1
Optimal Betting in Casino Blackjack III: Table-Hopping
N. RICHARD WERTHAMER New York, USA
Abstract The casino blackjack technique usually called back-counting, or wonging, consists of beginning play (“entry”) at a table only after an indicator of favourability exceeds a certain threshold. The back-counter is also advised to leave the table at a threshold of unfavourability, most usually after entry (here termed “exit”) but sometimes also before entry (here termed “departure”). I have analysed these thresholds previously and obtained optimal entry and exit criteria based on maximising the total cash value of the table between successive shuffles, recognizing that only some of its rounds are actually played. Here I extend and complete that investigation to include the value from a second, freshly-shuffled table, played after either departure or exit, until the first table is reshuffled; this extension is here termed “table-hopping”. An optimal departure point arises in table-hopping that does not appear when considering only a single table. Optimal table-hopping offers an important incremental advantage to its practitioner, here quantified for several representative game conditions, sufficient to make attractive several well-camouflaged betting methods.
Introduction A blackjack player welcomes any technique by which he can estimate his odds on the next hand, each time he places his bet. If the estimation is unfavourable, he then bets only a minimum amount; if favourable, he bets more than the minimum, depending on the degree of favourability. Such a technique, called “card-counting”, was first developed by Thorp (1962) and subsequently refined by others. In general, card counting involves observing every card as it is dealt. The player mentally maintains a numerical register, beginning at the pack reshuffle with a prescribed initial amount and incrementing the register as each card is dealt by an amount assigned to that card’s value. Although slightly variant counting amounts are used by different players, most divide the current register amount (“running count”) by the number of decks remaining undealt, to obtain the so-called “true count”. A positive true count indicates a favourable situation, a negative true count an unfavourable one. Werthamer (2005) determines the optimal amount to bet, based on the current true count and other factors. Stanford Wong, in his book “Professional Blackjack” (1975), suggests a further technique in which a card-counting player observes a table but begins betting at it (“entry”) only when its count indicates he has an advantage. Wong also suggests the player stop playing at that table (“exit”) whenever the count comes to indicate a disadvantage. Wong was not specific on these entry and exit counts, or on the player’s resulting performance improvement; but it is clear that betting only on advantageous rounds provides a major edge, despite the consequent time spent merely observing. A few later authors reiterate Wong’s suggestions, although with little further quantitative specification. Vancura and Fuchs (1998), for example, advise entry at the equivalent of a true count of +1 and exit at a true count of about -1 (slightly depth dependent) for multi-deck games, but do not quantify the resulting improvement. Schlesinger (2005), however, while calling it “the most complicated blackjack problem I 2
have ever studied”, reports at length on departure and exit thresholds. Although he gives little indication of how the results were derived, they are reviewed and compared with mine in the Discussion and Conclusions section. But even when understood only qualitatively, the entry/exit technique is widely used among card counters. In a previous publication, Werthamer (2006) analysed the technique. It was described there as 1) observing – but not yet betting at – a table while tracking its true count, and starting to bet only when a certain positive entry threshold is reached; and 2) leaving that table, after entry, when an exit threshold is reached. An additional (or alternative) step, not mentioned by Wong or Vancura and Fuchs, is 3) leaving that table when a “departure” threshold is reached prior to entry. Thus the technique is specified by parameters for the three independent thresholds: for entry, for exit and for departure. In general, each threshold should be identified by both its true count and the pack depth at which that count occurs, jointly; this is the viewpoint taken by Schlesinger, and also noted by Vancura and Fuchs. Instead, Werthamer (2006) argued that tracking these thresholds based on a combination of both true count and depth together may be too challenging for most players under actual casino conditions. Rather, Werthamer sought thresholds based on just one of the two, averaging over the other. Threshold optimality then results from the best balance between avoiding rounds with negative or only weakly positive expectation, and also avoiding excessive time spent merely observing. Werthamer reported several significant results: a) the optimal true count threshold, independent of depth, for entry when neither exiting nor departing; b) the optimal joint entry and exit true count thresholds, when not departing; and c) a qualitative guideline for departing, based on depth and independent of true count. Important yield advantages were found for these methods individually, and even larger when used in combination. Entry and exit together are usually called ‘back-counting’, or sometimes ‘wonging’ or ‘table-hopping’ (the names have been used inconsistently and almost interchangeably). Werthamer (2006) specified the term “back-counting” as entry, exit and departure from a single table. But no additional value was assigned to playing at a second table following either exit or departure from the first; in actuality the exiting or departing back-counter could shift his attention (“table-hop”) to another table being reshuffled and start again at step 1) above. The most appropriate measure of the incremental value of table-hopping is the expected cash flow from the two tables together (or others as well, if exiting or departing the second) relative to that from playing only the first, with the multi- table value totalled over the interval between successive reshuffles of the first table. Thus each value being inter-compared is accrued over the same average number of rounds. Here I generalize to include explicitly the contribution of a second table. This extension gives a more precise treatment for the optimal entry and exit thresholds, and departure is now found to have an optimal threshold as well. Also provided is a quantification of the large yield improvement table-hopping confers. Table-hopping is so valuable that the player can use it advantageously even with flat betting of a multi-deck shoe or with a well-chosen exit after entry at a shuffle; either variant provides excellent camouflage for card-counting. 3
Analysis Werthamer (2006) is built on the statistical foundation that a pack’s true count executes a so-called Brownian bridge, a random walk constrained to return to its starting point after a fixed interval. Specifically, a balanced true count starts at zero following a shuffle, and is again zero if and when all cards are dealt. Also, the true count and depth, which strictly are discrete variables, as approximated as continuous ones; this allows the random walk to be modelled as a diffusion process, the key to the classic analysis of Brownian motion. Since diffusion is described by a simple partial differential equation, the powerful array of mathematical tools for its solution under various initial and/or boundary conditions can be called upon. These tools are also capable of addressing the more general questions investigated here. Although the tools are commonly used throughout the physical sciences, they are relatively unfamiliar in the context of gambling; yet they provide deep insights not readily obtainable with conventional probability methods. For those not fully conversant with partial differential equations, the good news with respect to diffusion is that once a putative solution is put forward, it can be confirmed as the unique solution merely by verifying that it satisfies both the differential equation and the initial/boundary conditions. This Analysis section uses such a verification approach throughout.
Entry Although entry was previously analysed in Werthamer (2006), it is reviewed here in order to cover certain additional features which become important in the extension to table-hopping. I refer to equations from there by the prefix II. Equations (II-5) and (II-7), but with slightly compressed notation, showed that the probability of first reaching the entry true count γ≥E 0 at a ‘time’ τE is given by 2 γγEE⎛⎞γE ργτ=1(,)EE ργτ= (,)EE exp⎜ − ⎟. (1) ττ3 2 EE2πτE ⎝⎠ Because of the Brownian bridge condition, the variable τ (here called ‘time’ since it plays the same mathematical role as actual time does in physical diffusion) is related to the depth f of a pack with D decks by the nonlinear τ =−52f Df (1 ) . Also, the probability of true count γ at any ‘time’ τ prior to entry is
ργτ=ργτ−ργ−γτΘγ−γEE(,)[ (,) (2 ,)] (E ), (2) using arguments similar to equation (II-15); Θ is the standard unit step function,
Θ≡()xx 1, ≥≡ 0;0, x < 0. Note that ρE (,γτ) satisfies the diffusion equation ⎛⎞∂∂1 2 ⎜⎟−ργτ2 E (),=0 as per equation (II-6), along with the boundary condition ⎝⎠∂τ2 ∂γ
ργτ=EE(,)0 appropriate for diffusion to a barrier. This pair of expressions is further justified through showing that they satisfy conservation of probability. Thus the total probability of having entered at any τE prior to τ is, from equation (1), 4
τ ⎛⎞γ Pd ()τ= τρ ( γ , τ ) = 2erfE , (3) EEE∫ 1 E⎜⎟ 0 ⎝⎠τ ∞ where erf(xd )≡π() 2−1 y exp −y2 2 is the familiar error function; whereas the total ∫x () probability of not having entered by τ is, from equation (2), ∞ ⎛⎞γ Pd()τ= γργτ=− (,) 1 2erf E . (4) EE∫ ⎜⎟ −∞ ⎝⎠τ These indeed sum to unity, as required by conservation of probability. Furthermore, the two expressions are also linked by the expected ‘chain rule’ for contingent probabilities, 1 τE ∞ ddτγργ−γτ−τργτ=ργτ(,)(,)(,); (5) ∫∫ 1EE E1 EE τE 0 −∞ the integration steps needed for the proof are demonstrated in Appendix A, and serve as a prototype for similar convolutions in succeeding sections. Once having entered, the back-counter bets an amount B()γ per round based on an expected return R()γ from that round, each a function of the current true count γ . He continues playing until the next reshuffle, at penetration F and corresponding ‘time’
τ=F 52F D( 1 −F), so that the total value of the shoe subsequent to an entry at τE is F ∞ VdfdBR()τ= γγ() ()( γργ−γτ−τ , ). (6) EE∫∫ E fE −∞ Then his “yield”, or expected cash flow per round averaged over the shoe, is given by weighting that value by the probability of entry, 11τF F ∞ Yd=τργττ=( , )( V ) dfdBR γγγρΓγ()() ,τ , (7) EEE∫∫1 EE ∫(E() ) FF00−∞ where the convolution on τE evaluates as in equation (II-12), with ΓEE(γ≡γ−γ+γ) E.
The optimal entry threshold γ E maximises the yield. Note that the yield expression here omits any effect of ruin, a factor previously included in both Werthamer (2005) and (2006). Because, for a given bet ramp, table- hopping reduces risk of ruin vs. “play-all” (i.e., not back-counting), it is reasonable to set ruin aside; this simplification then relieves some of the problem’s heavy computational burden. As a consequence, certain results from Werthamer (2006), which should otherwise match comparable ones here, in fact differ by small amounts. The emphasis here is on relative yields between different techniques, rather than their precise absolute magnitudes.
Entry and Exit The back-counter can choose to exit, or leave the table prior to reshuffle, if the true count drops below a lower threshold, γ τX Pd()τ= τργ−γτ−τργτ=ργ−γτ ( , )(,) (2 ,). (8) XX∫ E11 X EX E EE1 E XX 0 5 The second equality follows by substituting from equation (1) for the ρ1 functions and performing the integration using the methods of Appendix A. Also, the probability distribution of true counts γ at τ subsequent to entry, equation (II-15), is ργτ=ρΓγτ−ρΓγ−γτΘγ−γEX (,)⎣⎡ (E () ,) (EX( 2) ,)⎦⎤ (X ), (9) satisfying the exit boundary condition ρEX(γτ= X , ) 0 . These two expressions, like equations (3) and (4) above, similarly satisfy conservation of probability but now conditional on entry: τ∞ ⎛γ E ⎞ dPττ+γργτ=XX() X dEX (,)2erf⎜⎟, (10) ∫∫ τ τ−E ∞ ⎝⎠ where the right hand side, by equation (3), is the probability of entry prior to τ . The yield from this first table is analogous to equation (7): 1 F ∞ YdfdBR(1) =γγγρ() () (,)γτ. (11) EX ∫∫ EX F 0 −∞ But upon exit, a table-hopper moves to a second table with a freshly shuffled shoe and repeats the entry process. Hence the more appropriate assessment of his yield is to include the cash flow from the second table during those rounds of the first that follow the exit and precede its reshuffle. Thus the yield from the second table alone, weighting with the probability of exit from the first, is τ−FX1 Ff ∞ YdP(2) =τ() τ dfdBR γγγργτ() () (,). (12) EX ∫∫∫XXX EX 00F −∞ Combining the yields from the two tables, and carrying out the intermediate τX integration, the total yield becomes 1 F ⎡⎤⎛⎞2γ−γ ∞ YYY=+=(1) (2) df12erf + EX dBRγγγργτ()() (,), (13) EX EX EX ∫∫⎢⎥⎜⎟ EX F 0 ⎣⎦⎝⎠τ −∞ where τ≡ 52(F −fD ) (1 − Ff + ) . Although contributions from additional tables beyond the second should in principle be included, computationally these are negligible: the probability of exit from the second table is quite small and the number of rounds played at a third is typically too few to generate much additional value. The optimal entry and exit thresholds, as in the entry-only case, jointly maximise the total yield. Entry and Departure Apart from a possible exit following entry, the back-counter may independently choose to depart the table prior to entry. This might occur, for example, if much of the shoe has been dealt without reaching the entry threshold; or if the true count becomes decidedly negative and the probability of it swinging sufficiently positive to trigger entry is correspondingly low. Ideally, the departure decision should involve a combination of these two circumstances. But Werthamer (2006) argued that a decision predicated on both true count and depth together is too difficult in practise, so instead it focused on just depth alone. The yield from the first table was found to decline monotonically with decreasing departure depth, without a clear-cut optimal threshold. 6 Here the analysis is extended to include the yield contribution from a second table. Again, no optimal departure boundary is found in the depth variable, i.e., no threshold that increases the yield over that without any departure. A qualitative understanding of this behaviour is that prior to entry the back-counter is expending no money – merely observing does not require an ante – during a time when the true count ranges from only slightly favourable, at best, to decidedly unfavourable. If a table-hopper switches to a second table and again begins to observe, he can hope only to shorten the time until he can start to play; he saves no money. An additional effect, though, is that the freshly-shuffled shoe at the second table has much lower volatility in true count than does the first at mid-shoe, so even if entry at the second is sooner it may offer fewer possibilities for advantageous rounds. In short, departure based on depth does not improve the table-hopper’s yield; the analysis leading to that conclusion is omitted here. Instead, we change direction and instead consider departure based exclusively on true count, averaging over depth. Although this approach may seem exactly like exit as discussed above, there are significant differences: departure is a decision prior to entry, during which the back-counter does not ante, while exit is a decision following entry, where an ante is at risk on every round. Departure under these conditions can improve yield, even if slightly, and does have an associated optimum. When departure at a threshold γ τE prior to departure extends equation (1) to γγ−EEγD ργτ≅1D (,)EE ργτ− (,)EE ργ−γτ (2D EE ,); (14) ττEE see Appendix B for the derivation. Then the corresponding probability distribution after entry, extending equation (9), is τ ργτ=τργ−γτ−τργτ(,)d ( , ) ( , ) ED ∫ E EEDEE1 0 (15) γ−γED ≅ρ() ΓEE() γ,2 τ − ρ() Γ() γ − γD, τ , γ−γED2 and the yield from the first table is 1 F ∞ YdfdBR(1) =γγγρ() () (,)γτ. (16) ED ∫∫ ED F 0 −∞ Conversely to equation (14), the probability of departure at τD prior to entry is similarly approximated as −γDEγ − γD PD ()τ≅DD ργτ− (,)DE ργ−γτ (2D ,D). (17) ττDD Then the yield from a second table following departure becomes, analogous to equation (12), τ−FD1 Ff ∞ YdP(2) =τ() τ dfdBR γγγργτ()()(,). (18) ED ∫∫∫DDD ED 00F −∞ Thus, analogous to equation (13), the total yield from both tables is 7 (1) (2) YYYED=+ ED ED 1 F ⎡⎤⎛⎞−γ γ − γ ⎛2 γ − γ ⎞∞ (19) ≅+df 1 2erfDEDED − 2 erf dγγγργ B( ) R ( ) ( ,τ ). ∫∫⎢⎥⎜⎟ ⎜ ⎟ ED F 0 ⎣⎦⎝⎠ττ2γ−γED ⎝ ⎠−∞ As with exit, the optimal entry and departure thresholds jointly maximise the yield. Entry, Exit and Departure When the table-hopper employs both departure (prior to entry) and exit (subsequent to entry), the probability distribution of true counts at τ combines equations (9) and (15): τ ρ(,)(,)(2,)(, γ τ =d τ ρ γ−γ τ−τ −ρ γ −γ−γ τ−τ ρ γ τ ) EDX ∫ E []EE X EEDEE1 0 (20) =ρ[]ED (,) γτ−ρED (2 γ X −γτ ,) Θγ−γ (X ); the yield from the first table is 1 F ∞ YdfdBR(1) =γγγρ() () (,)γτ. (21) EDX ∫∫ EDX F 0 −∞ Since the probability of departure is unaffected by any possibility of later exit, hence the departure contribution to the second table’s yield is still given by equation (18); but the probability of exit is influenced by the possibility of previous departure, so that equation (8) generalises to τX Pd()τ= τργ−γτ−τργτ, , XD X ∫ EXEXEDEE11()() 0 . (22) γ−γED =ργ−γτ11(2EXX , ) − ρ(2 γ−γ−γEX 2 DX , τ ). 2γ−γED Thus the total yield from the first two tables combined, extending equations (13) and (19), is 1 F ⎪⎧ ⎡⎤⎛⎞−γ ⎛2 γ − γ ⎞ Ydf=+12erfDE + erf X EDX ∫ ⎨ ⎢⎥⎜⎟ ⎜ ⎟ F 0 ⎩⎪ ⎣⎦⎝⎠ττ ⎝ ⎠ ⎫ γ−γED⎡⎤⎛⎞⎛222 γ−γ ED γ−γ−γEXD⎞⎪ −+2erferf⎢⎜⎟⎜ ⎟⎥⎬ (23) 2γ−γED⎣⎦⎝⎠⎝ττ⎠⎪⎭ ∞ ×γγγργτdB() R () (,). ∫ EDX −∞ As before, the three threshold true counts jointly maximise the yield. Computation Despite the complexity of the expressions developed here, a late-model personal computer is still able to give meaningful results within a reasonable run-time. Although we do omit risk of ruin, we continue to deal with the game conditions considered previously in Werthamer (2005) and (2006): 6 decks, dealt to a penetration of 0.8, and a set of rules (dealer stands on all 17s, no DAS, no resplits, no surrender) such that the first 8 hand after a shuffle has an expected return of -.0058 and a variance of 1.26. Also, we assume the player uses the Hi-Lo recipe for counting, a count-independent Basic Strategy, and either of the betting styles labelled in Werthamer (2006) as a) “Lifetimer” (capital of 1000 times the base bet, spread of 10, linear bet ramp with the HJY slope of s = 1.12 and risk of ruin over 106 rounds of .168), or b) “Weekender” (capital of 100 times the base bet, spread of 10, linear bet ramp with the MM slope of s = 3.33 and risk of ruin over 103 rounds of .144). All computations are implemented within the Mathematica software package. A further refinement used here is to recognize that the expected return on a round is actually a non-linear function of true count, whereas Werthamer (2006) approximated the relationship as linear. As a consequence of the non-linearity, the expected return also depends on the round’s depth, a generalization developed in detail in Werthamer (2007). As there, the exact functional dependence is approximated computationally through curve-fitting to a truncated Hermite polynomial series. The results for these two betting styles are shown in Table 1. The yield is normalized relative to the base bet, corresponding to an expected return per round averaged over an entire shoe; as in Werthamer (2006), we call this the “effective yield”. Table 1. Optimal thresholds and effective yield for table-hopping Effective # tables γ E γ D γ X yield Lifetimer Play-all 1 0 −∞ −∞ .0180 Optimal entry only 1 2.09 −∞ −∞ .0238 Optimal entry & exit 1 1.89 −∞ -3.57 .0242 Optimal entry & exit 2 1.81 −∞ -3.18 .0243 Optimal exit only 2 0 −∞ -1.91 .0224 Optimal entry & departure 2 2.08 -3.49 −∞ .0240 Optimal entry, exit & departure 2 1.85 -3.29 -3.45 .0245 Entry at +1 2 1.0 −∞ -2.95 .0239 Weekender Play-all 1 0 −∞ −∞ .0052 Optimal entry only 1 2.60 −∞ −∞ .0113 Optimal entry & exit 1 2.21 −∞ -2.72 .0116 Optimal entry & exit 2 2.15 −∞ -2.57 .0117 Optimal exit only 2 0 −∞ -1.54 .0095 Optimal entry & departure 2 2.58 -5.03 −∞ .0113 Optimal entry, exit & departure 2 2.14 -5.13 -2.61 .0117 Entry at +1 2 1.0 −∞ -2.40 .0111 Flat Bet Play-all 1 0 −∞ −∞ -.0040 Optimal entry & exit 2 1.87 -1.32 −∞ .0026 9 For each of the two styles, the first line gives results in the absence of any back-counting, i.e., “play-all”; for both, optimal betting produces a positive effective yield. The second line gives the optimal entry true count, without any exit or departure; entry improves the effective yield substantially, by about .006 in each case. The third and fourth lines give the jointly optimal entry and exit true counts, without and with table-hopping, respectively; including exit makes only a small additional yield improvement, and the contribution of the second table slightly lowers the two thresholds. The fifth line gives the optimal exit true count for a table-hopper who always enters at the shuffle; the improvement is roughly 70% of that from entry without exit. The sixth line gives the jointly optimal entry and departure true counts when table-hopping; departure is a near substitute for exit. The seventh line combines all table-hopping procedures into a single jointly optimal set of entry, exit and departure true counts; the effective yield improvement from exit and departure combined is roughly the sum of the contributions from each alone. The Flat Bet section of Table 1 gives results for the table-hopper who never varies his bet, a technique highlighted by Wong. The first line shows the play-all effective yield, in this case differing slightly from the off-the-top expected return of -.0058 because of the approximation of fitting a low-order polynomial to the return function over a wide range of possible true counts. For the flat-betting table-hopper who enters and exits optimally, the effective yield becomes slightly positive, although considerably less than with optimal bet variation; but the yield improvement from table-hopping is roughly the same as with optimal betting. Thus the table-hopper develops an advantage even when flat betting against a multi-deck shoe, thereby quantifying and justifying Wong’s original suggestion. Discussion and Conclusions Optimal betting, even without back-counting, is sufficiently powerful that it gives a positive effective yield. Adding entry, at an optimal true count in the range of +2.1 to +2.6, increases effective yield by another .006, so is a valuable addition to the counter’s repertoire. Exit after entry slightly improves effective yield, by another .0004; the optimal exit true count is in the range of -2.6 to -3.2, while the optimal entry shifts slightly downward to around +2.0. Remarkably, the table-hopper who merely exits – i.e., always begins play at a table just after a shuffle – accomplishes much of what entry and exit together could! Exit by itself improves effective yield by more that .004. As also noted in Werthamer (2006), this technique could be particularly difficult for casino surveillance to spot and so provides the table-hopper excellent camouflage with a relatively small sacrifice in performance. Departure, on the other hand, is of only slight value. Combined with entry alone, or together with exit as well, departure’s improvement in effective yield, even when optimised, is small. We remarked above that since observing a table and waiting for entry costs nothing, there is little to be gained from switching to another table and starting over. Some table-hoppers may feel that occasional departures provide some camouflage for their manoeuvres, but few experts advise it as a matter of course. In summary, the simplest rule of thumb for the optimal table-hopping thresholds is entry at a true count of +2 to +2.5 together with exit at -2.5 to -3.5, thereby increasing 10 effective yield for a 6-deck shoe by about .006. Departure, at true counts of around -3 to -5, is of minor further advantage. Finally, we comment on the extensive back-counting investigation reported in Schlesinger (2005, pp. 344-369). This work, a collaboration with several other researchers including (alphabetically) John Auston, Chris Cummings, James Grosjean, Karel Janacek, Kim Lee and Richard Reid, differs from ours in a number of significant ways: • The analysis focuses entirely on ‘departure’ while arbitrarily fixing the entry threshold (p. 348) at a true count of +1. We instead find the optimal entry to be in the range of +2 to +2.5, with exit and departure optimized jointly; and we find that the effective yield is more sensitive to entry than to the other two parameters. The eighth lines of Table 1, Lifetimer and Weekender, give our results for entry at +1; comparison with the fourth lines shows that this guideline is suboptimal. • Although exit and departure are recognized to be distinct (see note inserted at p. 347), they are regarded in practice as identical. Thus, both are called by the same name – ‘departure’ – and identical numerical values are always quoted for them. • Exit/‘departure’ true count thresholds are dependent on depth, as they strictly should be. But the charts presented for optimal exit/‘departure’ are complicated; these two-parameter guidelines may be difficult to execute accurately under actual casino conditions. For instance, the charts show the true count departure point swinging from negative to substantially positive as the depth nears the reshuffle point. On the other hand, they also show departure as roughly depth-independent for lesser depths, supporting our simplification. • Only a cursory description is given of the methodology used to obtain the results, and they have not been published elsewhere. Although the description (p. 346) is of Lee’s recursive approach to the problem, some of the results (p. 353) are from a proprietary simulation code due to Grosjean. As a consequence, the work cannot be independently verified. • Careful attention is given to including a lag time between leaving one table and starting to observe the next, while we simplify the transition to be (unrealistically) instantaneous. On the other hand, if we had included some reasonable lag, the resulting effective yield would no doubt lie somewhere between that of one table only (equivalent to a lag of all the rounds left before its reshuffle) and that of two tables with zero lag. Since our results for entry and exit show only small differences between just one table and inclusion of a second, we conclude that our results are insensitive to the duration of the lag. • Results are given for both 6 and 8-deck games, dealt to penetrations of 75% or 83%, compared with our one example of 6 decks, 80% penetration. • Consideration is explicitly given to the table-hopping technique (dubbed the “White Rabbit”) of exit even while entering at a shuffle; it is found, as do we, to give unexpectedly good performance considering its simplicity of execution. Appendix A: Integration methods As a prototype for the various integrals evaluated here in closed form, we demonstrate methods for the chain rule, equation (5). Begin with the double-integral expression 11 1 τE ∞ Jdd≡τγργ−γτ−τργτργτ(,)(,)(,). (A1) ∫∫ 1EE E1 EE τE 0 −∞ After substitution from equations (1) and (2), the integrand can rearranged into the form 1()γτEE γ−γτ2 Jdd=γτEE ∫∫ 3 γπEE−∞ 0 2(τ−τ )τ (A2) ⎧⎫⎡⎤22⎡ ⎤ ⎪⎪11⎛⎞⎛⎞γγEE11 ⎛ 2γ−γγE ⎞⎛⎞11 ×−−−−−−−⎨⎬exp ⎢⎥⎜⎟⎜⎟exp ⎢ ⎜ ⎟⎜⎟⎥. 22ττ ττ τ τ ττ ⎪⎪⎩⎭⎢⎥⎣⎦⎝⎠⎝⎠EE⎣⎢ ⎝EE ⎠⎝⎠⎦⎥ Next, change integration variable: for the first term in the braces substitute 21−− 1 21−− 1 u =γ( τ −τE ), while for the second term substitute u '(2= γ−γτ−τEE )( ). Then γ 1(E ⎧⎫⎪⎪∞∞ββ⎛⎞uu+β)22' ⎛('+β')⎞ J=γ d⎨⎬ du exp⎜⎟ − −du ' exp ⎜ − ⎟, (A3) γ ∫∫ 3322uu∫ ' E −∞ ⎩⎭⎪⎪0022ππuu⎝⎠' ⎝ ⎠ where β≡(),'()(2) γ−γEE γ τ β ≡ γ E −γ γ E −γ τE. But ∞ β ⎛⎞()u +β 2 du exp −=−β−exp(β ) , (A4) ∫ 3 ⎜⎟ 0 2πu ⎝⎠2u as listed, e.g., in Mathematica or derivable via the further change of variable vu=+β() u. Then shift the γ integration to γ=−γ for the first term in the braces and to γ=γ−γ ' E in the second, revealing a major cancellation between the two terms while the un-cancelled part of the γ integral becomes trivial. The result is just J =1, QED. Appendix B: Entry and departure When analysing exit, which can only occur after entry, it suffices to account for the single boundary constraint γ≥γX by the device of an inversion about γ X , as illustrated in equation (9). However, in the case of departure, which occurs alternatively to entry, the two simultaneous boundary constraints γ≤γ≤γD E require the more complex structure of an infinite array of inversions. A real-world analogy is that if one looks in a mirror one sees a single reflection; but if one looks in a mirror with another one parallel and facing it, one sees infinitely many, receding reflections. The solution of the diffusion equation subject to the two boundary conditions is the generalisation of equation (2), ∞ ργτ=ED (,) ∑ ⎣⎦⎡⎤ ργ−−ργ−γ−()(aaamm 2E m ) , m ≡ 2 ( γ−γED ). (B1) m=−∞ Then the probability of departure at τD is given by the generalisation of the chain rule, equation (5), 1 τγEE ργτ=,,dd τγργ−γτ−τργτ, 11DDD()∫∫ ()()E E ED τE 0 γD (B2) ∞ γγ−EEγD = ργτ−()EE,,∑ ⎣⎡ ρ()(aam −γτ−ρ+γτ EE m EE,)⎦⎤; ττEEm=1 12 the integrations use the same methods as Appendix A, although the algebra is lengthier. Since the terms in the sum decrease exponentially with increasing m , a valid approximation is to keep just the m =1 contribution from the first term in the summand; the result is equation (14). The corresponding result for the distribution of γ at τ subsequent to entry is the integration from equation (15), resulting in ∞ γ−γ ργτ=ρΓτ−,,ED⎡ ργ−γ+−γτ−ργ+γ++γτaa, ,⎤ .(B3) ED () (E )∑ ⎣ ()E m E ()E m E ⎦ m=1 amE−γ Keeping just the leading m =1 term gives the approximate result in equation (15). Finally, the exact expression for the probability of departure is the converse of equation (B2), ∞ −γDEγ − γD PaDD()τ= ργτ−()DD,,∑ ⎣⎡ ρ()(m +γτ−ρ−γτ DDa m DD,)⎦⎤ , (B4) ττDDm=1 so the yield from another table following departure is proportional to τ ⎛⎞−γ∞ ⎡ γ − γ ⎛aa+γ ⎞γ − γ ⎛−γ ⎞⎤ dPττ=2erfDED − 2 erf mD−EDerf mD. (B5) ∫ DD() ⎜⎟∑ ⎢ ⎜ ⎟ ⎜ ⎟⎥ 0 ⎝⎠ττm=1 ⎣aamD+γ ⎝ ⎠mD−γ ⎝τ ⎠⎦ Keeping, as before, just the leading m =1 term in equations (B4) and (B5) gives the approximate equations (17) and (19), respectively. Acknowledgements It is a pleasure to recognize a series of communications with Kim Lee regarding his approach to the departure problem. Although we’ve apparently used different analytical techniques, I suspect we’re reached similar end-points. I’m grateful for his encouragement and guidance through a number of my conceptual difficulties. References Thorp, Edward O. 1962. Beat the Dealer: A Winning Strategy for the Game of Twenty- One, Vintage Press, New York. Schlesinger, Don 2005. Blackjack Attack: Playing the Pro’s Way (3rd edn.), RGE Publishing, Las Vegas. Vancura, Olaf and Ken Fuchs 1998. Knock-Out Blackjack, Huntington Press, Las Vegas. Werthamer, N. Richard 2005. ‘Optimal Betting in Casino Blackjack’, International Gambling Studies, 5(2), pp. 253-270. (Also viewable at the website http://home.nyc.rr.com/werthamer.) Werthamer, N. Richard 2006. ‘Optimal Betting in Casino Blackjack II: Back-counting’, International Gambling Studies, 6(2), pp. 111-122. (Also viewable op. cit.) Werthamer, N. Richard 2007. ‘Basic Strategy for Card-Counters: An Analytic Approach’ in Ethier, S.N., Cornelius, J.A., and Eadington, W.R. (eds.), Optimal Play: Mathematical Studies of Games and Gambling, Institute of the Study of Gambling and Commercial Gaming, University of Nevada, Reno (in press). (Also viewable op.cit.) Wong, Stanford 1975. Professional Blackjack, Pi Yee Press, La Jolla. (For the 1994 and later editions, this material instead appears in Wong’s Blackjack Secrets, Pi Yee Press, La Jolla.)