完本神战

Chapter 6 Games people play（第1页）

Chapter6Gamespeopleplay

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。

Theoneywasbilledasthemoststressfulshowosurvivedonlyafewepisodesin2009。ButitdoesgiveasplendidopportunitytoillustrateusesoftheAdditionandMultipliLawsinfindingaprobability。