Risk and Reward: The Science of Casino Blackjack

162 Pages·2018·1.913 MB·English

by  N. Richard Werthamer

N. Richard Werthamer
Risk and
Reward
The Science of Casino Blackjack
Second Edition
Risk and Reward
N. Richard Werthamer
Risk and Reward
The Science of Casino Blackjack
Second Edition
123
N.RichardWerthamer
ChelseaTechnologies
SagHarbor,NY,USA
ISBN978-3-319-91384-1 ISBN978-3-319-91385-8 (eBook)
https://doi.org/10.1007/978-3-319-91385-8
LibraryofCongressControlNumber:2018942354
1stedition:©SpringerScience+BusinessMedia,LLC2009
2ndedition:©SpringerInternationalPublishingAG,partofSpringerNature2018
Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof
thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,
broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation
storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology
nowknownorhereafterdeveloped.
Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication
doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant
protectivelawsandregulationsandthereforefreeforgeneraluse.
Thepublisher,theauthorsandtheeditorsaresafetoassumethattheadviceandinformationinthisbook
arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor
theeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinorforany
errorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregardtojurisdictional
claimsinpublishedmapsandinstitutionalaffiliations.
Printedonacid-freepaper
ThisSpringerimprintispublishedbytheregisteredcompanySpringerInternationalPublishingAGpart
ofSpringerNature.
Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland
Preface
How did a physicist like me get involved with blackjack? I skied in. As a young
doctoral candidate at the University of California, Berkeley, I would occasionally
join my friends on ski trips to the Lake Tahoe area. Always on a student’s
tight budget, we found that we could stay inexpensively in Reno motels and that
the cheapest meals were at the coffee shops astutely placed at the back of the
town’s casinos. I found it hard to pass all the gambling frenzy without joining in.
Fortunatelyforthatbudgetandmyfinancialpeaceofmind,Iwasusuallyluckyin
winningafewdollars. Once,Ipaidformyentiretripwithanunlikelyhitatroulette.
When I learned that an optimized method for playing blackjack (what is now
calledBasicStrategy)hadbeenpublishedinascholarlystatisticsjournal,Iquickly
looked it up in the university library. Blackjack became my game of choice for
subsequent Tahoe excursions and, in later years after skiing and I had parted
company, for trips to other destinations with casinos. But my real recreation of
choicebecameblackjackanalysis!
My original motive for exploring its mathematics was to see if the articles and
books I had chanced upon were correct, or if I could find a better way to win.
Visionsofbigmoneydancedinmyhead!Later,asIcametoappreciatetherealities
involved,myinterestshiftedtothemathematicsforitsownsake,complexenough
tobeachallengeevenfortheprofessionalscientistIhadbecomeyeteasyenough
to yield to a sustained effort on almost every issue. Also, I wanted to verify (or,
as sometimes happened, falsify) claims about the best way to play, asserted in the
literaturewithlittleornoproof.
When I had finally addressed all the questions that occurred to me, my first
instinct as a scientist was to write up the results and publish them in a suitable
scholarly journal: an archival record for other practitioners of applied math. Yet I
came to realize that the results were sufficiently insightful about actually playing
blackjack in the real world that they would be even more valuable to the wider
casino-goingpublic.Toreconcilemyinternaldebatebetweenthesetworeaderships,
Idecidedtoservebothatonce.Theoutcomeisthissingletwo-partbook,whichfirst
describesallitsconclusionsincompletelynonmathematicaltermsforthemanywho
v
vi Preface
enjoygamblingandthenalsoprovidesdetailedderivationsofthoseconclusionsso
thattheycanbefollowedandcheckedbythemoremathematicallyinclined.Some
ofthederivationsrelyonlyonalgebra;otherstakevaryinglevelsofcalculusand/or
probabilitytheory.
Whentalkingtofriendsaboutmyblackjackproject,theyinvariablyaskwhether
Ihave“testedmytheoriesinacasino.”Irespondthattheconclusionsdrawnfrom
a mathematical proof are intrinsically true and don’t need field testing to be valid.
I suggest that they should rather be asking whether I have “applied” the theories,
and I add that analyzing blackjack is at least as entertaining for me as playing
it. I also point out that the cash outcome of playing several hundred hands at an
actual blackjack table, which would occupy a full evening, is little more than an
unpredictablestatisticalfluctuationratherthanatestofthemethods.Manymillions
ofhands,simulatedonacomputer,areneededtoapproachsignificantconclusions.
I’m happy to have examined the game in comprehensive detail, without having
riskedasingledime—eventhoughIhaven’twonone,either!
Several leading authorities on blackjack have been generous in sharing their
time and knowledge with me. Early on, Stanford Wong and Anthony Curtis
were encouraging and introduced me to several other prominent figures. One of
them, Don Schlesinger, intensively reviewed more than one preliminary draft of
my manuscript, thereby prompting major revisions and expansions. Subsequently,
Stewart Ethier graciously did the same, with similar results. It’s a pleasure to
acknowledge their help and support. Additionally, I’ve benefited from commu-
nications with Sergei Maslov, Kim Lee, Steve Jacobs, Michael Canjar, Norm
Wattenberger,andNathanielTilton.
SagHarbor,NY,USA N.RichardWerthamer
Introduction
Considering the large number of blackjack books published since that original
statistics paper more than 60 years ago, it may well be wondered what I could
possibly add. After all, the Basic Strategy recipe for playing one’s hand has been
knownandunchallengedfordecades.Cardcountingmethodsforsizingone’sbet,
as a complement to Basic Strategy, have also been fully developed. But the many
previous publications have shortcomings and missing pieces, which I fill in and
clarify.Theresultisasystematicdescriptionofeachmajoraspectofoptimalplay:
howbesttoplaythecardsinyourhand,howbesttoassesstheoddsexpectedforthe
nextround,andhowbesttousetheoddstoadjusttheamountyoubetonthatround.
Furthermore, I emphasize the trade-off between simplicity and performance:
I discuss how easy each aspect of optimal play is to actually use in a casino,
and I recommend simplified versions, where desirable, which give nearly optimal
performance.Anyseriousplayershouldbecarryingoutanumberofdistinctmental
processessimultaneously,rapidly,andaccurately;hemaywellwishtodroptheleast
valuableones,toensurehisfidelitytothosemoreworthwhile—aswellastolower
hisstresslevel.
I contribute much new material as well. The most significant is to that third
majoraspect—howyou,asaplayer,shouldmanageyourcash.(I’llusuallyreferto
youinthethirdperson,asPlayer,withmasculinegenderassumed,andthedealer,
correspondingly,asDealer,femininegender.)Ifyouareadvisedtoincreaseyourbet
when you’ve detected favorable odds, then it’slikely you’ll immediately ask,“By
howmuch”?Closelyrelatedishowyoucancontroltheriskofalosingstreakthat
wipesoutthetotalcapitalyoubroughttothecasino.Thisriskdependscriticallyon
the scale of your betting relative to your capital; most players probably lose more
moneyfrominadequatecapitalizationthanfromnon-optimumplay.
Almost all previous authors on blackjack either avoid the capital management
issue altogether or give vague, hand-waving impressions. Almost no one supplies
an answer that seems convincing, let alone thoroughly justified, and the better
workislargelyinrestrictedWebchatrooms.Incontrast,Idemonstratetheoptimal
relationshipbetweentheamountbetonahandanditsindicatedodds,arelationship
linkedtoPlayer’srisktolerance.
vii
viii Introduction
Also original is the important concept of Counter Basic Strategy, with the
procedure described in Sect.5.2 and the derivation sketched in Sect.10.2. And
I fully analyze several underappreciated refinements to betting strategy. Two of
these, referred to in the literature as back-counting and Kelly betting, are usually
mentioned only in passing. A third is betting on multiple simultaneous hands. I
showthatthesecaneachsignificantlyenhanceperformance.Theiruse,infact,isa
keytothesuccessofblackjackteams.
My original contributions as described here were first published in a series of
journal articles over the period 2005–2008, cited in the References. The articles
maybeviewedinfull,alongwithrelatedmaterial,onmywebsite,www.blackjack-
science.net.
Part I of Risk and Reward gives a straightforward, self-contained guide to
blackjack for a general reader, even a beginning or occasional player. Its narrative
is entirely nonmathematical: it recommends how best to play the game and what
resultstoexpect.
Chapter1reviewstherulesofblackjack,alongwithdefinitionsofthekeyterms
specifictothegame.Chapter2thencoversthesimplestandmost-usedtechniquefor
playingahand,widelyknownasBasicStrategy,inwhichPlayerbaseshisactions
onjustthecardsinhisownhandandtheonecardvisibleinDealer’shand.Limited
toonlythisinformation,Playershouldalwaysbetthesameamountoneachround.
Chapter3goesontodescribethemoresophisticatedgroupofstrategiesknownas
cardcounting.Inthese,Playertracksthecardsdealtonthepreviousroundsinorder
to estimate the odds on the next round and decide how much to bet. A number of
variantcountingschemeshavebeenproposedandadvocatedovertheyears;Chap.3
comparesandevaluatesseveralofthemoreaccurateandcurrentlypopular.
Chapter 4 is in many ways the most significant (and original) in the book.
Section 4.1 relates the scale of Player’s bet size, the rate at which he raises and
lowers his bet dependent on the count, and his tolerance for the risk of losing his
entire capital. The way in which these factors and others influence whether or not
Player has an edge over the casino is carefully explained. Section 4.2 examines a
modifiedbettingschemethatacceleratestheexpectedrateofwinning.Italsopoints
out the advantages of playing multiple hands simultaneously. Section 4.3 shows
howPlayerfurtherbenefitsfromproperlychoosingthemomentstobeginbettingat
atableandtoleaveit.
Chapter 5 revisits play of the hand, to examine ways in which it interacts with
card counting. Section 5.1 considers modifications to the counting technique that
bestsupportaplaystrategygeneralizedtobecomecountdependent.Section5.2,in
contrast,identifiesthebestcount-independentplaystrategy,termedCounterBasic
Strategy,foraPlayerwhocountscardsandsystematicallyvarieshisbets.
Chapter6collectsthekeypointsofoptimalstrategyfromtheearlierchapters.It
summarizesthetechniquesofplaystrategy,ofcardcounting,andofbetsizing,and
it reviews the complex of factors, including casino countermeasures, determining
the amount of money Player might win or lose. It presents the case that casino
blackjack for a single Player is better thought of as an entertainment, which can
be enjoyed with only moderate skill at relatively tolerable expense, rather than as
Introduction ix
aconsistentmoneymaker.Ablackjackteam,ontheotherhand,canusetechniques
not available to an individual that make it a true business, with attractive rates of
return.Chapter6shouldbescannedbyeveryreader,whetherornothehasdipped
intotheotherchapters.
Part II then details the analysis justifying those recommendations, intended for
thesubsetofreaderswhowishtounderstandthegamemoredeeplyand/orfollow
its mathematics. Part II gives detailed, self-contained derivations of the assertions
andqualitativedescriptionsofPartI,withthetopicsofChaps.7–10corresponding
respectivelytothoseofChaps.2–5.
The tools used in Chap.7 are almost entirely algebra and basic probability, so
more advanced mathematics should not be needed to follow it. Chapters 8–10,
however, bring in the machinery of vector calculus; college training equivalent to
a science or engineering degree should be sufficient to understand some of the
derivations, while the more sophisticated procedures of the others are detailed in
step-by-stepappendices.Forthosereaderswhodesireaddedsupport,Irecommend
Morse and Feshbach (1953), my own guidebook to applied mathematics and even
todayaleadingauthority.Amongrecentbutmorenarrowlyfocusedtexts,formulti-
variable calculus I suggest Marsden and Anthony (2007), and for matrix algebra I
suggestShores(2007).
Some further comments are in order on the style of the mathematics. First of
all, I have avoided the use of theorems, formally stated and rigorously proved as
intypicalmathematicspublications.Rather,Itaketheapproachofmostphysicists,
which is to narrate a derivation assuming, without explicit formal proof, that each
stepisvalidunlessthereissomeevidentreasontoquestionit.
Furthermore,atmanypointsinChaps.8–10whereaquantityis,strictlyspeaking,
arationalnumber(ratioofintegers),Iapproachitsanalysisbyapproximatingitby
a decimal, justified because the number of cards shuffled into the pack is large.
(Illustration: the fraction 17/52 is a rational, while the decimal 0.327 is a close
approximation to it.) The move from the exact discrete math of earlier blackjack
analysts, primarily based in probability and statistics, to closely approximated
continuum math opens the door to the powerful tools of calculus, and facilitates
new results difficult to obtain in the traditional way. As an example, this second
edition has added an important analysis of the probability distribution of Player’s
capitalevolvingwiththehandsheplays,illustratedwithaninformativenewfigure;
theseresultsarevirtuallyunobtainablewithoutthecalculus.
Supplementing the text is an index of terms, each citing its first use, and an
extensivelistofreferences.
AccessingInteractivity
Aspecialfeatureofthissecondeditionisthatanumberofitsfiguresandtableshave
beenmadeinteractiveusingMathematicaandtheWolframLanguage,aremarkably
powerfulsoftwaresystem,andaremountedontheWolframCloud.Thecaptionof
x Introduction
each such figure or table gives the URL, www.wolfr.am/BlackjackScience, of the
mainpageintheWolframCloudfromwhichitsinteractiveversioncanbeviewed.
Instructions for use are given in the note below each figure. Please allow time for
eachfiguretorecompute(thisprocessoccursintheCloudinrealtime;theimages
arenotpre-calculated).Somefiguresoccupya3Dspaceandmaybeclickedonand
thenrotated.
I am grateful to Wolfram Research, creators of Mathematica, for agreeing to
mount my interactive material on the Wolfram Cloud. I am also indebted to the
manytalentedstaffmembersofWolframResearchfortheiractivehelpandguidance
with Mathematica, all along the way. Among them, I particularly want to single
outAndreKuzniarekforhislong-termcontinuedsupportandJeremySykesforhis
activefacilitationwiththeWolframCloud.
I’m similarly grateful for the dedicated efforts of editorial and production
personnel at Springer. Special thanks to Paul Drougas for his encouragement and
directeditorialinvolvementand,withJenniferEvans,forpersistenteffortstomeld
interactivityfromWolframwithSpringer’sownpublishingendeavors.