Investigation and simulation of card counting methods

Transcription

1 Registration number 59 Investigation and simulation of card counting methods Supervised by Dr Ben Milner University of East Anglia Faculty of Science School of Computing Sciences

2 Abstract This project is an investigation into Blackjack, concentrated on investigating a variety of known card counting techniques and their impact on the game. Furthermore investigating the impact deck size has into card counting techniques. This investigation is driven by a simulation of card counting using the Java programming language to test a large number of Blackjack shoe games in order to obtain accurate data to assess the advantage that is provided to the player from card counting.

3 Contents Contents 1. Introduction 1 2. Basic Blackjack Terminology Play Rule variations Basic strategy 4 4. Card counting Terminology Card counting strategies Hi Lo Red Seven Uston Strongest and Simplest (Uston SS) Uston Advanced Point Count (APC) Keep It Short and Simple 3 (KISS 3) Zen Count Illustrious Related Work Implementation Results Strategy Mimic the Dealer Basic Strategy without doubling down or splitting Basic Strategy with either doubling down or splitting Complete basic strategy Reg: 59 iii

4 Contents 7.2. Card counting with one deck Hi Lo Zen Count Uston APC Red Seven KISS Uston SS Conclusion Card counting with six decks Hi Lo Zen Count Uston APC Red Seven KISS Uston SS Conclusion Card counting with eight decks Hi Lo Zen Count Uston APC Red Seven KISS Uston SS Conclusion References 23 A. Appendix 25 Reg: 59 iv

5 List of Figures List of Figures A.1. Comparison of Blackjack Strategies A.2. Comparison of card counting techniques used on a single deck A.3. Comparison of card counting techniques used on six deck A.4. Comparison of card counting techniques used on eight deck A.5. Hi Lo comparison of one, six and eight decks A.6. Red Seven comparison of one, six and eight decks A.7. Uston SS comparison of one, six and eight decks A.8. Kiss 3 comparison of one, six and eight decks A.9. Zen Count comparison of one, six and eight decks A.10.Uston APC comparison of one, six and eight decks Reg: 59 v

6 List of Tables List of Tables 2.1. The value given to each card in Blackjack A table of rule variations and if they are favourable to the player Basic stratergy based on single deck, the dealer stands on soft 17 and you cannot double down after splitting Hi Lo card counting values Red Seven card counting values Uston SS card counting values Uston APC card counting values Kiss 3 card counting values Zen Count card counting values Table of average loss or gain of a Blackjack Strategy Table of average balance and profit while card counting with one deck Table of average balance and profit while card counting with six decks Table of average balance and profit while card counting with eight decks 21 A.1. A table of card counting strategies A.2. The Illustrious A.3. Comparison of all averages for strategies and counting techniques Reg: 59 vi

7 1. Introduction 1. Introduction Blackjack is a popular casino game that involves the player trying to obtain a hand value closer to 21 than the dealer. Players bet before the hands are dealt and should they beat the dealer they will be paid for winning. Card counting was invented to help combat the advantage the dealer has over the players.this project intends to investigate and simulate card counting techniques to determine how beneficial these techniques are. Card counting is the broad term to describe strategies used to aid the player in determining if the next hand will give an advantage. Most strategies are based on the assumption that high value cards are good for the player and low values cards are bad. (HomePokerGames, 2003) By having higher value cards in the deck this increase the probability that a dealer will draw a high value card while playing and bust. The count gives an indication to the chance that the dealer will draw a high value card and bust. This is the advantage card counting provides the player. By taking note of played cards the player can alter bets and play style in order to maximum their potential gain. The point of card counting is to enable the player to have a long term statistical advantage at blackjack and balance the advantage the casino will have. This project will perform an investigation into Blackjack and a selection of well known card counting techniques. Using a simulation created in the Java programing language a large number of shoe games can be simulated to obtain an accurate set of data which can then be evaluated to show the advantage gained by card counting. 2. Basic Blackjack Blackjack is a popular casino game believed to have originated in French casinos around the year 1700 when is was called "Vingt-un" which means 21. Though there is also reason to believe it may have been created based on the Italian game "Sette e Mezzo" which translates to seven and half. Or that it was based on the Spanish game "Trente-un" meaning 31. It is likely an amalgamation of all the aforementioned games(blackjackapprenticeship.com, Reg: 59 1

8 2. Basic Blackjack 2014). This section will explain some common terminology used in the game Blackjack as well as describing a normal round of play Terminology Blackjack has its own vocabulary associated with it so these terms will help make sense of language mentioned thought out the report. Hit - Player takes another card from the deck Stand - The play takes no more cards, also known as "stick" or "stay" Blackjack - A Blackjack is when the players opening hand is a card of value ten and an ace. Generally the player is paid his original bet plus 50%(3:2). Alternatively if the dealer has Blackjack then the player loses their original bet. If both the player and the dealer have blackjack then the hand is a standoff or a "push". Split - This can occur when you re opening cards are the same value. The player can treat each cards as a new hand and double his bet. Double down - The player can increase his bet to twice his original bet and take exactly one more card. Surrender - The player can forfeit half the bet and give up on the hand. This option isn t always available and varies from casino to casino. Insurance - If the dealer s up card is an ace the player may make a side bet known as "insurance". The bet can be as much as half the player s current bet. If the dealer has Blackjack then the player s insurance bet is paid with two to one odds. Bust - The player s cards exceed the value of 21. The player is out of the round and loses their bet. Hole card - The dealer s face down card. In most European games this card isn t drawn until player s have made their choices. Reg: 59 2

9 2. Basic Blackjack Up card - Dealer s face up card. The shoe - Contains the decks where the cards are dealt from. Each card in the deck has a value. Two to ten are worth the value shown on the card. Jack, queen, king are also worth ten. Ace can be worth one or 11, a hand made from an ace being used as 11 is often called a "soft" hand. For example an ace and a six is a "soft" Table 2.1: The value given to each card in Blackjack Card A Ten Jack Queen King Value 11 or Play A normal round of Blackjack starts with the player s initial bets. The dealer will give each player, including them self, two cards. The dealer will have one of their two cards face up providing the players with the key bit of information to play out their hand. The dealer will then move round the table and the players will play out their hand by either standing, splitting, doubling down, surrendering, hitting or going bust. Any players with "Blackjack" automatically win unless the dealer has "Blackjack" as well. Unless the dealer has an ace in which case the player can take even money or wait. If the dealer hits "Blackjack" then result is a push. After the player s have made all their decisions the dealer will flip their hole card and hit until they have at least 17. If the dealer busts all remaining players win and are paid out. Players who push with the dealer will receive back their original bet.(graham Kendall, 2003) 2.3. Rule variations Depending on casino the house may vary the rules of Blackjack slightly. Some common variations to the rules are: Reg: 59 3

10 3. Basic strategy Dealer soft 17 - Some games make the dealer hit on a soft 17 and stand on all hard 17 s. This rule gives more favour to the house. Number of decks - Games of blackjack can have shoes containing one, two, four, six or up to eight decks. The less decks being used the more favourable the game is to the card counter. Payout for Blackjack - Different casinos can offer different odds to players who acquire Blackjack. The standard is 3:2. But some casinos only offer 6:5 or 1:1 on Blackjack increasing the houses edge over the player quite considerably. Lance Humble and Carl Cooper produced a table outlining which rules helped or hindered the player. 2.2 Table 2.2: A table of rule variations and if they are favourable to the player Favourable rule Unfavourable rule Early surrender Hard doubling restricted to 11 Doubling on any number of cards Two or more decks Doubling on any three cards Dealer hits soft 17 Doubling on any two cards Hard doubling restricted to 10 or 11 Drawing any number of cards to split Aces No soft doubling Surrender No dealer hole card Insurance No re splitting of Aces Adapted from (Lance Humble, 1998) 3. Basic strategy When the following conditions are met it has been proven that there can only be one correct way to play out a hand (Dalton, 1993): 1. The player has knowledge of the dealer s up-card. 2. The player has knowledge of his own hand. Reg: 59 4

11 4. Card counting 3. The player has no additional knowledge of the situation. 4. The remaining cards in the deck or shoe are randomly distributed. This providers a player with the optimum way to play Blackjack before card counting is applied. Once card counting is applied based on the count the optimum way to play will change.(wong, 1993) An example table of basic strategy can be found here 3.1 Due to the mathematical nature of Blackjack these strategy tables are considered to be the optimal way to play Blackjack. Other strategies have been created such as: Never bust - This involves the player never hitting a hand of 12 or larger to prevent busting. Every other choice is based on basic strategy. Mimic the dealer- This strategy involves the player playing exactly like the dealer. They will never double or split as the dealer does not. The player will stand with hard 17 or more. Assume a ten in the hole - The player plays the optimal basic strategy under the assumption the dealer s face down card is 10 unless the face up card is an ace. These are valid ways to play Blackjack and could be tested, although they are considered inferior to basic strategy.(wizardofodds, 2013) 4. Card counting Modern card counting is believed to originate from Edward O. Thorp and his book "Beat the Dealer" first published in This book wrote about a very basic strategy known as ten-count. This book caused casinos to alter shoe sizes to two or four decks in order to combat the methods that had been exposed though "Beat the Dealer". After casinos changed rule in order to restore a house edge a revised edition of "Beat the Dealer" was released in 1966 involving a counting system known as Hi Lo which had been developed by Harvey Dubner. (blackjackapprenticeship.com, 2014) This constant struggle is what pushes people to develop improved card counting techniques. While casinos introduce new rules and larger shoes of six and eight decks in order to entice players to try and beat the game. Reg: 59 5

12 4. Card counting Table 3.1: Basic stratergy based on single deck, the dealer stands on soft 17 and you cannot double down after splitting Player s hand Up card T A 5,3 H H H D D H H H H H 9 D D D D D H H H H H 10 D D D D D D D D H H 11 D D D D D D D D D D 12 H H S S S H H H H H 13 S S S S S H H H H H 14 S S S S S H H H H H 15 S S S S S H H H H H 16 S S S S S H H H H H A,2 H H D D D H H H H H A,3 H H D D D H H H H H A,4 H H D D D H H H H H A,5 H H D D D H H H H H A,6 D D D D D H H H H H A,7 S D D D D S S H H S A,8 S S S S D S S S S S A,A P P P P P P P P P P 2,2 H P P P P P H H H H 3,3 H H P P P P H H H H 4,4 H H H D D H H H H H 6,6 P P P P P H H H H H 7,7 P P P P P P H H S H 8,8 P P P P P P P P P P 9,9 P P P P P S P P S S T,T S S S S S S S S S S H = Hit S = Stand D = Double P = Split The player will hit on any hand value below 8. Adapted from (Dalton, 1993). Reg: 59 6

13 4. Card counting 4.1. Terminology Just like Blackjack has its own terminology, card counting has its own vocabulary too. These are a list of the common terms used in association with card counting. Balanced - A type of card counting technique where the count begins at zero and after all cards are dealt the count will be zero. Usually involves a truecount. Unbalanced - A type of card counting technique that after all cards in a deck have been dealt the count will not be zero. Does not usually make use of a truecount. Running count - A running sum kept as cards are seen. Systems only involving a running count are usually easier. Truecount - A true count is produced by using a method to convert the running sum. A true count could be produced by multiplying, tables, division and is used mainly for betting choices. Truecount strategies are usually more accurate but are harder to use. Side counts - Sometimes a strategy may make use of a side count. A common side count in strategies is used to count Ace s. Deck Penetration - Normally a percentage, refers to the depth reached though the deck Card counting strategies The main difference between card counting is balanced strategies and unbalanced strategies. Balanced strategies are considered to be more accurate but are usually much more difficult to use. Balanced strategies are commonly used in true count systems. Furthermore all the points values sum to the value of zero(qfit, 2013). Unbalanced strategies points values do not sum to zero making betting strategies harder. The advantage to unbalanced strategies is they can be used in easier running count systems. Some strategies such as K-O are only shown to be marginally less successful(olaf Vancura PhD, 1998). Reg: 59 7

14 4. Card counting A balanced strategy usually starts a new deck with a count of zero and makes use of a truecount. True count is usually derived by dividing the running count by the number of full or half full decks remaining. When dividing by full deck this makes betting much more accurate while being harder to compute while playing. Some strategies such as Hi-Lo Lite divide by quarter decks remaining but this makes betting more inaccurate. In contrast unbalanced strategies do not make use of truecount and instead just use the running count to assess how favourable the deck is. This can result in a loss of accuracy so to counter act this the count for a new deck is usually adjusted. An example is Red Seven which multiplies the number of decks in the shoe by -2 and assigns that as the starting count. This makes unbalanced strategies much easier to learn but in large shoe games can affect the accuracy of the method. The primary focus of card counting is to provide a player with a long term advantage over the house at the game of Blackjack. It achieves this by helping the player gauge when the correct time to bet more is. Quite a large number of studies into card counting and Blackjack look at optimizing betting.(werthamer, 2005) Table A.1 shows a large number of card counting strategies. This project will not be looking at all of these card counting techniques but a variety of level one, two, three, balanced, unbalanced and suit-specific counts. The list of counting techniques I have chosen to investigate are: Hi Lo 4.1 Hi Lo is an easy balanced technique created by Harvey Dubner. The Hi Lo strategy has been covered by numerous people such as Edward O. Thorp in "Beat the Dealer", Lawrence Revere in "Playing Blackjack", Julian Braun in "How to Play Winning Blackjack" and Stanford Wong in "Professional Blackjack"(Wong, 1994). As the name of the system would suggest this counting technique focuses on just the high and low numbers disregarding seven, eight and nine. As this is balanced the count will start at zero and if every card is dealt the count will be zero again. The count also makes used of a truecount which can be the hardest thing for a beginner at card counting. The truecount is derived by dividing the running count by the remaining decks. Betting is establishing based on the truecount.(wong, 1994) Reg: 59 8

15 4. Card counting Table 4.1: Hi Lo card counting values A T Hi-Lo Red Seven 4.2 Red Seven is an unbalanced technique involving a suit-specific count with sevens created by Arnold Snyder(Snyder, 2005). The unique part of the Red Seven count is the suit-specific count used on the sevens. Black sevens are left as zero while red sevens are given the value of plus one. It should be noted that the sevens can be given a value of +0.5 instead and should have no impact on the overall effectiveness of the card counting technique. These values assigned to the sevens give the counting system it s unbalanced nature. Due to the unbalance in this counting system the starting count has to be altered to accommodate. This is done by multiplying the number of decks by -2, for example if their are six decks being used then the starting count will be -12. The simulation will be using the version of Red Seven that counts red sevens as +1 and leaves black sevens as zero. Red Seven claims to achieve 80% of the potential gain of the Hi Lo strategy.(snyder, 2005) Table 4.2: Red Seven card counting values A Red 7 Black T Red Seven Uston Strongest and Simplest (Uston SS) 4.3 Uston SS is a level three unbalanced card counting technique created by Ken Uston in the 1980 s(uston, 2009). Like the Red Seven count the starting count is dependent on the number of decks. Instead of multiplying by -2 the count is multiplied by -4. Reg: 59 9

16 4. Card counting Table 4.3: Uston SS card counting values A T Uston SS Uston Advanced Point Count (APC) 4.4 Uston APC is a level 3 balanced card counting system created by Ken Uston and introduced in "Million Dollar Blackjack" in Uston APC is an advancement on Uston Advanced Plus or Minus (APM). Unlike most card counting techniques cards eight and nine are assigned values of +1 and -1 respectively and aces are left at zero.(uston, 1992) Table 4.4: Uston APC card counting values A T Uston SS Keep It Short and Simple 3 (KISS 3) 4.5 KISS 3 is the third revision of Keep It Short and Simple and is a suit-specific unbalanced strategy made popular by Fred Renzey(Renzey, 2006). KISS 3 is a counting system optimised for low deck games. KISS 3 is believed to provide the player with +0.7% edge against the house(countingedge.com, 2014). Table 4.5: Kiss 3 card counting values A Black 2 Red T Kiss Reg: 59 10

17 5. Related Work Zen Count 4.6 Zen Count is an intermediate level 2 balanced card counting technique also introduced by Arnold Snyder(Snyder, 2005). Arnold created this counting system to target the intermediate level card counters. Table 4.6: Zen Count card counting values A T Zen Count These card counting systems were selected in order to provide an equal representation from level one, two and three systems as well as unbalanced and balanced systems Illustrious 18 When card counting is included this also allows the inclusion of whats now know as the Illustrious 18(WizardofOdds, 2013). The Illustrious 18 A.1 is a matrix of 18 important situations to change play from basic strategy based on the count(blackjackchoice.com, 2014). The most important of the Illustrious 18 is the first which dictates when to take insurance. Playing without card counting means the player can not accurately assess when the correct time to take an insurance bet is, this solves that. 5. Related Work N. Richard Werthammer was a re occurring name while researching card counting and has performed studies into optimal betting in Blackjack(Werthamer, 2005). Also a systematic description of optimal play(werthamer, 2009). As well as an analytic look at Basic Strategy when used with card counting(werthamer, 2011). Reg: 59 11

18 6. Implementation Patrick Sileo conducted a study into the profitability of Blackjack games and creates an index that can be used by an average player to become more profitable at Blackjack.(Sileo, 1992) Charlie H. Cooke has performed a study into probability tables to determine the percentage outcome of each result to the dealer s hand(cooke, 2009). B. B. van der Genugten conducts a study into optimal play of Blackjack of Holland casinos. Due to the majority of reading being based on America Blackjack rules, primarily Las Vegas rules, this provides a look into how optimal plays are altered with a different rule set(van der Genugten, 1997). A plethora of books are available for Card Counting such as Ken Uston s "Million Dollar Blackjack" (Uston, 1992). Fred Renzey s "Blackbelt Blackbelt II" which has a number of card counting strategies(renzey, 2006). Stanford Wong s "Professional Blackjack" (Wong, 1994) and Arnold Snyder s "Blackbelt in Blackjack"(Snyder, 2005). 6. Implementation The simulation was created using the Java programing language in the Netbeans developing environment. The simulation is created from five classes and one main class that runs though the simulation in the sequence of a normal casino game. The simulation takes the balance of the player after each round and stores it. After the conclusion of the simulation the program outputs the data into a excel spreadsheet to be evaluated. To gain accurate data the simulation runs though shoe games of 100 rounds. This means each simulation plays Blackjack times. From this data an average is produced for balance and profit. Also a line graph is plotted based on the average balanced of the player after each round of Blackjack over the shoe games. 7. Results The investigation tests the very fundamentals of card counting by investigation into the best method of playing Blackjack. Finding the best method of playing Blackjack means that this can be used as a base for the card counting techniques to be applied on to. Reg: 59 12

19 7. Results 7.1. Strategy To keep the following simulations fair they will abide by the following rules: There is only one deck being used The player will bet 10 units each hand The player can not re-split The player can double down after a split The dealer will hit on soft 17 Table 7.1: Table of average loss or gain of a Blackjack Strategy Strategy Average balance Average profit Mimic the Dealer Basic Strategy without Double Down or Splitting Basic Strategy with Double Down without Splitting Basic Strategy with Splitting without Double Down Basic Strategy with Double Down and Splitting Graph A.1 shows that these strategies by themselves lose money over the lifetime of the simulation. The well established Basic Strategy was tested without double down or splitting and then with them individually to see how much of an impact these had on Basic Strategy and assess how advantageous they are to a player Mimic the Dealer Mimic the dealer shows a steep loss over the simulation and estimates about 0.62 units is lost with each hand.a.1 The average balance of (7.1) is almost twice as bad as playing basic strategy without double downing or splitting. These results make it clear that Mimic the dealer is not a profitable way to play blackjack and therefore should not be used in conjunction with card counting as this will provide a large handicap to the player. Reg: 59 13

20 7. Results Basic Strategy without doubling down or splitting By removing the option to double down and split the impact on the player can be measured from the simulation. The results show that while still operating at a loss this version of basic strategy offers a large performance increase with the average profit being only and an average balance of Basic Strategy with either doubling down or splitting Introducing doubling down has another increase in profit boosting the profit to per round. Splitting also provides the player with an advantage as implementing splitting gives an average profit of Splitting and doubling down are very close in performance with only 0.4 difference in profit between them. Looking at graph A.1 we can also see that the lines are very close during the first 50 hands of the game. Doubling down does have a slight edge over splitting making that more favorable to the player Complete basic strategy As expected basic strategy including both splitting and doubling down has the best result. A.1 On the graph this version of basic strategy is clearly above the rest of the strategies and also offers the most profit of per round along with the largest balance of This makes this version of basic strategy the correct choice to use when applying card counting techniques in the further simulations as it is the most advantage Blackjack strategy for the player. Reg: 59 14

21 7. Results Table 7.2: Table of average balance and profit while card counting with one deck Counting technique Average balance Average profit % of rounds won Hi Lo Red Uston SS KISS Zen Count Uston APC Card counting with one deck Using basic strategy as a platform to build card counting on the simulation then runs Blackjack with the previously mentioned card counting techniques to identify their advantage to the player. Table 7.2 shows the average balance and average profit of the simulations. The simulation is able to reach a 75% penetration of the deck before it must be reset Hi Lo The Hi Lo count offers an increase in profit over basic strategy with a average profit of With less unfavorable rules such as not being able to re-split and the dealer hitting on soft 17 Hi Lo could have produced a positive profit. The graph shows that a positive balance is achieved on numinous occasions and has a trend to slowly rise and fall though out the rounds. Hi Lo has a peak balance of 3.56 and a lowest balance of A.2 With only one deck there is going to be numerous occasions of the deck being reset and therefore the count being reset. When the count resets the counting method will switch back to the basic strategy rules of blackjack and wont benefit from card counting as much. As you get deeper into the deck the count will give a better representation of how favorable the deck is and cause the player to bet accordingly. Reg: 59 15

22 7. Results Zen Count The Zen count had a significantly better average profit, -0.02, than Hi Lo which is to be expected considering that Zen count is a level two card counting strategy. Despite the negative average profit the average balance is 0.12 which is reflected in the graph as the line is frequently above zero. 7.2 But like Hi Lo it has a trend of slowing rising and falling which probably due to same reason as Hi Lo with the deck being reset. The peak balance of Zen is 2.9 and the lowest balance is -2.8 this gives a much smaller range meaning Zen Count is a more consistent counting system. A Uston APC This simulation was unexpected as Uston APC actually produced an average loss of profit greater than basic strategy at This could be down to how much of an impact each card has on the running count. Most cards impact the running count by -2 or +2 causing a large shift in the running count and then on the true count. This could mean that the running count might predict the deck to look extremely favorable and cause the player to bet much larger amounts that other strategies would suggest causing much larger losses. This is shown on graph A.2 as a huge downward slope Red Seven Red Seven is the only card counting strategy to have a positive average balance of 2.88 and a positive average profit of This is expected as this card counting technique is refined to work better with a smaller number of decks. This reflects in graph A.2 as through out the rounds the balance remains above all other counting techniques apart from a small number of times. Red Seven also has the highest percentage of winning rounds which can also help explain its success in the simulation KISS 3 KISS similarly to other counting methods has a trend to slope downwards then increase as can quite clearly be seen in graph A.2. Unlike other counting methods of similar average profit, -0.03, KISS has a surprisingly low average balance of (7.2) this Reg: 59 16

23 7. Results can clearly be explained by the fact that it never manages to raise above zero to attain a positive balance as shown on graph A Uston SS In comparison to Uston APC this level three counting method performs significantly better with an average profit and of and respectively.7.2 This is likely due to the starting count beginning at -4 for a single deck of cards. This means that the player will only start betting larger sums of units when the deck is really stacked in the players favour. In comparison to the other techniques Uston SS is of a similar performance Conclusion All card counting techniques offer a advantage to the player over just playing basic strategy apart from Uston APC. This proves that most card counting methods do offer the player an advantage over just using basic strategy. Counting techniques follow a pattern of peaking and dipping which is likely to do with the fact the deck will reset a lot with the low number of cards available in a single deck. The number of rounds won seems to have little effect on the overall effectiveness of the counting techniques which is probably to do with the fact that the counting techniques are designed to predict when you are most likely to win and doesn t actually effect the cards you draw. Reg: 59 17

24 7. Results Table 7.3: Table of average balance and profit while card counting with six decks Counting technique Average balance Average profit % of rounds won Hi Lo Red Uston SS KISS Zen Count Uston APC Card counting with six decks Most, if not all, casinos do not use a single deck for Blackjack. Many casinos will operate with shoes of six or more decks therefore it is important find out how effective these card counting techniques are in a more real world application. The card counting simulations were run under the same conditions as previously, but this time with a shoe consisting of six decks. The simulation is able to reach 95.8% penetration of the shoe before it must reset Hi Lo Unlike the single deck simulation of Hi Lo there is a significant drop within the first 40 rounds. A.5 This will have something to do with the penetration of the shoe required for Hi Lo to prove effective. As the line begins to straighten out the player must be quite deep into the six decks and is therefore able to ascertain a good count and work out the favour of the deck. Then when the deck resets the count is as well and the line begins to drop quicker again. A.5 Compared to other counting techniques on six decks Hi Lo doesn t perform very well with an average balance of and average profit of Based on those values this makes Hi Lo the second worst counting technique. Reg: 59 18

25 7. Results Zen Count In comparison to Zen count on a single deck there is a reduction in performance but this can easily be explained by the favour gained by the house for increasing the number of decks. A.9 When Zen count is compared to other counting techniques it shows an extremely good performance in this simulation with the best average balance and profit of and respectively. 7.3 With most other card counting techniques winning round percentages being within plus or minus 0.04 it is pretty clear that this is not a factor that will hugely effect the performance of the counting technique Uston APC When compared to the single deck simulation Uston APC performs almost identically with a average balance of and average profit of A.3 This would draw the conclusion that Uston APC was designed with shoes consisting of a large number of decks as the performance of the counting technique is almost unaffected by the increased amount of decks. Then compared to the other counting techniques simulated with six deck Uston APC actually performs much better than in the single deck simulation. A.3 Further supporting the fact that Uston APC was designed for larger shoe sizes Red Seven As expected we see a drop in performance from Red Seven in comparison to its single deck simulation. A.6. This is due to the fact unbalanced strategies such as Red Seven are aimed to shoes of smaller decks. This can be seen clearly by the difference in average balance between the single deck simulation and the six deck one. A.3 Looking at the results for all six card simulations it is made clear that Red Seven is the best performing unbalanced strategy out of the three chosen. 7.3 There is not as quite a steep drop in comparison to the other counting techniques during the start of the simulation. A.3 This is probably explained by the fact that Red Seven s starting could for six decks will be -12 so will not start betting until the deck is really favourable. Reg: 59 19

26 7. Results KISS 3 As most simulations have shown there is a drop in performance from the counting techniques from a single deck to a shoe of six decks. This increased edge for the house has caused KISS to have a lower average profit and balance than the single card simulation. A.3 This again is likely due to unbalanced counting techniques favouring a shoe made of a smaller number of decks. The fact KISS starts on a -12 count seems to not be effective enough to counter the advantage gained by the house. Compared to the other six deck simulations KISS appears to have a moderate performance Uston SS Unsurprisingly there is a performance loss from a single deck to six decks. With also the largest jump in average balance from with one deck to with six decks. A.3 Despite altering starting count to be -24 for six decks this seems to have had no effect in countering the larger number of decks. Furthermore when compared to the other simulations on the same deck size it appears that Uston SS has the worst overall performance. A.3 This is unusual as this is a level three strategy and should theoretically out perform the level two and level one strategies though, as is shown, this is not always the case Conclusion The advantage provided to the house by increasing the shoe size to six decks has a significant impact on the performance of the card counting strategies with all counting techniques showing a drop in performance. Using six decks shows an advantage to using a balanced counting method for a shoe of this size as the two best techniques, Zen Count and Uston APC, were both balanced. This would indicate a balanced count provides the player with a more accurate evaluation of the decks bias. Furthermore increasing the number of decks has had an impact on the percentage of rounds won by the player reducing all of them by about 0.3%. This likely will not have a huge impact on the balance and profit but will reduce the maximum yield from any of the card counting strategies. 7.3 Reg: 59 20

27 7. Results Table 7.4: Table of average balance and profit while card counting with eight decks Counting technique Average balance Average profit % of rounds won Hi Lo Red Uston SS KISS Zen Count Uston APC Card counting with eight decks Casinos often use shoes consisting of eight decks. This makes simulating the card counting techniques on shoes of eight decks important in assessing their performance on real world situations. This simulation is able to reach a penetration of 96.88% of the shoe before it must reset Hi Lo Between six and eight decks there is a much smaller gap between the average balance and profit. Between the balance there is only a difference of 2.56 and between profit there is only a 0.04 difference. A.3 This is expected as increasing the deck from six to eight should result in an increased house edge. An interesting note is the curve towards the end of the line graph when it starts to straighten out. This is most probably the point in which the counting method is able to very accurately predict how favourable the deck is due to how much of the deck has been seen. This is where the house advantage has been neutralised by the count. A Zen Count Comparing the simulation results of six and eight decks for Zen count shows that they are practically the same. A.9 There is only a very slight difference in average profit and average balance of 0.29 between balances and 0.02 between profit. This is a testament Reg: 59 21

28 7. Results to the Zen Count s ability to accurately predict the favour of the cards remaining in the shoe. Once again Zen out performs the rest of the card counting techniques confirming that balanced strategies are superior to unbalanced techniques when it comes to card counting with a large shoe Uston APC The simulation on eight decks actually caused the Uston APC average balance to raise slightly from to A.3 Average profit on the other hand did not increase. This simulation would work towards confirming that this particular card counting technique is aimed towards larger shoes. When compared against the previous two simulations it becomes quite clear that all the simulations result in a similar performance this would prove to make this counting strategy quite consistent. A.10 Uston APC has a very similar performance to the six deck simulation once again being the second best card counting technique with an average balance of and average profit of This would confirm that balanced strategies are more likely to perform better than unbalanced strategies with large shoes Red Seven Starting the count for Red Seven at -16 due to the increase in shoe size seems to prove useful as during the beginning simulated rounds the player is prevented from betting until the deck is supposed to be in their favour. This seems the keep the player s balance slightly higher than the simulation of six decks until it drops rapidly. The line then curves out as the deck becomes favorable to the player again and the house advantage from the increased deck size is mitigated. A.6 The Red Seven count again is the third best counting method and is the best unbalanced counting technique with an average profit and balanced of and Reg: 59 22

29 REFERENCES KISS 3 KISS 3 follows a similar pattern to Red Seven when comparing the six and eight deck simulations which is unsurprising as they are both unbalanced strategies. A.8 This simulation in conjunction with the results from Red Seven serve the idea that unbalanced strategies are better suited towards shoes of less decks. When KISS is compared to the other counts in the eight deck simulations we see a similar result to the six deck simulation in which KISS sits pretty central in performance neither performing badly or standing out. A Uston SS Comparing the Uston SS single deck simulation with the Uston APC eight deck simulation shows almost a complete swap in the performance of the two card counting strategies. A.3 Uston SS seems to be the most affected counting method affected by the addition of two decks with average balance jump from to Conclusion Adding two decks to the shoe seems to have a minor affect on most strategies. The winning round percentage drops in some cases for some of the card counting techniques but the difference between the six (7.3) and eight deck (7.4) simulation percentages are negligible. Balanced strategies also prove again to be the superior choice with shoes containing a large number of decks. References blackjackapprenticeship.com (2014). Blackjackapprenticeship. www. blackjackapprenticeship.com/. BlackjackChoice.com (2014). Blackjackchoice. Cooke, C. H. (2009). Probability models for blackjack pok0er. Computers and Mathematics with Applications. Reg: 59 23

30 REFERENCES countingedge.com (2014). Countingedge. Dalton, M. (1993). Blackjack: A Professional Reference. Spur of the Moment Publishing. Graham Kendall, C. S. (2003). The evolution of blackjack strategies. HomePokerGames (2003). Card counting. cardcounting.php. Lance Humble, C. C. (1998). The World s Greatest Blackjack Book. Doubleday. Olaf Vancura PhD, K. F. (1998). Knock-Out Blackjack The Easiest Card-Counting System Ever Devised. Huntington Press. qfit (2013). Card counting. card-counting.htm. Renzey, F. (2006). Blackjack Bluebook II: The Simplest Winning Strategies Ever Published. Blackjack Mentor. Sileo, P. (1992). The evaluation of blackjack games. Snyder, A. (2005). Blackbelt in Blackjack : Playing 21 as a Martial Art. Cardoza. Uston, K. (1992). Million dollar Blackjack. Carol Publishing Group. Uston, K. (2009). The Uston SS Count. BAUM Productions. van der Genugten, B. B. (1997). Blackjack in hallnd casino s: basic, optimal and winning strategies. Werthamer, N. R. (2005). Optimal betting in casino blackjack. Routledge. Werthamer, N. R. (2009). Risk and reward the science of casino blackjack. Springer. Werthamer, N. R. (2011). Basic strategy for card-counters an analytic approach. wp-content/ uploads/ 2011/ 06/ Basic- Strategy-for-Card-Counters.pdf. Reg: 59 24

31 A. Appendix WizardofOdds (2013). Blackjack. wizardofodds.com/ games/ blackjack/. Wong, S. (1993). Blackjack Secrets. Pi Yee Press. Wong, S. (1994). Professional Blackjack. Pi Yee Press. A. Appendix Reg: 59 25

32 A. Appendix Table A.1: A table of card counting strategies Strategy Value assigned to card A T Canfield Expert Canfield Master Hi-Lo Hi-Opt I Hi-Opt II KISS 2 0 0/ KISS 3-1 0/ K-O Mentor Omega II Red Seven / REKO Revere Adv. Plus-Minus Revere Point Count Revere RAPC Revere 14 Count Silver Fox Unbalanced Zen Uston Adv. Plus-Minus Uston APC Uston SS Wong Halves Zen Count Adapted from (qfit, 2013). Reg: 59 26

33 A. Appendix Table A.2: The Illustrious 18 Order Play Index 1 Insurance Vs Vs ,10 Vs ,10 Vs Vs Vs Vs Vs. A Vs Vs. A Vs Vs Vs Vs Vs Vs Vs. 3-2 Adapted from (WizardofOdds, 2013). Reg: 59 27

34 A. Appendix Table A.3: Comparison of all averages for strategies and counting techniques Counting technique / Strategy Average balance Average profit Mimic the Dealer Basic Strategy without Double Down or Splitting Basic Strategy with Double Down without Splitting Basic Strategy with Splitting without Double Down Basic Strategy with Double Down and Splitting Hi Lo one deck Hi Lo six decks Hi Lo eight decks Red 7 one deck Red 7 six decks Red 7 eight decks Uston SS one deck Uston SS six decks Uston SS eight decks KISS 3 one deck KISS 3 six decks KISS 3 eight decks Zen Count one deck Zen Count six decks Zen Count eight decks Uston APC one deck Uston APC six decks Uston APC eight decks Reg: 59 28

35 A. Appendix Figure A.1: Comparison of Blackjack Strategies Reg: 59 29

36 A. Appendix Figure A.2: Comparison of card counting techniques used on a single deck Reg: 59 30

37 A. Appendix Figure A.3: Comparison of card counting techniques used on six deck Reg: 59 31

38 A. Appendix Figure A.4: Comparison of card counting techniques used on eight deck Reg: 59 32

39 A. Appendix Figure A.5: Hi Lo comparison of one, six and eight decks Reg: 59 33

40 A. Appendix Figure A.6: Red Seven comparison of one, six and eight decks Reg: 59 34

41 A. Appendix Figure A.7: Uston SS comparison of one, six and eight decks Reg: 59 35

42 A. Appendix Figure A.8: Kiss 3 comparison of one, six and eight decks Reg: 59 36

43 A. Appendix Figure A.9: Zen Count comparison of one, six and eight decks Reg: 59 37

44 A. Appendix Figure A.10: Uston APC comparison of one, six and eight decks Reg: 59 38