarerootofadditionaldatacounts;thisisnotthecasehere).

Thesecond–moreserious–errorPinkermadeinhisconclusionistobelievethattaileventsandthemeanaresomehowdifferentanimals,notrealizingthatthemeanincludesthesetailevents.Further,forfat-tailedvariables,themeanisalmostentirelydeterminedbyextremes.Ifyouareuncertainaboutthetails,thenyouareuncertainaboutthemean.Itisthusincoherenttosaythatviolencehasdroppedbutmaybenottheriskoftailevents;itwouldbelikesayingthat someone is "extremely virtuous except during the school shooting episode when he

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killed30students",orthatnuclearweaponsareverysafeastheyonlykillasmallpercentageofthetime.

Ourmethodology

TheData:Weselectedallavailableobservationsovertheperiod1-2015AD,usuallyarmedconflictswithmorethan3,000casualtiesinabsoluteterms,countingbothsoldiersandcivilians.

Thereareafewexceptionsinourdataset,someeventsthatcannotbeconsideredstandardarmedconflictsinthedefinitionofWallensteenandSollenberg(2001).Theyaresomeofthebloodiestdictatorshipsofhistory,suchasStalin’sregime.Thischoicehasbeenmadetobeconsistentwithotherworksaboutwarvictimsandviolence(e.g.Mueller,1989;Pinker,2011).

Thedatacomefromdifferentsources,suchasPhillipsandAxelrod(2004) EncyclopediaofWars,andNecrometrics(2015),withsomeconsiderationsofselectionsbyBerlinski(2009),Goldstein (2011), Mueller (1989), Pinker (2011), and White (2013). For some onlineresourceslikeNecrometrics(2015),wedouble-checkedthedataagainstthecitedreferences.

ThefirstobservationinourcollectionistheBoudicca’sRevoltof60-61AD,whilethelastoneisthestill-openinternationalarmedconflictagainsttheIslamicStateofIraqandtheLevant.Weendedconcentratingon565pieces.

A natural question is why we have chosen to impose a 3000-casualties threshold, whencollectingtheobservationsaboutarmedconflicts.Wehavethreemainreasons:

• Wedonotneedsmallercasualtiestogettheproperties,assmallercasualtiesdonotaffecttheaverage(Richardsonhimselfnoted,“Anyonewhotriestomakealistof"allthewars"encountersthedifficultythattherearesomanysmallincidents,thatsomerulehastobemadetoexcludethem.”(Richardson,1948).Thehigherthethreshold,thefewertheobservationsandthelowerthenoiseandimprecision.

• Themainobjectofourconcernistailrisk,that’stheriskofmajordestructiveconflicts.Thestatisticaltechniquesweusetostudythistypeofextremeeventsrequiretheimpositionofthresholdsforalltheapproximationstohold.

• Conflictswithmanyvictimsaremorelikelytoberegisteredandstudiedbyhistorians.Itwouldbeimpossibletohavereliableinformationabout“small”battleswithtensofvictims.Empirically,3000victimsprovedtobeagoodselectionthresholdforotheraspectsoftheanalysis.

• A3000-casualtiesthresholdgivesusabetterconfidenceabouttheestimatednumberofcasualties,thankstothepossiblylargernumberofsourcestocompare.However,theriskofover-exaggeration,especiallyforthelargeconflictsofantiquityissomethingwehadtotakeintoconsideration.

Inordertobeconsistentwiththesociologicalliteratureonarmedconflicts,wehaveuseddifferenttypesofdatainouranalyses:

• Actualdata,i.e.dataascollectedbyhistorians.Statisticallyspeakingthesearetherawdata.

• Rescaleddata,i.e.casualtiesexpressedintermsoftoday’sworldpopulation,inordertohavecomparabilityintermsofrelativeimpactofwars.Rescaleddataareobtainedbydividingthenumberofcasualtiesinagivenyearby

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theworldpopulationinthatyear,andthenmultiplyingeverythingbytoday’sworldpopulation.Whenrescalingdata,wehaveusedthepopulationestimatesofKleinandvanDrechts(2006),andUnitedNations(1999,2015).

• Transformed-rescaleddata,thatisdataobtainedviatheso-calleddualtransformation,whoseaimistodealwiththeboundess/unboundenessofthesupportofthedistributionofwarcasualties(Taleb,2016;TalebandCirillo,2015).Thistransformationismoretechnical,but,aswebrieflyexplaininthesectionaboutMethods,itismeanttocorrectforaninterpretationerrorabouttheapparentinfinite-meannatureofwarcasualties’data.

Interestingly,ourresultsholdnotwithstandingthedefinitionofdata.Numericalestimatesmayvary,butthequalitativeinterpretationstaysthesame.

DataProblems:Firstofall,itisimportanttonoticethatourdata,especiallyforwhatconcernsantiquity,arelikelytosufferfromselection/historiographicalbias.ItwasinfactnotpossibletocollectobservationsabouttheconflictstakingplaceintheAmericasandAustralia,beforetheir“discovery”byEuropeanconquerors.Naturally,thislackofevidencedoesnotmeanthatnothinghappenedinthoseareasinthepast.

Similarly,becauseofproblemswithsources,weprobablymisssomeconflictsofantiquityinEurope,or,say,inChinainthesixteenthcentury.However,wecanassumethatthemajorityoftheseconflictsarenotintheverytailofthedistributionofcasualties,sayinthetop10or20%.Itisinfactnotreallyplausiblethathistorianshavenotreportedaconflictof1millioncasualties(ormore),sothatsuchaneventisnotpresentinoursources.

Dealingwith historical data, some dating back to the first centuryAD, also requires someattention,becauseof theprobablyproblemsof inconsistencyand lackofuniformity in theattributionofcasualtiesbyhistorians. Itcanbedifficult–ifnotimpossible–todistinguishcasualties from direct violence from those arising from such side effects as contagiousdiseasesandhunger.

Wementioned earlier that reportswere highly source dependentwith an impossibility tofact-check.Somedata,suchastheAnLushanrebellion,estimatedbyPinker(2011)tohavekilled 36 million people (around 430 million by today’s population) are highly dubious(Durant,1960)–andhelpperpetuatetheimpressionthattheworldis“lessviolent”.Itmayhave been 13 million, and the numbers were the result of the census and dispersion ofofficialsinrevenuedepartment.(Fitzgerald,1935,BBC2012)

Data aggregation is another issue.We said that conflicts such as the so-called “MongolianInvasions”arenothingmorethanartificialdesignations,whichneedtobetreatedcarefully,assyntheticobservations.Theseeventsareinfactartificialcontainerscreatedbyhistoriansto aggregate those battles sharing important historical, geographical and politicalcharacteristics, but that never really existed as a single event. Forhistorical/historiographicalreasons, theseevents tendtobemorepresent inantiquityandtheMiddleAges,thuspossiblycausinganaiveoverestimationoftheseverityofconflicts inthe past. Even among these aggregations there are major differences: WW1 and WW2naturally also involved several tens of battles in very different locations, but these battlestookplaceinamuchshortertimeperiod,withnomajortimeseparationamongthem.

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Curing thedataproblems:Allthesedataproblemscanbedealtwithbyconsideringeachsingle observation in our collection as an imprecise estimate, in the definition of Viertl(1995). Our technique for robustness is as follows. Using Monte Carlo methods , andassumingthattherealnumberofcasualtiesinaconflictisuniformlydistributedbetweentheminimumandthemaximumestimateintheavailablehistoricalrecords,thetailexponentξ,the quantity that governs the tail of ourwar casualties’ distribution, is not affected, apartfromthenegligibledifferencesinthesmallerdecimals.Wedidanotherbatteryoftestsforothervariables.(SeeFigures1and2foranideaofhowweconductedthetests.)

Fromastatisticalpointofview,themethodsweusetostudytailriskarerobust. Insimplewords,thismeansthatourresults–andtherelativeinterpretations–areimmunetosmallchanges in the data. Even more: our results cannot be reversed on the basis of a fewobservations,added,removedor“corrected”.Athoroughanalysisofrobustnessshowsthatourestimatesarepreservedandwouldreplicateevenifwemissedonethirdofthedata.

Toconclude this sectionabout “dataproblems”,we think it is important to stress thatourdataset,despiteitsevidenttemporalconnotation,doesnotformapropertimeseries.Itisinfact trivial to notice that the different conflicts of humanity do not share the same set ofcauses.Battlesbelongingtodifferentcenturiesandcontinentsarenotonlyindependent,butalsosurelyhavedifferentorigins. Instatisticalwords,wecannotassumetheexistenceofauniqueconflictgeneratorprocess,asifconflictswerecomingfromthesamesource.

For this reason, we believe that performing time series analysis on this kind of data isuseless, if not dangerous, given that one could extrapolatemisleading trends, as done forexample in Pinker (2011). How could the An Lushan rebellion in China (755 AD) bedependentontheSiegeofConstantinoplebytheArabs(717AD),orhaveanimpactontheVikingRaidsinIreland(from795ADon)?

Noticethatwearenotsayingthatallconflictsareindependent:duringWW2,theattackonPearl Harbor and the Battle of Francewere not independent, notwithstanding the spatio-temporal divide, and that’s why historiansmerge them into one single event, as we havealreadynoticedbefore.Andwhilewecanacceptthat,historically,mostofthecausesofWW2are related to WW1, it is better to avoid translating this dependence when studying thenumberofcasualties:itwouldbequiteabsurdtobelievethatthenumberofvictimsin1944hadanythingtosowiththedeathtollin1917.HowcouldthemagnitudeofWW2dependonWW1?

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Figure1Howwetestedourrobustnesstothereliabilityofhistoricalreports.Wecreate100,000differenthistoriesasuniformrandomnumbersbetweenhighandlowestimatesfromthedatasets(whichunderaggregationappearGaussian)andcheckifre-combinationsleadstodifferentresults(weusedthep-value,actually1-p-valueforthescaleparameternotbecausewerelyonp-valuesbutbecausep-valuesareextremelysensitivetochangesindata).SomehistoriesincludePinker’sexaggeratednumbersfortheAnnLushanrebellion,othersdon’t.

Figure2Thetailexponentfrommaximumlikelihood,notEVT,isinvarianttoerrorsinthereportingofconflicts.

Methods

Sinceweareinterestedinestimatingthetailriskofviolentconflict,thatistheriskoflargedestructive wars and armed conflicts, we use tools from extreme value theory (EVT) to

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understandthebehavioroftherighttailofthedistributionofwarcasualties.EVTisabranchof statistics dealing with extreme and rare events, in the form of maxima and minima(Gumbel,1938;Embrechtsetal.,2003;deHaanandFerreira,2006).

Within the broad field of EVT, we mainly use the so-called Generalized Pareto (GP)approximation (Balkema and de Haan, 1974; Pickands, 1975), according to which all theexceedancesaboveahighthreshold,ifthisthresholdiscorrectlychosen,tendtofollowaGPdistribution,askeweddistribution,whichcanbecharacterizedbyfattails.

ThefunctionformofaGPdistributionis

𝐺𝑃𝐷 𝑧; 𝜉,𝜎, 𝑢 = 1 − 1 + 𝜉 !!!!

!!! 𝜉 ≠ 0

1 − 𝑒!!!!! 𝜉 = 0

,

where𝑧 ≥ 𝑢for𝜉 ≥ 0,and𝑢 ≤ 𝑧 ≤ 𝑢 − 𝜎/𝜉for𝜉 < 0,with𝑢, 𝜉 ∈ ℝand𝜎 > 0.

Theparameter𝜉is themost importantone forus,as itcontrols for the fatnessof therighttail.Thelarger𝜉,thefatterthetail.For𝜉 > 0.5,theGPdistributionhasaninfinitevariance.For𝜉 > 1,eventhemeanisnotfinite.

Forwhatconcernswarcasualties,wefindthattheParetiantailisactuallysofatthat,fromatheoreticalpointofview,themeanofthedistributionisnotfinite(𝜉 > 1).Insimplewords,thismeansthatthetailriskissolargethatonesingleevent,onesinglewar,coulddestroythewholehumanity(7.3billionpeople).

Inreality,thingsarealittlebitmorecomplicated,becausedatacanbemisleading,evenwhenapproachingaproblemusingthecorrectmethodology(EVT,whenstudyingtails).Whileweshowthatthetailriskofviolentconflictisactuallylarge–muchlargerthanwhatonecouldsimply infer using standard descriptive and inferential statistics (not appropriate in thiscase),wealsopointoutthatitcannotbeinfinite,asdatatendtosuggestnaivelyusingEVT.In fact, no single conflict can killmore than thewholeworld population. This implies thepresenceofanupperboundthatwecanusetocorrectourestimates.Thedualdistributionapproach, basedona special log-transformationofdata, is theway inwhichwedealwithapparently infinite mean phenomena like war casualties. For more details about ourmethodology, especially for the use of the dual distribution, we refer to Cirillo and Taleb(2016).

Onceagain,itisworthunderliningthatallthestatisticalmethodsweusearerobust,thatistosaytheytendtobeimmunetoeventonon-trivialchangesinthedata(athirdofreportedeventsinourdatacouldchangesignificantly,andstillourresultsarepreserved).

Thedistributionofwarcasualties:basicfacts

Whenlookingatthedistributionofwarcasualties,thefirstthingwenoticeisthatitishighlyskewedwith a very fat right tail. Figure 3 contains a simple histogram using actual data:

12

whilemost armed conflict generate a few thousands victims2, on the left-hand side of thepicture,afewconflictscausemillionsofcasualties,withWW2totalingbetween48.5and85millionvictims,dependingonthesource(inthegraphweshowtheaverage:73mio).

Figure3:Histogramofwarcasualtiesusingrawdata.

The average number of casualties in our sample is 1,067,568. The in-sample standarddeviation is 5,738,541. However, since the standard deviation is not a reliable measureunderfattails,giventhatthetheoreticalvariancemaynotexist(Embrechtetal.,2003),wealso provide the mean absolute deviation, or MAD3: 1,747,869. This number shows theextremevolatilityofwarcasualties,somethingcompatiblewithafat-tailedphenomenon.

2Pleasenoticethatwearenotgivinganyethicaljudgement.Whenwesay«afewvictims»,wethinkofthemasstatisticaldata,numbers.Fromanethicalpointofview,onevictimisalreadytoomuch.3TheMADisthemeanofalltheabsolutedeviationsofeachdatapointfromthesamplemean.Formally:MAD(x)=E|X-E[X]|.

13

Other useful statistics are the median (40,000), the first quartile (13,538) and the thirdquartile(182,000):75%ofallarmedconflictsgeneratelessthen182,000casualties,andstillthesamplemeanisalmost6timeslarger!

Figures 4 and 5 show the number, and the relative magnitude with respect to worldpopulation,ofwarcasualtiesovertime,whenusingtwodifferentdefinitionsofdata:actualandrescaledobservations.

Figure4:Warcasualtiesovertimew.r.t.worldpopulationusingactualdata.

It is interesting to notice how the choice of the type of data may lead to differentinterpretationoftrendsandpatterns.Fromrescaleddata(Figure3),onecouldforexamplesuperficiallyinferadecreaseinthenumberofcasualtiesovertime.Itisinfactobviousthatrescalingdatawill tend to inflate past observations, as alreadynoticedbyEpstein (2011),who also object that rescaling may generate paradoxical situations. Citing him: “[...] whyshouldwebe contentwithonly a relativedecrease?By this logic,whenwe reach aworldpopulationofninebillionin2050,Pinkerwillconceivablybesatisfiedifameretwomillionpeoplearekilledinwarthatyear”.

As to the number of armed conflicts, both figure 5 and 7 seem to suggest an increase ofbelligerenceovertime,sincemosteventsareconcentratedinthelast500yearsorso.Thisisverylikelyjustanillusion,probablyduetoareportingbiasfortheconflictsofantiquityandearly Middle Ages. It is certainly easier to obtain decent information about more recent

14

conflicts, that is why we have many rather precise observations in the last decades andcenturies,withrespecttowhathappenedinthethirdcenturyAD.

Tocorrectlystudythetailriskofarmedconflicts,weneedtounderstandwhetherourdatareallyexhibitaParetianrighttail,assuggestedbyFigure4. Theanswerisaffirmative.Forexample,inFigure6,weseethat,onalog-logscale,thedistributionofwarcasualtiesshowsa clear linear behavior in the right tail. Figure 6 is a Zipf plot (mentioned earlier), and itrepresentsaheuristictooltolookforParetianityindata(Cirillo,2013),byusingapropertyof the survival function of a Pareto distributed random variable.Many other plots can beused to verify the presence of a fat right tail, such asMaximum-to-Sum andmean excessplots,andwerefertoCirilloandTaleb(2016)formoredetailsanddiscussions.

Figure5:Warcasualtiesovertimew.r.t.worldpopulationusingrescaleddata.

Figure6notonlyconfirmstheideaofaParetianrighttail,butitalsosuggeststhatthewholedistributionofwarcasualtiesmaybeinthedomainofattractionofafat-taileddistribution.Thelinearityofthesurvivalfunctionstartsindeedfromtheveryleft-handsideoftheplot.Inaddition Figure 6 shows how the right tail tends to close down a little bit. This is aphenomenoncommonlyknownasfinitesamplebias.Itisinfacthighlyimprobabletobeabletoobserveasufficientnumberofmaximainthedata,sothatthetaildecreaseslinearlyuntiltheend.

Table1 contains some informationabout theoccurrenceof armedconflictsover time.Forexample, ifwe look at events generating at least 500,000 victims,we discover that,when

15

using raw data, we have to wait an average of 24 years to observe such events. Thecorrespondingmeanabsolutedeviationis33years.If,onthecontrary,weuserescaleddata,theaverageinter-arrivaltimeis10years,withaMADof12.

Threshold AvgRaw MADRaw AvgRescaled

MADRescaled

500k 24 33 10 12

1mil 34 48 13 16

2mil 57 73 20 24

5mil 93 117 34 43

10mil 136 139 52 61

20mil 252 267 73 86

50mil 372 362 104 114

Table1:Averageinter-arrivaltimesandtheirmeanabsolutedeviation,inintegeryears,fordifferentcasualties'thresholds.

16

Figure6:ZipfPlot(log-logplotofthesurvivalfunction)ofwarcasualtiesusingactualdata.

If we increase the threshold4, and consider conflictswith at least 5million casualties, weneed towait93or34years,dependingon thedatadefinition. Intuitively, thebloodier theconflict,thelongertheinter-arrivaltime.Foraconflictwithatleast50millionvictims,averyextremeandhopefullyrareevent,theaverageinter-arrivaltimeis372years,usingrawdata,withaMADof362.

All this tells us that the absence of a conflict generating more than – say – 20 millioncasualtiesinthelast20yearsishighlyinsufficienttostatethatitsoccurrenceprobabilityhasdecreasedovertime,giventhattheaverageinter-arrivaltimeis252years(73forrescaled),withaMADof267(86 forrescaled)years!Unfortunately,westillhavetowaitquitesometimetosaythatwearelivinginamorepeacefulera;theactualdatawehavearenotinfavornoragainstastructuralchangeinviolence,whenwedealwithwarcasualties.Verysimply:wecannotsay.

4ThethresholdsinTable1arejustarbitrary,andmeanttogiveusefulinformationinacompacttable.Otherthresholdscanbechosen,butonegeneralmonotonebehaviorcanbeobserved,inaccordancewithintuition:thehigherthethreshold,thelongertheaverageinter-arrivaltime.

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17

Thetailriskofarmedconflicts

As said before, using extreme value theory, and in particular the generalized Paretoapproximation,wecanextrapolateinformationaboutthetailriskofarmedconflicts.

TheestimationoftheparametersofaGPDcanbeperformedindifferentways,andwerefertoEmbrechtsetal.(2003),ordeHaanandFerreira(2006)formoredetails.Withourdata,thebestresultsareobtainedusingmaximumlikelihood(CirilloandTaleb,2016).

Table2containsourestimatesforthegeneralizedParetoapproximationforthedistributionofwarcasualties,usingbothactualandrescaleddata.Forbothdefinitions,wecanseethatthe thresholds above which the GPD approximation holds are definitely larger than theoriginal 3000 casualtieswe have imposed in collecting the data. However, in both cases,almost60%ofalltheobservationlieabovethetwothresholds.Sinceweareinterestedintheriskofvery largeconflicts,dealingwiththetop60%ofallconflicts isdefinitelymorethansufficientforourpurposes5.

ThemostinterestingresultofTable2isthequalitativeinformationwecanextrapolatefrom𝜉(decimals aremuch less relevant). Forbothactual and rescaleddata,we clearly see that𝜉 > 1. This indicates thepresenceof an infinitemeanphenomenon, that is aphenomenonwhoserealizationscanbesolargeanderraticthatthemeanisnotareliablequantity.AndifthemeanoftheGPDisnotfinite,thenthemeanofthewholedistributionofwarcasualtiesmustbeinfinite.Infact,sincethetailmeanisacomponentofthewholedistributionmean,iftheformerisnotfinite,thesameholdsforthelatter.

Data u(threshold) ξ σ

Raw 25k 1.4985 90620

Rescaled 145k 1.5868 497436

5InmostapplicationsofEVT,onlythetop5%(orless)oftheobservationslieabovethegiventhreshold.Our60%isdefinitelyverylarge,confirmingtheideaofawholedistributioninthedomainofattractionofafat-taileddistribution,liketheFréchet(Embrechtsetal.,2003).

18

Table2:MaximumlikelihoodestimatesfortheparametersoftheGPDapproximationoftherighttailofthedistributionofwarcasualties.AllestimatesaresignificantwithatypeIerrorof5%.

As said, these results are robust tomissing ormisspecified data. As shown in Cirillo andTaleb (2016),using tools likebootstrap,up to20%of theobservations (in the tailornot)couldchange,withoutaffectingtheresultsoftheanalysis.

Butifthemeanofthedistributionofwarcasualtiesisnotfinite,thenitmeansthatthetailriskofarmedconflictsisnotfinite.Inotherwords,wecouldexperienceanysecondasingleeventannihilatinghumanity.Anuclearholocaust,orevenworse.

Butcanthisreallybethecase?

Whendealingwithtails,extremevaluetheoryistherightapproach.Itwouldbewrongandhighly misleading to approach tails using other techniques, mainly relying on normality.However, when using EVT, it is extremely important to take into consideration the realnatureofdata.Canaconflictkillmorethanthewholeworldpopulation?

Theanswerisclearlyno.Andthisfactneedstobetakenintoaccount,ifwedonotwanttobefooledbydata.

Thedistributionofwar casualties isnecessarilybounded:we surely cannotkill anegativeamount of people, but, on the other side, we cannot kill more than the whole worldpopulation(atpresent,7.3billionpeople,accordingtotheUnitedNations).Fromastatisticalpointofview,boundnesshasoneimportantimplication:allthemomentsofthedistributionneedtobefinite,thusincludingthemeanandthevariance.Thesearetheshadowmoments,intheterminologyofTalebandCirillo(2015),i.e.momentsthatcannotbecorrectlyinferredfromdata,unlesswetakeintoconsiderationtheexistenceofanupperbound,andwecorrectforit.

Using the so-called dual distribution (Cirillo and Taleb, 2016), that is a particular log-transformationoftheoriginaldata,tomapthemontheboundedsupport,onecanobtaintheactualmomentsofthedistributionofwarcasualties.Thesemomentsarenaturallyfinite,butthey tend to be much larger than those one could estimate from data using their simpleempirical counteparts (whichare thereforenot reliable).Forexample,usingrescaleddata,wediscoverthatthetailmeanofwarcasualtiesabovethe1millionthresholdis6.21million,againstacorrespondingsamplemeanof3.95million(1.57timeslarger).Forathresholdof50millionvictims,thesamplemeanis28.22million,whilethetrue(shadow)meanis67.17(2.38timeslarger).

Whendealingwithtailrisk,anothersetofimportantstatisticsisrepresentedbyquantiles.Aquantile is the value above which a certain percentage of observations lie. The top 5%quantileisthevalueabovewhichwecanfind5%ofalltheobservations.Table3containsthetopquantilesof thedistributionofwar casualties,using thedualdistributionapproachonbothactual and rescaleddata.The results are frightening: there is a5%probability, usingactual data, of obseving a conflict generating at least 2,380,000 casualties; and a 1%probabilityofconflictswithatleast26.8millionvictims.Evenworse:ourdataalsosupporta0.1%probabilityofawarkillingsomething like800millionpeople,more than10%of thewholeworldpopulation.Thesefiguresareevenscarierwhenweuserescaleddata.

19

%above

RawData×107

RescaledData×107

5% 2,380,000 15,400,000

1% 26,800,000 198,200,000

0.1% 801,100,000 4,751,500,000

Table3:Topquantilesforthedistributionofwarcasualties,asobtainedviathedualdistribution.

Conclusion:Isthereanytrend?

Theshortanswerisno.

Our data do not support the presence of any particular trend in the number of armedconflictsovertime.Humanityseemstobeasbelligerentasalways.Noincrease,nordecrease.

Naturally we are speaking about the type of conflicts for which we have performed ouranalysis,thatistosaythelargestandmostdestructiveones.Wecannotsayanythingaboutsmall fightswith a few casualties, since they do not belong to our data set –however it iscrucial that,asacentralpropertyofthefat-tailednessoftheprocess,adecline inhomicidedoesnotaffect the totalpropertiesofviolenceandanyone’s riskofdeath. Aswesaid, themeanistaildriven.

At the best of our knowledge no available data set contains enough information tomakecrediblestatementsaboutstatisticallysignificanttrendsinthenumberofconflictsovertime,unlesswereally think it isreasonabletoextrapolate long-termtrendsonthebasisofsixtyyearsofobservations,likethoseafterWW2.Giventheinter-arrivaltimeswehaveobservedabove,itwouldbequitenaïvetoactthatway.

Ifwefocusourattentiononourdataset,andinparticularontheobservationsbelongingtothelast600years(from1500ADon),forwhichmissingobservationsshouldbefewerandreportingerrorssmaller,ouranalysessuggest that thenumberof largeconflictsover timefollows a homogeneous Poisson process. In a similar process, the number of observationsover time, oncewe fix a given time interval (say50years), follows aPoissondistribution.The number of expected data points only depends on the length of the time interval wechoose. For intervals of the same size, the expected number of observations is the same,because the intensity of theprocessdoesnot varyover time. In simple terms, this findingsupportstheideathatwarsarerandomlydistributedaccidentsovertime,notfollowinganyparticulartrend,asalreadypointedoutbyRichardson(1960).

Interestingly,similarconclusionscanalsobederivedbyasimpledescriptiveanalysisofdata,somethingthatshouldmakeourresultsaccessiblealsotonon-statisticians.

InFigure7and8weshowtheaveragenumberofcasualtiesandthenumberofconflitsintheperiod 1500-2015, using a non-overlappingmovingwindow of 20 years. Thismeans that

20

boththeaveragecasualtiesandthenumberofconflictsarecomputedfortheperiods1501-1520(thereisnoobservationin1500),1521-1540,1541-1560andsoon.

In Figure 7, the red dots represent rescaled data, while the green ones are the actualobservations.Itisquiteevidentthat,forwhatconcernstheaveragenumberofcasualties,nocleartrendisobservable.

Figure7Averagenumberofcasualtiesintheperiod1500-2015usinganon-overlappingmovingwindowof20year.Reddots:rescaleddata.Greendots:actualdata.

In Figure 8,we show the number of armed conflicts in the samemovingwindows as perFigure7.Inthiscase, thenumberofconflictsseemstobeincreasingovertime,evenifthevolatility itself appears to be higher. This is an interesting phenomenon from a statisticalpointofview,asitmakesthesimpleinferencebasedonafewyearsofdata(namelythelast60 ones, as in the “Long Peace” theory) not at all reliable. While we are aware that thisbehaviorcouldbeduetoahistoriographicalreportingbias,accordingtowhichmorerecentconflictsaremorelikelytoberecordedindata,wewouldliketostresshow,inanycase,noLongPeace is observed. Inparticular, the last 200yearsprove to bequite belligerent and“stable”.

1500 1600 1700 1800 1900 2000

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Figure8Numberofconflictsbetween1500and2015,usinganon-overlapping20-yearmovingwindow.Thenumberofconflictsisclearlyincreasingovertime,togetherwiththeirvolatility.

Onefinalcommentonthefiner-grainedperiodsinceWW2.Figure9illustratesthemistakemade in theorizing aboutwhat has happened since 1945. Aside from the fact thatwe arepickingaspikeandthatadropisnaturalaftereveryspike.Thewaytoproperlylookattheissueistoconsidertheentirehistoryofwars,simulatingfromtheprocess,andcheckinghowmanyperiodsdonot look likethestretchsince1945–(bygeneratingsimulatedregressioncoefficients).Alas,weareabout .37standarddeviationsawayfromabsenceoftrend. Notethatweareignoringthesurvivorbias.

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Figure9Violencedropsince1945.Divergencefromtheprocesstocallita“trend”ispatentlynotstatisticallysignificant,.37%ofastandarddeviationaway.Noscientistbuildsatheoryfrom.37standarddeviations.

Figure10Shadowmeanatdifferentthreshold.

1960 1980 2000

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Figure11Samplemean(journalistic)andmaximumlikelihoodmeanatdifferentvaluesofthetailexponent.Wenoticethatadropinthetailexponentcausesanasymmetriceffect,henceuncertaintyandlackofprecisionaboutthetailmeansworsemean(Taleb2016).

Thefactthatthe“shadowmean”ormaximumlikelihoodmeanisinexcessofthesamplemeanacrosspotentialvaluesoftheparameters(Figures10and11)isnottrivial.Itmeansthatevena“real”statisticallysignificantdropinviolencewouldnotalterbymuchthegravityofthesituation:theworldisevenmoredangerousthanitlooks.

Weconnectwiththerestofthechaptersasfollows.OurpapermayaddsomeargumentsortakesidesintothedemocraticpeacedebateentailingRussett(2017,thisvolume)andGowaandPratt(2017,thisvolume),asfollows:therehasbeen,atthemacrolevel,ariseindemocracy,butnoevidenceofdeclineofwarsandgeneralviolence.Itisalwaystemptingtoassumethattheriseininstitutionshascontributedtochangeinthestructureoftheworld–justasmanyargumentshavebeenmadethatthecreationofthefederalreserveandotherfinancialinstitutionshavecontributedtostability.Infinance,thisargumentturnedouttobewrong:extremeeventshavebeenatleastassevere(andifanythinghaverisen)inspiteofthedevelopmentofsuchinstitutions.Itmaybethesamewithviolence.

References

Sample ("journalistic") MeanMax Likelihood Mean

Range of �

0.40 0.45 0.50 0.55 0.60 0.65 0.70�

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Balkema,A.A.,deHaan,L.Residuallifetimeatgreatage.TheAnnalsofProbability5,792-804(1974)

BBC(2012).Inourtimes:TheAnLushanRebellion.http://www.bbc.co.uk/programmes/b01by8ms.

Berlinski,D.TheDevil’sDelusion:AtheismanditsScientificPretensions.BasicBooks,NewYork,(2009)

Buckle,H.T.HistoryofCivilizationinEngland,Vol.1,London:JohnW.ParkerandSon(1858)

Cederman,L.E.Modelingthesizeofwars:frombilliardballstosandpiles.TheAmericanPoliticalScienceReview97,135-150(2003)

Cirillo,P.AreyourdatareallyParetodistributed?PhysicaA:StatisticalMechanicsanditsApplications392,5947-5962(2013)

Cirillo,P.,Taleb,N.N.Onthestatisticalpropertiesandtailriskofviolentconflicts.PhysicaA:StatisticalMechanicsanditsApplications452,29-45(2016).

Clauset,A.,Woodard,R.Estimatingthehistoricalandfutureprobabilitiesoflargeterroristevents.TheAnnalsofAppliedStatistics7,1838-1865(2013)

Contestation.JournalofConflictResolution53,934-50(2009)

Cox,L.A.,Greenberg,M.R.(eds.)SpecialIssue:AdvancesinTerrorismRiskAnalysis.RiskAnalysis,onlineISSN1539-6924,(2013)

deHaan,L.,Ferreira,A.ExtremeValueTheory:AnIntroduction.Berlin,SpringerSeriesinOperationsResearchandFinancialEngineering,2006

Durand,JD,“ThepopulationstatisticsofChina,AD2–1953,”PopulationStudies(1960),Vol.13,No.3,p.209,223

Embrechts,P.,Kluppelberg,C.,MikoschT.ModellingExtremalEvents.Springer,Berlin,

Epstein,R.BookReview:TheBetterAngelsofOurNature:WhyViolenceHasDeclined.ScientificAmerican,October7,(2011)

Falk,M.,Hüsler,J.,Reiss,R.D.Lawsofsmallnumbers:extremesandrareevents,Birkhäuser,Berlin,(2004)

Fitzgerald,CharlesPatrick,China:aShortCulturalHistory, Cresset Press, ,1935

Friedman,J.A.Usingpowerlawstoestimateconflictsize.JournalofConflictResolution59,1216-1241(2015)35

Gnedenko,D.V.Surladistributionlimitedutermed’uneseriealeatoire.AnnalsofMathematics44,423-453(1943)

Goldstein,J.L.Winningthewaronwar:thedeclineofarmedconflictworldwide.Penguin,NewYork,(2011)

Gumbel,E.J.StatisticsofExtremes.CambridgeUniversityPress,Cam-bridge,(1958)

Hayes,B.StatisticsofDeadlyQuarrels.AmericanScientist90,10-14(2002)

Horne,A.ASavageWarofPeace.NewYorkReviewBooksClassics,NewYork,(2006),34

Inkester,N.(ed.)ArmedConflictSurvey2015.IISS,London,(2015)

25

Kleiber,C.,Kotz,S.StatisticalSizeDistributioninEconomicsandActuarialSciences,Wiley,NewYork,(2003)

KleinGoldewijk,K.,vanDrecht,G.HYDE3.1:Currentandhistoricalpopulationandlandcover.InBouwman,A.F,Kram,T.,KleinGoldewijk,K.(eds.)Integratedmodellingofglobalenvironmentalchange.AnoverviewofIMAGE2.4.NetherlandsEnvironmentalAssessmentAgency,TheHague(2006)

Mandelbrot,B.B.-TheFractalGeometryofNature,Freeman5(1983)

Maronna,R.,Martin,R.D.,Yohai,V.RobustStatistics-TheoryandMethods.Wiley,NewYork,(2006)

Mueller,J.E.RetreatfromDoomsday:theobsolescenceofmajorwar.BasicBooks,NewYork,(1989)

Necrometrics,(2015)Availableat:http://necrometrics.com.

Norton-Taylor,R.Globalarmedconflictsbecomingmoredeadly,majorstudyfinds.TheGuardian,Wednesday20May,(2015)Availableat:http://www.theguardian.com/world/2015/may/20/armed-conflict-deaths-increase-syria-iraq-afghanistan-yemen

Phillips,C.,Axelrod,A.EncyclopediaofWars3-VolumeSet.InfobasePub-lishing,NewYork(2004)

Pickands,J.Statisticalinferenceusingextremeorderstatistics.theAnnalsofStatistics,1,119-131(1975)

Pinker,S.TheBetterAngelsofOurNature:WhyViolenceHasDeclined.VikingPress,NewYork,(2011)

Pinker,S.,TamingtheBeastWithinUs,Nature,478,309-311(2011b)

Reiss,R.,Thomas,M.StatisticalAnalysisofExtremeValues,Birkhauser,Berlin,(2001)

Richardson,L.F.VariationoftheFrequencyofFatalQuarrelswithMagnitude.JournaloftheAmericanStatisticalAssociation43,523-46(1948)

Richardson,L.F.StatisticsofDeadlyQuarrels.QuadrangleBook,Chicago(1960)

Scharpf,A.,Schneider,G.,Noh,A.,Clauset,A.ForecastingoftheriskofextrememassacresinSyria.EuropeanReviewofInternationalStudies1,50-68(2014)

Seybolt,T.B.,Aronson,J.D.,Fischhoff,B.(eds.)CountingCivilianCasualties,AnIntroductiontoRecordingandEstimatingNonmilitaryDeathsinConflict.OxfordUniversityPress,Oxford,(2013)

Taleb,N.N.-TheBlackSwan:TheImpactoftheHighlyImprobable.RandomHouseandPenguin,(2007/2010).

Taleb,N.N.SilentRisk(2016).Availableonline:www.fooledbyrandomness.com/FatTails.html

Taleb,N.N.,andP.Cirillo."Ontheshadowmomentsofapparentlyinfinite-meanphenomena."arXivpreprintarXiv:1510.06731(2015).

UnitedNations-DepartmentofEconomicandSocialAffairs.2015RevisionofWorldPopulationProspects.UNPress,NewYork,(2015)

26

UnitedNations-DepartmentofEconomicandSocialAffairs.TheWorldatSixBillion.UNPress,NewYork,(1999)

vanderWijk,J.Inkomens-enVermogensverdeling.PublicationoftheNed-erlandsch

Viertl,R..StatisticalMethodsforNon-PreciseData.CRCPress,BocaRa-ton,(1995)

Villasenor-Alva,J.A.,Gonzalez-Estrada,E.AbootstrapgoodnessoffittestforthegeneralizedParetodistribution.ComputationalStatisticsandDataAnalysis53,3835-3841(2009)

Wallensteen,P.,Sollenberg,M.ArmedConflict1989-2000.JournalofPeaceResearch38,629-644(2001)

White,M.TheGreatBigBookofHorribleThings.W.W.NortonandCompany,NewYork(2011)

White,M.Atrocities.W.W.NortonandCompany,NewYork,(2013)

Wolfram,S.ANewKindofScience,WolframResearch(2002)