Basic Strategy or Car-Counters: An Analytic Approach by N. Richar Werthamer Abstract We evelop an eicient metho, similar to that o Marcus (7) who calls it Counter Basic Strategy, or casino blackack car counters to play the cars o each han. It is as easy to use as Basic Strategy, unlike elaborate schemes involving strategy inices; an captures much o the yiel improvement they oer. This metho is taken to be that set o count- an composition-inepenent play parameters (like Basic Strategy in this regar) that maximizes the yiel, the expecte cash low average over all rouns between shules. Basic Strategy, in contrast, maximizes the expecte return o ust the irst roun ater a shule. 1. Introuction Most players who use a car counting technique o so in orer to gauge the appropriate amount to bet on the next han. But it is well known that the ieal way to play that han epens on the un-ealt cars, or which counting also gives a reasonable inicator. Yet this ieal way is quite complex, involving many changes o the ecision parameters as the count varies through its wie range o statistically likely values. The parameters involve - altogether more than two ozen in number - are: when to stan vs. raw (or har an sot hans separately); which hans (har an sot) to ouble own; which pairs to split; an which hans to insure or surrener. Because the ieal is generally regare as iicult to carry out in a casino approaching, some might say, the super-human various simpliications have been propose; one or another is use by almost all car counters. These simpliications can be classiie by how much inormation the player uses to make his playing ecisions. The ieal is one in which he employs the iniviual ientities o every car he is ealt, along with that o the ealer s upcar. With only slightly reuce perormance, the player uses the ientities o ust the irst two cars in his han, plus the upcar; the ecision rules or subsequent cars are ixe. Such a class is usually reerre to as composition-epenent. Easier to manage in practice is the class ( total-epenent ) using ust the total value o the two initial cars, not their ientities (although still recognizing opportunities to ouble sot hans an split pairs), again together with the upcar. Complicating the ieal strategy even urther is the issue o what count(s) to use in guiing play. Usually the car counter selects his counting metho to approximate as closely as practical the optimum or betting, an tracks that single count. I he also wants a counting inicator or playing his han, he has several options. Most ieally, he woul simultaneously maintain nine separate an istinct counting registers (in eect, knowing the exact composition o the remaining pack); but this is clearly beyon human capability. Less ieal is to keep a secon count, istinct rom the bet count, on which to base play ecisions. Authorities ier on the single play count that best balances
accuracy with simplicity, but generally suggest that it s airly similar to the bet count. Even more o a compromise is to use ust one count or both betting an playing ecisions, with the bet count itsel being reasonably close to the best possible choice. It can be estimate that the single-count compromise captures roughly hal the maximum (i.e., with complete inormation via nine istinct counts) return increase rom play variation; but even the maximum is much smaller than the improvement available rom optimal betting. Our analysis here conorms to this compromise. In the absence o a car count, the best total-epenent playing strategy or a given number o ecks best, or the moment, meaning greatest expecte return or the irst roun ater a shule, with zero true count an pack epth we ll call Optimal Basic, or OBS, in an attempt to istinguish it rom among the various extant uses o the simpler term Basic Strategy. It s nearly ientical to the best composition-epenent strategy or games with our or more ecks. In act, OBS or our ecks is also best or any o the larger numbers (six an eight) oun in casinos. This particular OBS has come to be calle Generic (or sometimes Generic Basic ); it is requently suggeste as an approximation to OBS or games with ewer than our ecks (one or two eck games can still be oun in some casinos). The expecte returns known or these classes o play strategy are assemble in Table 1. Table 1. Expecte return, irst han ollowing a shule, rom Composition-Depenent, Optimal Basic an Generic Basic play, or various numbers o ecks (with rules incluing ealer stans on sot 17, no oubling or re-splitting o split pairs, ouble any two-car han, no surrener) Composition- Total-Depenent Decks Depenent Optimal Generic 1 +.3 -.147 -.418 -.3489 -.361 -.3657 4 -.5 -.574 -.574 6 -.5791 -.5819 -.5819 8 -.674 -.691 -.691 But blackack analysis has been absorbe or ecaes by the urther question: How shoul the play parameters (whether composition-epenent or total-epenent) be varie to epen on the current true count, an how vali are the simpliications o either truncating the array o variations or ignoring variation altogether? An example o a truncate scheme, the Illustrious 18, is etaile in Schlesinger (5); Blackack Historian (5) gives a perspective on the I18 s origins. Here we revisit these ol questions, with some new conclusions. To begin, we put asie composition-epenent strategies: we consier them iicult to implement in practice an o incremental beneit so moest as to not warrant the eort. We rather ocus our analysis on the total-epenent class, an on the single true count optimize or betting. We in that the player neen t learn the complex
proceure where each play parameter, inepenently o the others, has its own variation with true count (requently calle strategy inices ), a scheme we ll label Count- Depenent Play. Instea we show that much o the possible avantage a car-counter coul gain in this way is achieve by the proper choice o a count-inepenent strategy. Such a play strategy is easy to use in a casino ust like any basic strategy an yet improves consierably on OBS. A scheme o this type has been propose by Marcus (7) an calle Counter Basic Strategy (CBS); we ll aopt his terminology. Begin by recalling that the expecte return varies - an rather strongly - with true count even i the play parameters remain ixe; varying the play parameters improves return by only a moest aitional amount. Thus a goo strategy approximation is a count-inepenent one that - importantly - matches the (slightly higher return) countepenent one at the right choice o count. The most sensible right or proper choice is the one that maximizes the player s average expecte cash low per roun (his yiel ), recognizing that he is betting ierent amounts on ierent rouns guie by the true count. Since he makes higher bets or higher true counts, the probability istribution o his bet sizes is peake at a true count that is signiicantly positive. Play that is optimal at or near that peak - not at zero epth an count - is the right or proper choice or the otherwise count-inepenent strategy! This is Counter Basic Strategy.. Analysis Our analysis here, an our notation as well, is largely base on our previous work concerning optimal betting, Werthamer (5) an (6), which we ll cite as OB-I an OB-II. The player s yiel, Y, is eine as his expecte cash low per roun, average over all rouns between successive shules; rom OB-I, assuming no risk o ruin, F 1 Y = B(R)R p{} F Here is the pack epth, up to a reshule penetration F ; is the true count with the Gaussian probability istribution. (1) p{ } o equation A3 an OB-I equation (5) ; an B(R) is the optimal bet size, speciie later in Results. The quantity R R ( ) is the expectation, or that count an epth, o the roun s return over the probabilities o rawing a car o value, with = 1,,1. The contingent expecte return expression seems quite ormiable at irst. But we are able to show that it can be constructe to at least a close approximation rom the R. This is the return rom a much simpler (an computationally easible) orm ( ) han at zero epth, with the probability o rawing value on each car o the han given by the count-epenent expectation = 1 α 5 ; () ( )
here α is the counting vector an is the probability o car value rom a reshly shule shoe. The rather lengthy erivation o these results, an some aitional einitions, are given in the Appenix. But the expecte return also epens on the play parameters: ieally they shoul be auste at each true count an epth so as to maximize the return there, although typically any epth epenence is neglecte. We esignate such a maximal array o play parameters as π( ); an we make this play epenence explicit in the expecte return expression, as R ( ; π ( ) ). But this proceure orces parameter changes, or strategy inices, as the count varies. Depening on how the very wie range o possible counts is truncate to rop the more improbable values, the total number o such changes can be well over a hunre (since some parameters change several times over even a truncate count range). The inices are a complicate roster to remember an apply uner casino conitions; the Illustrious 18 is ust a particular subset o the most inluential. Much easier or the player, o course, is a total-epenent play strategy that has no variation with count. We are ree to test the total-epenent play that is optimal at any true count; at zero, equation () shows this is ust the OBS, with expecte return R ( ; π ()). In particular, note that although p{ } is symmetrical about zero true count, B( R ) optimally ramps upwar or increasingly positive returns an counts; so the probability istribution o bet sizes, BR ( ) p { }, is skewe towar positive counts an shows a peak, uner many conitions, or at least a shouler. Thus we expect that the yiel will be maximize or a true count near the peak or shouler in the bet size istribution. We ll enote the true count that maximizes the yiel in this way as *. Then the count-inepenent play parameters o CBS are π( *), with resulting maximal yiel obtaine via integration over 3. Computations ( ) R ; π ( *) as per equation (1). Our computational program comprises a sequence o steps. In outline, the irst step writes an algorithmic coe (we use Visual Basic as the ramework) to compute the expecte return rom the irst han ater a shule o a pack with a speciic number o ecks D. In this step, since the epth at the beginning o the han is zero, the irst car ealt has value with probability (see Appenix). The play parameters are auste to maximize the return at zero count, corresponing to π (). The output, R ( ; π ()), successully reprouces results well known in the literature, such as by Griin (1999) or one eck an by Manson, et al. (1975) or our ecks. The secon step generalizes to a non-vanishing count. The Visual Basic coe is extene so that the probabilities o the cars rawn to the han relect the true count at R ; π (). The Count-Depenent play its start as per equation (), giving ( )
parameters π( ) emerge rom maximizing the expecte return at that true count, an ( ) give the count-epenent return R ; π ( ). As anticipate, these expecte returns increase with increasing, although with signiicant curvature away rom linear. The thir step its the compute results or the count-epenent return o the previous step to a low-orer polynomial in ; a quartic is suicient or accuracy to about 3 signiicant igures throughout. The ive coeicients resulting rom the it are then transcribe into a ourth orer truncation o the corresponing Hermite polynomial expansion (see the Appenix) giving an excellent numerical approximation to R ( ; π ( ) ) at non-zero true count an epth. Because o the curvature o R ( ; π ( ) ) with, the Hermite terms o orer -4 convey a epenence on epth. The ourth step selects several positive trial values or true count *. For each it generates π ( *) rom the results o the secon step (i.e., maximizing R ( ; π * ) with respect to π ) an then computes R ( ; π( *) ) as a unction o. A truncate Hermite polynomial expression is generate ust as in the thir step, approximating the corresponing ; π ( *). R ( ) The ith step aopts an optimal bet size B( R ), speciie below; combines it with the expecte return approximation rom the ourth step; an (having switche to Mathematica as the ramework) computes the yiel Y, via equation (1), by integrating over an averaging over the epth, weighte by p{ }. Lastly, the yiel values obtaine or the several trial values o * are interpolate to arrive at that proviing the maximum yiel. The corresponing π(*) then represents our esire CBS. Note, however, that CBS is epenent, at least ormally, on the particular choice o bet strategy, as well as on the counting vector an the penetration. It remains or the uture to streamline these computations, here assemble a-hoc rom pieces o earlier work, by consoliating them all into a single Mathematica coe. 4. Results We o not report an exhaustive set o computations or all possible choices o parameters. We rather select a ew representative examples to illustrate our methoology an results. We ocus on a penetration o F =.8 (as throughout OB-I an -II); an use the optimal counting vector
= 1 3 4 5 6 7 8 9 1 α = -1.8 +.8 +.94 +1.1 +1.5 +.98 +.57 -.6 -.4-1.7 rather than one or another o its more popular (an practical) approximations. This vector is roughly equivalent to Griin (1999, page 45, last line o Table) an Epstein (1995, page 44, last line o Table 7-1, labele Thorp Ultimate) but is closer to Epstein (1995, Table 7-11, last line). Further, we isplay results only or games with a single eck (where yiel is the most sensitive to choice o play strategy) an with our ecks (representative o six an eight eck games, as well). We consier the same two betting patterns as in OB-I an -II: the Weekener an the Lietimer, iering in their risk/rewar proile. For each example the bet is a linear ramp, B( R) = sr σ, where σ is the variance o the return, but cappe on the lower en by a positive base bet an on the upper en by a maximum bet (or sprea ) o 1 times the base. The coeicients o proportionality (rom criteria evelope in OB-I) are s σ =.78 or the Lietimer an.61 or the Weekener, with 4 ecks. These 3 coeicients give the Lietimer, with a capitalization assume to be 1 base bets, a.13 6 risk o ruin over 1 rouns; the Weekener, with capitalization assume at only 1 3 base bets, has a.19 risk o ruin in 1 rouns. For single eck, the corresponing coeicients are.9 an.43, respectively, an the risk o ruin is negligibly small. The results are shown in Table as the ratio o yiel to base bet. Table. Yiel ratio or Counter Basic, Optimal Basic an Count-Depenent strategies Betting style: Lietimer Weekener Number o ecks: 1 4 1 4 Count-Inepenent Optimal Basic +.76 +.19 +.6 +.8 Counter Basic +.91 +.3 +.9 +.11 Count-Depenent +.83 +. +.13 +.11 Eective true count, * 5.1 3. 4.6 3.5 Clearly, Counter Basic increases the yiel over that rom Optimal Basic by a meaningul amount, particularly against a single eck. Table 3 etails the Counter Basic Strategy or a single eck, consistent with an eective true count in the vicinity o +5 that is miway between the * o the Lietimer an the Weekener; this CBS is, by construction, ientical to Count-Depenent play at that true count value. The CBS parameters that ier rom those at zero true count (i.e., Optimal Basic) are bole an italicize.
Table 3. Counter Basic Strategy, single eck (same rules as Table 1) Upcar Stan on Double Split A 17; s18 1, 11 A, 8, 9 1; s18 9-11; s17, 18 A, 6-9 3 1; s18 9-11; s15, 16, 17, 18, 19 A,, 3, 6-9 4 1; s18 8, 9-11; s13-18, 19 A,, 3, 6-9 5 1; s18 8-11; s13-18, 19, A,, 3, 6-9 6 1; s18 8-11; s13-19 A,, 3, 6-9 7 17; s18 9, 1, 11 A,, 3, 7, 8 8 17; s18 1, 11 A, 8, 9 9 16; s19 1, 11 A, 8, 9 1 15; s19 1, 11 A, 8 Always take insurance 5. Discussion It is encouraging to compare our CBS result o Table 3 against those o Marcus (7). Although the respective methoologies are quite ierent (he uses a simulation program), the resulting plays are quite similar, in particular when interpolating his single-eck charts or sprea o 8 with penetrations o.6 an.7, an sprea o 1 with penetration.65, to our conitions o sprea 1 with penetration.8. Also, Marcus aopts a bet scheme that rises quaratically with true count rather than an optimal linear ramp as here. Note in Table that the yiel ratio or CBS in some cases is actually superior to Count-Depenent play. The comparison is even more ramatic vs. Illustrious 18 play, whose yiel is typically aroun 7% o Count-Depenent. This seeming paraox is unerstoo by recalling that Count-Depenent maximizes the expecte return at each true count, but at zero epth; it is not recalculate or each separate epth value. In contrast, CBS maximizes the yiel, the ominant contributions to which come rom sizable epths, where the true count is more likely to take on large, positive values an the expecte returns an bet sizes are corresponingly larger. The epth epenence o the expecte return an play strategy, although almost always ignore in the blackack literature, is a signiicant inluence in our computational results. It is intriguing to ollow this logic still urther an note that the optimal counting vector α o the previous section has been eine as maximizing the resulting expecte return at zero epth. A better version might be obtaine by instea maximizing the yiel, with greater inluence rom conitions o larger epths an higher true counts. Investigation o this generalization, incluing its quantitative implications, will be pursue separately.
Appenix We consier a han that begins ater one or more rouns ollowing a shule, the previous rouns having use m cars o value, totaling M = m in number, rom a pack o D ecks. Deine shule: as the probability o rawing value as the irst car ollowing a = 1/13, 1; = 4/13. Then the probability o rawing value on the 1 irst car o the latest roun is ( 5 D m) ( 5 D ) with a vector α (assume balance, α = the true count at the start o the roun is = M. The player is counting, an normalize, α = 1) so that m ( D M 5) 5 ( ). (A1) = α = α The probability istribution ρ{ } or the irst car rawn to the han in the absence o counting is given by ( ) 1 ρ {} = π Δ δ ( ) exp, (A) π Δ Δ (OB-I equation (4)), where 5D( 1 ) Then the istribution contingent on the true count [ p ] Δ parameterizes the istribution with. becomes ρ{ } = ρ{ } { } δ( + 5 α ( )), (A3) with normalization p { } = p{ } given by 1 exp πτ τ, (OB-I equation (5)) an τ 5Δ. The expecte car istribution, conitional on, becomes ( ) { } ( 1 ) = ρ = α 5. th The probability o the κ car, rawn rom a pack eplete to 5D M having value κ an conitional on the previous cars being 1,, 1 is ( ;,, ) κ 1 ( ) ε δ(, ) κ κ l= 1 κ 1 κ 1 = ε 1 ( κ 1) ( ) 1 κ l κ cars, (A4), (A5) where ε 5D M =ε +Δ increases with epth. Then the return rom any han o K cars can be expresse as a sum o terms, each corresponing to a ierent K coniguration o cars rawn an each with the probability actor κ. κ= 1
We now have the machinery to evelop a close orm expression or the expecte return rom the han. To make explicit the epenence o the return on, an on the speciic coniguration o the current pack, we enote it as ( ). Then ( ) ( ) ρ { } R R = R ( ) ( ) R. (A6) Ater substitutions o equations (A) - (A4), o the Taylor series ( ) exp ( ) R =, an o the Fourier representation δ ( x) = ϕ( 1 π) exp ( iϕx ) or the two Dirac elta unctions, the resulting multiple integrations can all be perorme straightorwarly, so that 1 R ( ) exp ( ) ( = Δ α α R ), (A7) where the caret variables help make the notation more compact: α α,, an 5. Furthermore, applying a Taylor series again ( l) l l on the variable, R ( ) = exp Δ + ( ˆ ) ˆ ˆ R ( But the sum-o-proucts orm o ( ) 1 ˆ ˆ ˆ α α ). (A8) ε R rom the previous paragraph enables the nearcancellation (emonstrate below) so that 1 1 Δ + R ( ). (A9) ε 5D Even or a single eck, ( 5D is negligibly small. As a consequence, to this orer o ) approximation, the expecte return, equation (A8), reuces to ust R ( ) 1 ( ) ( ) ( ) ( exp 1 exp. R R ) Δ = Δ α α α Equation (A1) establishes that the expecte return o every han in a shoe, in the absence o counting, is very nearly the same as the irst ollowing its shule. (Some authorities assert that these shoul actually all be ientical; but our erivation oes not reveal such an equality.) (A1) We still have to eal with the exponential actor in equation (A1a), to which we apply an expansion in a series o Hermite polynomials,. Thus H n 1 Δ Δ = n n ( α ) α R( ) Hn ( α ) R ( ) exp. n= n! Δ (A11)
We anticipate that truncating the series at a low orer provies an aequate R approximation or computational purposes. This is born out by computing ( ) (i.e., equation (A1b) with Δ =, corresponing to the irst roun ater a shule) as a unction o an ining that the resulting curve while not strictly linear as approximate in OB-I an -II - can instea be it reasonably with a quaratic, an with a quartic to an accuracy much better than 1%. We then insert the 5 coeicients o the it into a truncation o equation (A11) at orer 4; the result now incorporates correctly (an computably) the Δ epenence seen in equation (A1). In other wors, the leastsquares it c (returning to the original variables rom the 4 n caret ( ) R n= ones) translates into the approximation n ( ) 4 n / n= n R c τ Hn( τ ) ( ) ( ) ( = c + c + c τ + c 3τ + c 6 τ+ 3 τ. 3 4 1 3 4 ) (A1) To complete our arguments, we nee to provie a proo o equation (A9). Begin with a representative prouct term in the probability o a han o K cars: substituting equation (A5), such a prouct term is o the orm K κ 1 ( ε δ(, )) ( 1 ( 1) ) κ ι= 1 κ ι ε κ. (A13) κ= 1 1 Then apply the operator + to it an carry out the ierentiations. Although ε the manipulation is teious an the resulting expression is lengthy an cumbersome, rearrangement an careul attention to cancellations among terms shows that it is ε ( ) 1 = 5D proportional to. The co-actor remains complicate but seems not to vanish, either ientically or in the limit o ε ; nevertheless, we ve arrive at the esire result.
Reerences Epstein, R.A. (1995) The Theory o Gambling an Statistical Logic, rev. e. Acaemic Press. Blackack Historian (5) Blackack Forum XXIV, #1, 4/5 Griin, P.A. (1999) The Theory o Blackack, 6 th e. Huntington Press. Manson, A.R., Barr, A.J., Goonight, J.E. (1975) Optimum Zero-Memory Strategy an Exact Probabilities or 4-Deck Blackack. The American Statistician 9 84-88. Marcus, H.I. (7) New Blackack Strategy or Players who Moiy their Bets Base on the Count. In: Ethier, S.N., Cornelius, J.A., an Eaington, W.R. (es.) Optimal Play: Mathematical Stuies o Games an Gambling. Institute or the Stuy o Gambling an Commercial Gaming, University o Nevaa, Reno. Schlesinger, D. (5) Blackack Attack: Playing the Pros Way, 3r e. RGE Publishing. Werthamer, N.R. (5) Optimal Betting in Casino Blackack. International Gambling Stuies 5 53-7. Werthamer, N.R. (6) Optimal Betting in Casino Blackack II: Back-counting. International Gambling Stuies 6 111-1.