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AVariableThreatGameofRansom
AvinashDixit,PrincetonUniversity
IhadtheprivilegeandpleasureofJohnandAliciaNash’sfriendshipfor
almostaquartercentury.WhenIthinkofJohn,thewordsthatspringtomymind
aremodesty,courtesy,rationality,andprecision;whenIthinkofAlicia,thewords
arequietcourageandlove.
Forsomeonewhomadesuchpathbreakinganddeepcontributionsto
mathematicsandgametheory,Johnwasincrediblymodest.Atconferencesor
seminarshenevertriedtoattractattentiontohimself,andwassurprised(but
clearlyhappy)whenattentionfoundhim,whichitalwaysdid.Hewasunfailingly
courteousandattentivetoall–studentsandseniorprofessorsalike–andhadan
amazinglygoodmemoryfornamesandfacesofpeoplehehadmetsomeyearsago.
Inmattersofrationalityandprecision,Johnperhapsovercompensatedforwhathe
calledhis“yearsofirrationalthinking.”Indeed,hemayhavecarriedthislogictoofar
inhisadvocacyof“idealmoney,”amonetarystandardaspreciseasthestandard
meterandkilogrampreservedinParis.Theveryword“standard”connotes
precision,anditsimplicationshadtobefollowedthroughlogicallytotheend.Most
ofuswouldnotwantsucharigidmonetarystandardandwouldplacemorevalue
onflexibilitytorespondtoshockstotheeconomy,butJohn’slogicgivesusaclear
andrigorousstatementofthecaseforarigidcommitment.
OfcourseIknewofJohn’sfoundationalresearchingametheoryforalmost
halfacentury,andusedtheconceptsofNashequilibriumandtheNashbargaining
solutionmoretimesthanIcanrecall.InanearliertributeIsaidthatifJohngota
dollareverytimesomeonewroteorsaid“Nashequilibrium,”hewouldbearich
man.1ThesamegoesfortheNashbargainingsolution,whichisextensivelyusedin
1“JohnNash–FounderofModernGameTheory,”inGameTheory:AFestschriftinHonorofJohnNash,eds.ConstantinaKottaridiandGregoriosSiouroun,Athens:EurasiaPublications,2002,pp.98-100.ThebookisinGreek;theEnglishversionisonmywebsite,http://www.princeton.edu/~dixitak/home/nashenco.pdf
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laboreconomics,internationaltrade,andmanyotherareasofeconomicsand
politicalscience.
NowordscansufficetodescribeAlicia’scourageanddedicationtoJohnand
theirsonJohnny.IamsureJohn’srecoveryismostlyduetohercareandlove.Sheis
thetrueheroineoftheincrediblestoryoftheirlives.
AbstractionversusIllustration
Ithasbeensaidmanytimesthattheessenceofmathematicsisabstraction,2
andmanymathematiciansrevelinthemostabstractpossibledefinitionsand
propositionsonanysubject.ButJohnrosetoahigherlevelinrecognizingthatthe
bestentréeintotheworldofmathematicalconceptsisthroughspecific,simple,and
memorableexamples.Hisclassicpaperonbargainingillustratedthegeneral
solutionconceptwithanexchangeofgoodslikeaball,abat,apen,andahat.3Ina
veryhelpfulandperceptivequoteheprovidedformytextbook,hestatedthis
pedagogicalphilosophysuccinctly:“Thegenerousvarietyofillustrativecaseshas
theeffectthatwhatislearnedcanbemoreeasilyretainedthaniftherewereonly
theassertionsoftheoreticalconceptswithoutenlighteningexamples.”4
ThereforeIhavechosentowriteaboutanexamplethatvividlyillustrates
John’sextensionofhisclassicpaper,namelyvariablethreatbargaining.5Nash’s
originalbargainingsolutionwasformalizedasatwo-playercooperativegame𝐺,
wheretheplayerscommunicatetoagreeupontheirstrategies,andthesechoices
2Forexample,seeMathematics:AVeryShortIntroductionbyTimothyGowers,OxfordUniversityPress,2002.3JohnF.Nash,Jr.,“Thebargainingproblem,”Econometrica18:155-162,1950.4SeethebackcoverofAvinashDixit,SusanSkeathandDavidReiley,GamesofStrategy,NewYork:W.W.Norton,fourthedition2015.5JohnF.Nash,Jr.,“Two-personcooperativegames,”Econometrica21:128-140,1953.MyformalstatementfollowsR.DuncanLuceandHowardRaiffa,GamesandDecisions,NewYork:Wiley,1957,pp.140-141.
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areexternallyenforceable.Thereisaspecifiedpayoffvector(𝑣!, 𝑣!)whichwill
resultiftheplayersfailtoreachanagreement.Thisisoftencalledthethreatpoint,
orBATNA(BestAlternativeToNegotiatedAgreement)inthejargonoftheHarvard
BusinessSchool’snegotiationproject.6Thesetofallfeasiblepayoffsconstitutesa
compactset𝑃 ⊂ ℝ!;thesubsetconsistingofundominatedpayoffsisthebargaining
frontier𝐵.(Formally, 𝑏!, 𝑏! 𝜖 𝐵ifandonlyifthereisno 𝑝!,𝑝! 𝜖 𝑃suchthat
𝑝! ≥ 𝑏!and𝑝! ≥ 𝑏!withatleastoneoftheinequalitiesstrict;informally,𝐵isthe
north-eastfrontierof𝑃.)TheNashsolutionisthe(𝑥!, 𝑥!)in𝑃thatmaximizesthe
product 𝑥! − 𝑣! 𝑥! − 𝑣! .Ageneralizedversionhasthesolutionmaximizing
𝑥! − 𝑣! !(𝑥! − 𝑣!) !!! ,where0 < 𝜃 < 1, and𝜃, 1− 𝜃canrepresenttherelative
bargainingstrengthsofthetwoparties,ortheirrelativemeritsintheeyesofan
arbitrator.7(TheoriginalNashsolutionisequivalenttosetting𝜃 = 1/2.)Istrivialto
provethat(𝑥!, 𝑥!) 𝜖 𝐵.
Invariablethreatbargaining,anon-cooperativegame𝐺∗precedes𝐺.Its
Nashequilibriumpayoffs(𝑣!, 𝑣!)constitutetheBATNAof𝐺.Whenchoosinghis/her
strategyin𝐺∗,eachplayerwillseektoachievetheoutcomethatwillyieldthebest
payoffforhim/herintheensuingNashcooperativesolutionof𝐺.8
Whatisallthistellingusaboutthreatsinbargaining?Mostpeople,even
mathematicians,onafirstreadingwillbesomewhatbaffledbytheabstract
formulation.Avividandmemorableexamplewillclarifyit.
6RogerFisherandWilliamUry,GettingtoYes,Boston:HoughtonMifflin,1981.7SeeRogerMyerson,GameTheory,Cambridge,MA:HarvardUniversityPress,1991,pp.379,390.8Thisprocessoflookingaheadtotheoutcomeofalatergametochoosestrategiesinapriorgameisanearlyinstanceoftheconceptofsubgameperfectness,latermaderigorousandfamousbyReinhardSeltenin“Reexaminationoftheperfectnessconceptinextensivegames,”InternationalJournalofGameTheory4(1):25-55,1975.
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Ransom
InthemovieRansom,thesonofmultimillionaireTomMullen(playedbyMel
Gibson)hasbeenkidnapped.Themanholdinghimisdemandingaransomoftwo
milliondollars.MullengoesonliveTVwiththemoneyspreadoutonatablebefore
him,andmakesthefollowingannouncement:“Thewholeworldnowknows...my
son,SeanMullen,waskidnapped,forransom,threedaysago.Thisisarecent
photographofhim.Sean,ifyou'rewatching,weloveyou.Andthis...well,thisis
whatwaitsforthemanthattookhim.Thisisyourransom.Twomilliondollarsin
unmarkedbills,justlikeyouwanted.Butthisisascloseasyou'llevergettoit.You'll
neverseeonedollarofthismoney,becausenoransomwilleverbepaidformyson.
Notonedime,notonepenny.Instead,I'mofferingthismoneyasarewardonyour
head.Deadoralive,itdoesn'tmatter.Socongratulations,you'vejustbecomeatwo
milliondollarlotteryticket...excepttheoddsaremuch,muchbetter.Doyouknow
anyonethatwouldn'tturnyouinfortwomilliondollars?Idon'tthinkyoudo.I
doubtit.Sowhereveryougoandwhateveryoudo,thismoneywillbetrackingyou
downforalltime.Andtoensurethatitdoes,tokeepinterestalive,I'mrunninga
full-pageadineverymajornewspapereverySunday...foraslongasittakes.But...
andthisisyourlastchance...youreturnmyson,alive,uninjured,I'llwithdrawthe
bounty.Withanyluckyoucansimplydisappear.Understand...youwillneversee
thismoney.Notonedollar.Soyoustillhaveachancetodotherightthing.Ifyou
don't,well,then,Godbewithyou,becausenobodyelseonthisEarthwillbe.”9
Letusrepresentthisingame-theoreticlanguage.CallTomMullenplayer1;
thekidnapperJimmyShaker(playerbyGarySinise)isplayer2.Figure1showstheir
payoffs.Theoriginisatthepointoftheirinitialwealths.Beforethekidnapping,
Mullenalsohashisson;denotehisvalueorutilityfromthatinmoney-equivalent
termsby𝑎.Sothepayoffpointinthestatusquoexanteis 𝑎, 0 ,orthepointQin
9Thetextcomesfromhttp://www.imdb.com/title/tt0117438/?ref_=nv_sr_1,accessedFebruary16,2016.
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thefigure.ThelinethroughQwithslope-1showsallattainablepayoffpoints
startingatQandtransferringmoneybetweentheparties,andisthereforethe
bargainingfrontier.
Afterthekidnapping,ifnegotiationfails,MullenwilllosehissonbutShaker
won’tgetanymoney,soShaker’sthreatpointTistheorigin.Heasksfor$2million.
IfthisgoesthroughandMullengetshissonback,thepayoffswillbe(𝑎 − 2,2),
shownasthepointP.
Mullen’sstrategychangesthethreatpoint.Nowifthenegotiationfails,
Mullenwilllosehissonandenduppayingthe$2millionbountytothepersonwho
killsShaker(probablyoneofShaker’sconfederates),whileShakerwilllosehislife.
LetShaker’svaluationofhisownlifebedenotedby𝑏.Thenthepayoffsatthenew
BATNAare −2,−𝑏 . ThefigureshowsthisasthepointT*.
Figure1:VariouspayoffpointsintheRansomgame
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Withthisnewthreatpoint,MullenoffersShakerhissolutiontothe
bargainingproblem,namelygoingbacktothestatusquopointQ:bringmysonback
unharmedandIwillwithdrawthebountyonyourhead.10
ThefigureisdrawnasifthesesolutionsconformtotheNashbargaining
solution.Itiseasytoverifythatthisimplicitlysets𝑎 = 4and𝑏 = 6.Forothervalues
of𝑎and𝑏,theNashsolutionforMullen’scounterproposalneednotbeexactlyatQ.
AndthegameGmaybeplayedinsomewayotherthanNashbargaining,for
examplethethreatenermaybeabletomakeatake-it-or-leave-itoffertotheother
player.Butthefollowingreasoningyieldssomegeneralconclusionsapplicabletoall
suchvariants.
WhenwillMullen’sstrategygivehimanoutcomebetterthantheonehe
wouldgetbyaccedingtoShaker’soriginaldemand?IftheNashsolutionforthe
threatpointT*istothesouth-eastofthatforTalongthebargainingfrontier.This
happensifthelineT*QliesbelowthelineTP,thatis,if𝑏 > 2,thatis,ifShakervalues
hisownlifemorethantheransommoney.SinceT*istothesouth-westofT,in
changingthethreatpointfromTtoT*MullenworsensbothBATNAs.Hisstrategy
aimstoachieveShaker’sacceptanceofthealternativeproposal,becauseitcarries
thethreatofanevenbiggerlossforShakerthanforhimselfifthenegotiationfails.
Inotherwords,MullenisimplicitlysayingtoShaker:“Thiswillhurtyoumorethan
itwillhurtme.”Weoftenhearsuchstatementsmadeinargumentsanddisputes;
nowweseethestrategicroletheyplayinnegotiations.
If𝑏 > 6,thelinefromT*willmeetthebargainingfrontieratapointsouth-
eastof(𝑎, 0).ItwillthereforecorrespondtoanegativepayoffforShaker,thatis,he
willendupwithwealthbelowhisoriginallevel.ItisasifheispayingMullentotake
hissonback!Thismaybeimpossible,andifMullen’sthreatT*isthatsevere,the
10Movieshavetheirownrequirementsofdramatictensionanddenouementthatoverridegame-theoreticlogic.Toconformwiththosedemands,Ransomdoesnothaveanyefficientresolutiononthebargainingfrontierwhereoneoftheplayersaccedestotheother’sdemand,buttwiststhatendinchasesandgunfights.ButthatisnotmaterialtothebasicbargaininggameIwanttoillustrate.
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outcomemaybeacornersolutionatQ.However,itisnotoutsidetherealmof
possibilitiesthatkidnappersagreetopaytoberidoftheirhostage,atleastinsome
otherbranchesoffiction.11
Thevariablethreatstrategyof“Thiswillhurtyoumorethanitwillhurtme”
hasbeenusedinreallife.Forexample,smartlaborunionsthreatenorlaunch
strikesattimeswhenthatwilldeliverthebiggesthittothefirms’profits.British
coalminers’uniondidthisconsistentlyinthe1970s.Conversely,Mrs.Thatcher’s
strategyofprovokingtheunion’sleaderArthurScargillintostrikinginthespring
andsummerof1984wasinstrumentalinthecollapseofthestrike,andledtoa
collapseoftheunionitself.
Thesamestrategywasusedinthebaseballstrikeof1980.12Thestrike
startedduringtheexhibitiongamesofpreseason.Theplayersreturnedtowork
(actually,toplay)atthestartoftheregularseason,butresumedthestrikeafter
MemorialDay.Thiscuriousdiscontinuousstrikecanbeunderstoodwhenwe
examinethetime-varyingcostsofthestriketothetwosides.Duringtheexhibition
gamesperiod,theplayersarenotpaidsalariesbuttheownersearnsubstantial
revenuesfromfanswhocombineavacationinawarmerclimewithfollowingtheir
favoriteteam’sstarsandprospects.Oncetheregularseasonstarts,theplayersget
salaries,butattendancesatgames,andthereforetheowners’revenues,grow
substantiallyonlyafterMemorialDay.Thereforethediscontinuousstrikewasthe
players’correctstrategytomaximizetheowners’lossrelativetotheirown.
11InO.Henry’sshortstory“TheRansomofRedChief,”twosmall-timecrookskidnapabanker’s10-year-oldson.Heturnsouttobeabratwhomakestheirlivessoimpossiblethattheypaythefathertotakehimback.Thetextisavailablefromhttp://fiction.eserver.org/short/ransom_of_red_chief.html,accessedFebruary17,2016.12LawrenceM.DeBrockandAlvinE.Roth,“StrikeTwo:Labor-managementnegotiationsinmajorleaguebaseball,”BellJournalofEconomics12(2):413-425,1981.
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ConcludingComments
Youmayhavealreadyforgottentheformaldefinitionsofvariablethreat
bargaining,butIguaranteethatyouwillnotforgettheexamples.Andwiththe
examplesinmind,anymathematicianwilleasilybeabletoreconstructthe
formalism.ThereforeIhopeIhaveconvincedreadersofthemeritsofthe
pedagogicalphilosophyIamhappytohavesharedwithJohnNash:vividexamples
canconveyconceptsandevenformalmethodsofmathematicaltheoriesbetterand
morememorablythanpurelyalgebraicorsymbolicstatements.
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