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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.