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Chapter 6 Games people play（第1页）

Chapter6Gamespeopleplay

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Maionalgamesbineskillandce.Skillyouworkon，ceis，well，amatterofluck.Forallthe‘games’discussedhere，itiseasytopersuadeyourselfthatthereissomefiofoutes，allequallylikely.Thusinthischapter，ulystatedotherillusetheclassicalapproadingprobabilities:tthenumberofpossibleoutes，andtheprobabilityofaakeioeswheretheeventhappens.

Myaimistoshowhorobabilityhelpaplayermakegooddedersofuy.Aandingofprobabilityalsoaddtotheeertaiofspectators.

Lotteries

Aoisthatknownas649，asiionalLottery.Here49rubberspheres，paihdifferentnumbers，arelastictub，sixareatrandom.Gamblerspay￡1toseleumbers，andrizeiftheirseletaihreeofthosewinsinly50%ofthetakingsgointotheprizemourntoLotteryplayersisfarlessthaninos，orattheracetrack.

&raistheprospeote，ofanenormousprize–oickethaswonover￡22million，aheUSAhaveexillion.gtellsusthattheprobabilityofwinniopprizefromogallthewinningnumbers，isaboutonein14millionintheUK，lessthanonein116millionintheEuromillionsgame，andaroundonein176millionintheUSAMegaMillionsgame.

Toappreciatejusthowtinythesecesare，fixontheUKgame.Figuresshorobabilityofdeathwithinoneyearforarandom40-year-oldmanisarouhousand.Sotheceofhisdeathwithinadayisaboutonein365，000，withinanhouritisaboutonein9million，sotogetdowntoonein14milliaboutthecehedieswithi35miheMegaMilliohesameassumptions，thecehistisaJackpotshareisparabletohisceofdeathwithihreeminutes.

&helowreturnandfodds，‘utility’givesaratioitiexgefor￡1，youwillgetbaaverageanyway，aher50pbuysyhttodreamaboutyourfutureluxuriouslifestyle，yourphilanthropy，andthepossibleenvyofpeoplelikemewhoassuredyouitwasawasteofmossurelyhavesomeutility.

Weshallassumethatfuturelotterydraeofpastresults–aninanimaterubbersphereber’whetheritis‘due’tobe.Shortofg，thereisnowayofgtheiyou　fluehesizeofahoselotteriesroportionoftheprizefundissharedoutamongthewiier.Thereisanopportunitytoexercisealittleskill.

Itarisesbecauseumbers，typicallylow（birthdates）andodd，areoreoftenthanothers，ateryplayersspreadtheirchoilyovertheticket，perhapsiakedoitheyareseleg‘atrandom’.Inbinationswithseveralhighnumbers，orwitheredtogether，orontheedge，arelessoften.Ifyoutifythesortofchoicesotherplayersaremaking，ahi，yotaffected，butifyoudoihanaverage.

&ryioocrafty，likeseleg{1，2，3，4，5，6}，

orthewinhelastdraw，onthegroundsthat‘Nobodyelsewillthinkofthat’.Theywill.Wheerybegan，about10，000peopleweregthefirstsixnumbers.IhewinningitwosuccessivedrawsintheBulgarianLottery:hemthefirsttime，but18didsothesee.

Providedthatotherplayersuetomarktheirticketsmuchastheyhaveihefollowingprocedure，forUK-type649lotteries，willhelpyou.Takeanordinarydeckof52playingddiscardthreeofthem.Identifytheremainingcardswiththeo49，shufflewell，asixcards.Thisisawayofgsixnumberspletelyatrandom.Humanbeingsakesuchaseleaided，theyhissortofauxiliaryhelp.

Ahesesixnumbers，　providedthat

（a）theytotalatleast177（togiveabiastohighnumbers），and

（b）wheheticket，theyfallintotwo，three，four，orfiveclusters，and

（c）three，four，orfiveofthemareosideborderofthetid

（d）theydonotformanyobviouspatter.

Ifaionsfail，returnthesixcardstothepack，shuffieitthhly，ahissequence.

Ifyoufollowthisrecipe，youshouldstillexpeoheoverallpaybaly50%ofmoneyreceivedishardtoovere.ButyouarelesslikelytohavetoshareanyJackpotwiththeworldandhiswife.

TVgames

GoldenBallswasfirstairedin2007.ThelasttwoplayersarefacedwitheleveheGoldenBalls），someofwhioheKillers）areworthzero.Theplayersselectfiveofthesespherestogeialprizefund;anyKillerreducesthevalueofanypreviouslyselectedBalltoostvalue.ThustwoKillersaftera￡50，000Ballmakeitworthjust￡500.

Allthespheresareoutwardlyidentical，sotheplayersarepletelyatrandom.Thereare462waystochoosefiveobjeeleven，sotheceofpigthefivemostvaluableBallsis1462.I288shows，thisoccurredjustonce.

TakeaBallnominallyworth￡1，000:evenigntheKillers，theceofselegitisonly511，soitsrealmeanvalueis￡455.AnyKillerswillreducethissumevehthreeKillers，itsmeanvaluebecalculatedas￡255.

&hefiveBallshavebeeheactualprizefundisknolayermakesaprivatede，whetherto　Splitthefundwiththeotherplayer，orseekto　Stealallofit.Theyrevealtheirultaneously:ifbothSplit，theysharethefund，ifjustoeals，thatplayergetstheentirefund，ifbothSteal，hing.

Thissarioiswellknowheory，uhePrisoners’Dilemma’.SupposeyouroppochoosesSplit:theeroffifyouSteal.AchoosesSteal，youwohinganyway.Sowhateverchoiakes，yuethatgStealwillneverlose.Frequently，bothselectSteal，andtheoheTVprodupanywhopayoutzero.

VersionsofDealorNoDealhavebeenshowytries.Ihereare22sealedboxesgdifferentamfrom1pto￡250，000.Theboxesareallodomlyto22players，oneofwhom，Amy，willplaythatday.Herownboxremaiheeselectsfiveotherboxes，whosetsarerevealed.Abahenoffersasumofmoneyiheamouoacceptthis，shesays‘Deal’，endingthegame，whilethewords‘NoDeal’rejectthisoffer.Ifthegameues，moreboxesareopened，anewofferismade，andsoon.

Atthetimeofaheexatsstillinplayareknown，sotheirmeaniseasilycalculated.Intheearlyrounds，thebanker’sofferisnormallyfarlessthanthisamount，butAmymusthaveherutilityfunlyinsight:ifshestronglydesires￡5，000，andtheofferis￡5，400，shecouldratio，evenifthemeanamountiisover￡20，000–shemightendupwith1pifshehangson.

&imein22，Amywillowhthetopprize，butshewillwinthatamouen.Utilitiesgiveagexplanation.Atthefinaldepoint，twoboxesremaih￡250，000aherwithmaybe￡2.Ifthebankeroffers￡80，000，eventhoughthisiswellbelowthemeanvalueof￡125，001，oorrichestAmywillrejectit.Abirdinthehand....

ProvidedthebankeralwaysofferslessthaintheremaiheLawehat，inthelongruswhoDealtakehomelessthaintheirbox.Soarealbank，thatdidpayouttheofferbutalsoreceivedtheamountintheboxwheantDeals，wouldmakealong-runprofit.