Non-Random Shuffles and Winning or Losing Streaks
FROM ET FAN:
Man vs. Computer: The Non-Random Shuffle and StreaksBy Dr. John M. Gwynn, Jr. and Arnold Snyder
(Blackjack Forum Vol. VIII #1, March 1988)
© Blackjack Forum 1988
(This is a synopsis of a paper on non-random shuffles and computer simulation of blackjack presented at the 7th International Conference on Gambling and Risk Takng.)
[Arnold Snyder is the author of The Blackjack Shuffle Tracker's Cookbook: How Players Win (and Why They Lose) With Shuffle-Tracking.]
It has been public knowledge for more than 25 years now that the game of blackjack can be profitably exploited by players who diligently apply strategies based on card counting. First popularized by Dr. Edward Thorp's 1962 best-seller, Beat the Dealer, card-counting strategies have been expounded upon, refined, and extolled by dozens of authors since. The validity of card counting has been proven many times via computer simulation of the game.
In the past five years, however, some of the standard computer blackjack simulation techniques have been seriously questioned by a number of authors, players, and system sellers. Most authorities on the game believe that the blackjack simulation methods generally employed to test systems are applicable to the game as dealt in the casino, even though all casino procedures are not simulated. The prime complaint of those who question the simulation data concerns the methodology of computer "shuffling." In a computer simulation, the most time-efficient method of shuffling is to place cards in order via a "random number generator" that assigns each card in the deck or decks a new randomly chosen position with each shuffle.
Detractors of this methodology complain that in the casino shuffle — as applied by less than perfect human dealers — "clumps" of cards often remain together from shuffle to shuffle due to "lopsided" cuts and sloppy riffles; other procedures, such as the order in which dealers pick up played hands, tend to clump high cards with high cards in the discard rack; etc. Indeed, casino shuffles are not random. The most recent work on shuffling indicates that seven riffles are required to randomly rearrange a deck of cards.
In the past five years, numerous new blackjack systems based on these non-random shuffles have appeared. Players are being advised by some system sellers to avoid certain types of shuffles while seeking out others. Non-card-counting systems for beating the game are also being extolled as valid by some system sellers due to these non-random shuffles. Strategies based on "streaks" of wins and losses, allegedly caused directly or indirectly by the non-random shuffles, are also being sold.
Unfortunately, none of these new systems and methods has been accompanied by any published computer simulation data that would support the system sellers' claims. Nor is any logical mathematical theory presented that would support the validity of these new systems. Some of these systems inventors have claimed they have personally produced non-random blackjack computer simulations that have proven their systems to be valid, but none have published any hard data to support this claim.
Most of the non-random shuffle systems are primarily touted to be used in multiple-deck blackjack games. In this paper, we will limit our computer simulations and analyses to single-deck blackjack games. This is due solely to the time factor involved. It is our intention to continue testing many more of these non-random theories, claims, and systems in both single-deck and multiple-deck games; the results will be published as obtained.
Analyzing Live Casino Non-Random Shuffles
In order to write a computer simulation program that would mimic the shuffle of a human blackjack dealer, it was first necessary to analyze the types of shuffles that dealers in casinos employ.
The common casino shuffling procedures were all provided by Las Vegas author Steve Forte. Mr. Forte at one time owned and operated a professional casino dealing school in Las Vegas after having spent some years as a blackjack dealer, pit boss, and casino manager. In 1986 he personally surveyed all of the major casinos in Nevada; at this time he recorded each casino's standard series of cuts, breaks, riffles, strips, discard procedures, etc. in an effort to devise various "shuffle tracking" and "card location" strategies.
"Shuffle tracking" and "card location" are advanced strategies, based on card-counting theory, and are most often employed in conjunction with card counting. These tracking and location strategies do take advantage of "non-random" human shuffles. A player applying one of these techniques might do so by first breaking down and analyzing a casino's shuffling routine so as to learn how to follow the positions of specified cards in the discard rack when they are reshuffled. Although little has been written on card tracking of this type, this will not be a topic of study for this paper. In fact, the data from the simulations sheds no light on this type of strategy.
The purpose of this study is to simulate non-random shuffles and attempt to answer the following questions. Does a "poor" shuffle have a notable effect on the basic strategy player's win/loss rate? Does it cause lengthier and/or more frequent "streaks" of wins/losses? Does it lower the effectiveness of card counting in determining favorable hands? Are there "biases," independent of any conventional card counting information, induced by certain non-random shuffles, i.e., positive counts no longer indicate a player advantage because the deck is "dealer biased?" Are certain shuffles disadvantageous to players? Et cetera.
In short, how does nonrandom shuffling affect the game of blackjack? In this paper, we initially chose to simulate two of the most common single-deck shuffles, Riffle-Riffle-Strip-Riffle and Riffle-Strip-Riffle-Riffle. Mr. Forte indicated that most casinos require one of these as a minimum single-deck shuffling procedure. It was then decided to add Riffle-Riffle-Riffle, a procedure that would be used if the dealer omitted the intervening strip; this would mix the cards a little less.
In 1987, Anthony Curtis obtained data from four Las Vegas dealers to determine how they broke the deck for a riffle and how well their riffling mixed the cards. He asked each dealer to perform one riffle on a new deck; then he recorded data about the riffled deck. After each riffle, the deck was reordered and the dealer riffled again. Each dealer performed ten such riffles. Curtis reported the fol lowing data about these 40 riffles.
There were 906 singly interlaced cards, 358 2-card clumps, 72 3-card clumps (less than 2 per riffle), 28 4-card clumps (less than I per riffle), 10 5-card clumps (I per 4 riffles), and only two clumps larger than five cards (I per 20 riffles). On 33 out of the 40 trials, the break was within 2 cards of the center. Three breaks were 3 cards from the center, one break was 4 cards off center, two breaks were 5 cards off center, and one break was 7 cards off center (this break resulted in a 7-card clump after the riffle). Since dealers always break and riffle more than once during a shuffle, it is unlikely that any large clump would be preserved throughout a complete single-deck shuffle.
Richard Epstein, author of The Theory of Gambling and Statistical Logic, stated that when professional dealers riffle the cards, they drop one card 80% of the time, two cards 18% of the time, and three or more cards only 2% of the time. Applying Epstein's distribution to the same 40 breaks that Curtis reported on, a little approximation yields around 1100 singly interlaced cards, about 250 2-card clumps, and nearly 30 clumps of 3 or more cards. The Curtis data, small sample that it may be, indicates less precise riffling that Epstein reported.
This research deals entirely with single-deck blackjack under the further assumptions that a single player goes head up against the dealer and plays a single hand. The authors are aware of the non-random shuffle theory that multiple players at the table have a sianificant effect on the subsequent non-random ordering of the cards. Again, it is our intention to test the effect of multiple players, and to publish these results as obtained. This initial study is a starting point, not a definitive evaluation of all non-random theories. All simulatians used Las Vegas Strip rules: dealer stood on soft 17; doubling was permitted on any first two cards (not offer splits); only one card could be drawn to each split ace; aces could not be resplit.
Basic strategy governed the play of all hands, and insurance was never taken. The High-Low count was kept for certain measurements, but it was not used to vary strategy. The number of split hands was limited to four because a single deck was used and tens were never split. In this description, the player is identified as male and the dealer as female and right-handed.
Every attempt was made to come up with a realistic model of actual casino play, where the order of cards in the deck is possibly nonrandom. Variables that might influence this order included: was the dealer's up card dealt before or after her down card; in what order did she pick up the player's cards when he did not split a pair; did she pick up the player's split hands clockwise or counter-clockwise; were the player's cards or her cards placed in the discard rack first; in what order did she pick up her own cards; what rules governed reshuffling; did she place the unplayed cards on top of or below the cards from the discard rack when preparing to reshuffle; how did the player cut the shuffled pack before she dealt from it. A myriad of combinations was possible if one ran the gamut of reasonable choices; hence, a single realistic choice was made for each of the above variables.
When the dealer picked up the cards used in any round, she placed all of the player's cards in the discard rack before her own. In case a pair was split, she dealt the split hands left-to-right (clockwise) but picked up the unbusted split hands right-to-left (counter-clockwise). If any split hand busted, she placed its cards in the discard rack before play continued.
With these fixed, only the shuffling technique differed from simulation to simulation; for any one simulation, the same technique was used throughout. Seven different shuffles were selected for use in the simulations:
Three of these shuffling techniques (RRR, RSRR, RRSR) attempted to simulate casino dealer behavior; three others were obviously inadequate (NOSHUF, I PR, 2PR).
The reshuffling rules reflected dealing through 3/4 of every deck. Immediately after each shuffle, the player always cut the cards close to the center of the deck; then the dealer burned the top card by placing it in the discard rack without the player seeing it.
To begin each simulation, the deck was randomly ordered before an initial shuffle; this guaranteed no dependency on the original deck order. Hence, the simulations would only measure the differences between random and other shuffles after the cards were thoroughly mixed. Many non-random shuffle system proponents claim that the continual introduction of cards in "new deck order" is a major cause of the subsequent non-random ordering of the cards via imperfect shuffles. The authors intend to test this effect, and to publish these results as obtained.
Eight simulations were run, one using each type of shuffle and an additional one using the random shuffle. To avoid confusion, the two random simulations were denoted RANI and RAN2. Each simulation dealt 2,650,000 freshly shuffled decks, resulting in over 20,000,000 hands played (a pair split into two or more hands only counted as a single hand played). During each simulation, numerous data were gathered for later analysis; hopefully, this data would provide enough information to determine if random and simulated dealer shuffling produced significant differences in the game of blackjack.
The reason for running two simulations with a random shuffle was as follows. If the gathered data seemed to indicate any significant "difference" between two shuffling methods, the data from RAN1 and RAN2 could be compared to see if the same "differences" appeared. If so, this observed "difference" could be deemed spurious and simply the result of random fluctuations.
Results of the Non-Random Shuffle Simulations
Table I shows the results of the simulations for a flat bet of one unit, giving the total number of hands played, expectation per hand played, and expectation per unit bet. All expectations are given in percent; in addition, all expectations are per hand played unless specifically stated to be per unit bet.
The results for no shuffling agree approximately with those of Stanford Wong, who found that no shuffling gave an increase of about 0.75% over basic strategy for the 6-deck Atlantic City game. The results for 2PR are a mystery. Two standard errors for the difference in expectations between two runs is about 0 07% and 2PR's differs from others (excluding NOSHUF) by a little more than this. Extensive analysis of the voluminous gathered data led to the following conclusion: within the limitations of this study, point count basic strategy players cannot telI the difference between simulated dealer shuffling and random shuffling.
Point count systems presume that the unseen portion of the deck contains an essentially random mix of cards. The question here is whether non-random shuffles induce sufficiently many patterns in the unseen cards to perturb a count system's ability to determine which hands are favorable, and, more precisely, how favorable.
Based on the three runs using simulated dealer shuffles (RRR, RSRR, RRSR) and the two runs employing random shuffles (RAN1, RAN2), the High-Low count was an equally effective indicator of player advantage.
HypotheticalIy, assume that the player can sit at the table and not play some hands, i.e., bet 0. If the player bets 0 on the unfavorable hands and 1 on the favorable ones, he will obviously win. The player uses a point count to help decide which hands are favorable; there is a number K such that a true count K indicates favorability and a true count
Table 1. Flat Bet Results for the Eight Simulation Runs
Table 2. Per Hand 0-1 Betting Expectation as a Function of K
For each integer K from -I to 8, the difference in per hand 0-1 betting expectations was examined for all ten pairs of the simulations RAN1, RAN2, RSRR, RRSR, and RRR. None of these 100 expectation differences was significant at the 95% level. These results strongly imply that a point count system is just as effective for casino shuffling as it is in theory, i.e., when random shuffling is employed. For all eight of the simulations, Table 2 shows the observed 0- I betting expectation. For NOSHUF, the 0-1 betting expectations showed the greatest differences when compared to the other runs.
In the simulations, the player always rounded the true count to the nearest integer before using it to make a bet. In determining the player's expectation per hand, the number of hands played included those on which 0 was bet.
If one plots the expectation per hand at integer values of K, the function increases to a maximum and then decreases. It is conceivable that the maximum might not occur at the same integer value of K for all of the simulations runs, especialIy since non-random shuffling (which includes no shuffling) was used in six of them. However, the function's maximum occurred at the same value, K = 0, for every simulation except NOSHUF, for which it occurred at K = -1.
The hands were then partitioned into three classes: hands won, hands lost, and hands pushed. A Chi Square test with two degrees of freedom was used to compare the frequencies of hands in these three classes between pairs of simulations. At the 95% level of confidence, there was no significant difference between any two of RAN1, RAN2, RSRR, RRSR, and RRR. Hence, the frequencies of wins, losses, and pushes did not vary significantly between random shuffling and any of the three simulated casino shuffles.
Whereas the frequencies of wins, losses, and pushes seemed to be useful in measuring the differences between two shuffling procedures, the frequencies of hands dealt with positive counts, with negative counts, and with zero counts did not. Most important to the player are his expectations on hands in each of the three categories. If one considers only the hands dealt with positive counts, the per hand expectations did not differ significantly (at the 95% level) between any two of the simulations RAN1, RAN2, RRR, RSRR, RRSR; this was also the case for the difference in per unit expectations for these hands. Expectations for hands dealt with negative counts and for hands dealt with zero counts exhibited identical behavior.
Suppose the hands are partitioned into nine categories: wins with a positive count, wins with a negative count, wins with a zero count, losses with a positive count, losses with a negative count, losses with a zero count, pushes with a positive count, pushes with a negative count, and pushes with a zero count.
A Chi Square test with eight degrees of freedom was used to compare the frequencies of hands in the nine categories between each pair of simulations. For RAN1 versus RAN2, RAN1 versus RSRR, and RAN 2 versus RSRR, the difference was not significant at the 95% level; for the other 25 pairs of simulations, each difference was significant at the 99.5% level. It seems that considerable fluctuations are to be expected in the frequencies of hands in these categories.
As for expectations, however, there was little variation in each of the nine categories if one confined the discussion to RAN I, RAN2, RSRR, RRSR, and RRR. Clearly, hands in each of the three push categories always had expectation 0. For each of the other six categories, the differences in per hand and per unit expectations were examined for all ten pairs of these five simulations. Based on expectation per unit, none of these 60 differences was significant at the 95% level. If RRR was excluded from the list, the per hand expectations for hands in any category also did not significantly differ (at the 95% level) between any pair of the remaining four simulations.
Table 3. Percentages and Expectations of Hands Won and Lost with Positive, Negative and Zero Counts
For the five simulations using random and simulated casino shuffles, Table 3 shows the percentages (of all 20+ million hands) in each of the six non-push categories as well as the per hand and per unit expectations for these hands.
Some players contend that their casino observations indicate a higher percentage of player losses on positive counts than theory predicts. After examining Table 3, such players might feel vindicated by pointing out that RRSR showed a higher percentage of losses on positive counts than RAN1. If they ignored the fact that the expectations (both per hand and per unit) on these hands did not differ significantly at the 95% level, they would be disappointed that RRSR's losses on these hands were less than RAN1's in spite of RRSR's higher percentage of such hands.
This example has additional implications concerning casino observations. Between RRSR and RAN1, the difference in percentages of hands lost on negative counts is only 0.06% For such a small difference to be significant at the 95% level, the player would have to observe over 2 million total hands, assuming RAN1's percentage of losses on positive counts is the theoretical one. It seems doubtful that any player has made such extensive casino observations; even if some player has done so, it is even more unlikely that he kept accurate enough records to note this small difference.
The results for hands in the nine categories strongly imply that the simulated dealer shuffles RSRR, RRSR, and RRR do not perturb the per unit expectations of hands won or lost on positive, negative, or zero counts. They further strengthen an earlier statement that the count provides the same indication of the player's expectation for these simulated casino shuffles as it does for random shuffling. Also, since the per unit expectations of the hands in each category do not differ significantly in spite of the variations in the frequencies, any perturbations in the frequencies of the nine categories probably cannot be exploited.
Non-Random Shuffles and Winning/Losing Streaks
Analysis of the data on winning and losing streaks indicates no significant differences between simulated dealer shuffling and random shuffling. The analysis will be confined to streaks of 1, 2, ..., 9 consecutive losses and I, 2, ..., 9 consecutive wins. For the purposes of this study, a streak of consecutive wins encompasses all hands from the first win and continues to (but does not include) the next loss. This means that embedded or following pushes are included as hands in the streak. Hence, Win-Push-Win-Win-Push is defined as a streak of three consecutive wins during which five hands were played. Analogously, a streak of consecutive losses begins with the first loss, continues to but does not include the next win, and includes embedded and trailing pushes.
The first test involved the frequencies of I, 2, ..., 9 consecutive losses and of 1, 2, ..., 9 wins. Each frequency was first divided by the total number of streaks to obtain a probability. Then a Z test was used to compare corresponding probabilities (one for each of the 18 different types of streaks) for all 28 pairs of simulations.
Comparing each of RAN1, RAN2, RRR, RSRR, and RRSR with all of the other four runs required ten sets of 18 tests each. Of these 180 tests, one streak differed significantly at the 99% level, another at the 98% level, and two more at the 95% level; these four differences occurred in testing four different pairs of simulations. The remaining 176 tests did not differ significantly at the 95% level. Hence, apart from expected statistical variation, the streak probabilities did not differ at the 95% level between any two runs.
These results strongly indicate that streaks occur with the same probability whether casino shuffling or random shuffling is employed. However, one fact seems obvious: the less random the shuffle, the more the differences in streak probabilities.
Table 4, shows some of the win-loss streak data from RAN1. This will give an approximate indication of what the player might expect to see during actual play. Negative streak lengths indicate losing streaks and positive streak lengths indicate winning streaks. The percent of streaks is the number of streaks of this length divided by the total number of streaks, which was 9,170,446 in this simulation.
Hands is the total number of hands played, including pushes, during streaks of this length. Hands per streak is column 4 divided by column 2. Expectation per hand is the net win/loss divided by the total number of hands played during streaks of this length (column 4). Expectation per unit bet is the net win/loss divided by the total money bet on all streaks of this length.
Table 4. Streak Data for RAN1
Excepting streaks of length 1, there were always more losing streaks than winning streaks of any length; as the length increased, the ratio of losing streaks to winning streaks increased. In short, the player should expect considerably more losing than winning streaks. Since the table is truncated, the following data may be of interest.
There were 7144 streaks of 11 or more consecutive losses and 2729 of 11 or more consecutive wins. The longest number of consecutive losses was 23 and the longest number of consecutive wins was 21.
Similar to streaks of consecutive wins and losses, a streak of consecutive positive counts encompasses all hands from the first positive count and continues to (but does not include) the next negative count; embedded or following zero counts are included as hands in the streak. An analogous definition of streaks of consecutive negative counts also makes embedded and following zero counts part of the streak.
With minor exception, the authors could not find the same patterns in streaks of positive and negative counts that appeared in streaks of wins and losses. Even between two random runs there were significant differences (at the 95% level) between streak probabilities. These probabilities seem to show considerable fluctuation and are probably not a good measure of a shuffle's adequacy. This is not surprising since the frequencies of hands with positive, negative, and zero counts also do not seem to provide a reliable measure.
Table 5. Numbers of Streaks with 30 or More Like Counts
The minor exceptions had to do with the numbers of streaks of length 30 or more. The numbers of such streaks were enormous for no shuffling and decreased rapidly to reasonable values as the amount of shuffling increased. This implies that grossly imperfect shuffling increases the probability of long streaks of like counts.
However, it is not clear that the long streak probabilities provide a sufficiently fine measure of the shuffle's adequacy. There were significant differences (at the 95% level) in the frequencies of long negative streaks between RAN1 and RRR and between RAN2 and RRR.
In addition, there were significant differences in the frequencies of long positive streaks between RRSR and RSRR at the 98% level, between RRSR and RAN2 at the 99% level, and between RRSR and RRR at the 99% level. Table 5 shows the numbers of such streaks for each of the simulations. NEG indicates streaks of 30 or more hands with negative counts; POS indicates streaks of 30 or more hands with positive counts.
Data was gathered to allow computation of the correlation between outcomes of consecutive decks. After the play of each deck was completed, three accumulations took place. First, the mean expectation of this deck's hands was computed and added to the sum of the deck means. Second, the square of this deck's mean was added to the sum of squares of deck means. Finally, this deck's mean was multiplied by the mean of the previous deck and added to the sum of consecutive deck products.
Intuitively, one would expect a statistically insignificant correlation for random shuffling and a statistically significant one for no shuffling; this was indeed the case. Table 6 shows the correlation coefficients for all seven shuffling methods; the note below each coefficient tells whether or not it was significantly different from zero (SIG or NOT, respectively) and at what level. The table values were computed using expectation per hand played; those obtained using expectation per unit bet were similar.
The major conclusion is that for each of the simulated dealer shuffles, there was no significant correlation between the outcome of one deck and the outcome of the next. The lack of any significant correlation between consecutive decks adds to the evidence that each of RSRR, RRSR, and RRR is an adequate shuffle.
Table 6. Correlations for Outcomes of Consecutive Decks
The correlation for NOSHUF and 1PR are perfect examples of values that differ significantly from zero in the statistical sense but are otherwise insignificant. Even for NOSHUF, whose coefficient is larger, the correlation between consecutive deck outcomes seems too small to be exploitable.
For single-deck play, two projects immediately came to mind as natural extensions of this research. The first would be to rerun the three simulated casino shuffles with a new deck being introduced every 100-200 hands. Of course, each new deck would be introduced in its usual order, and a simulated new-deck shuffle would precede dealing the first hand. It is conceivable that hands dealt after the initial and next two or three shuffles might have different characteristics than those dealt after more shuffIing has taken place.
A second interesting project would be to use a less precise riffle in RSRR, RRSR, and RRR. Such a riffle would be governed by a distribution with a decreased frequency of dropping single cards and elevated frequencies of dropping two or more cards; Anthony Curtis' observations could be used to construct such a distribution. Simulations using this distribution for riffles would be run both with and without introducing a new deck every 100-200 rounds.
Acknowledgements: We owe much to Peter Griffin, who provided invaluable guidance toward proper statistical analysis of the gathered data and made several suggestions that significantly improved the paper.
We are also deeply indebted to Steve Forte, who isolated and explained each of the common shuffle techniques and procedures as employed in Nevada and Atlantic City casinos (even though we have not yet used the Atlantic City data).
In addition, we thank Anthony Curtis, who got four professional dealers to shuffle for him and provide new first hand data about how dealer riffling mixes the cards. ♠Return to Blackjack Forum Professional Gambling Library
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||Non-Random Shuffles and Winning/Losing Streaks
This Blackjack Forum article compares non-random casino shuffles to random computer simulation shuffles to study the effects of non-random shuffles on results of blackjack gambling systems, including card counting. While players can use professional gambling methods like shuffle tracking to exploit non-random shuffles on casino blackjack games, but found no evidence to support that computer shuffling produces different results for other types of gambling systems. This paper on non-random shuffles and computer simulation of blackjack was presented at the 7th International Conference on Gambling and Risk Taking by Arnold Snyder and John Gwynn.