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 .