Framework to Determine a Minimal Bet Spread to Eliminate House Edge
Whenever I have played blackjack in a casino I have always bet minimally. I have mainly concentrated on 1 and 2 deck games which I had determined to have a reasonable expectation of yielding a positive outcome by counting cards without having to bet too much. I have only dabbled in playing multi-deck shoes just for the experience without much expecation of a positive outcome. I have since developed some software tools that are an aid in analyzing blackjack through combinatorial analysis. Using these tools it has been my goal to be able to determine a minimal bet spread needed to eliminate the intrinsic house edge.
Benchmark Game
The benchmark analysis applies to a 6 deck shoe dealt 75% to a cut card with these rules: dealer hits soft 17, blackjack pays 3:2, full peek, double any 2 cards, split aces once : 1 card to split aces, split to up to 4 hands for other pairs with doubling afer splitting allowed, no surrender of any kind. This game was purposely chosen because it is not so easily beaten. The commonly accepted spread necessary to beat this game is a minimum of 1 to 15: minimum bet = 1 unit; maximum bet = 15 units. I suppose this includes ensuring that a minimum amount per hour is achieved. However, I would never subscribe to any spread even remotely close to that. I am not saying that this is wrong but only that for me personally such a spread would never be an option.
Balanced KO Counting System
This is the KO counting system. It has 3 groups of cards.
- 2, 3, 4, 5, 6, 7 --- L --- (low cards) --- tag = -1
- 8, 9 --- M --- (medium cards) --- tag = 0
- T, A --- H --- (high cards) --- tag = +1
When there are more low cards than high remaining to be dealt, running count is negative.
When there are more high cards than low remaining to be dealt, running count is positive.
When there are an equal number of low cards and high cards remaining to be dealt, running count is 0.
There are 4 more low cards than high cards for each number of full decks so
initial running count (IRC) = -4 * (number of decks)
I have created a system I call Balanced KO with the same 3 groups as KO.
- 2, 3, 4, 5, 6, 7 --- L --- (low cards) --- tag = -2
- 8, 9 --- M --- (medium cards) --- tag = +1
- T, A --- H --- (high cards) --- tag = +2
There are still 4 more low cards than high cards for each number of full decks but
initial running count (IRC) = 0 for any number of decks.
I am going to use above the Balanced KO system to establish a minimal betting/playing framework for the above benchmark game. The below outlined strategy calls for betting and wonging out based only on running count (RC) similar to KO. Betting/wonging according to RC utilizes a reference point which is 1/2 of the cards to the cut card. In this case that is 1/2*(6 decks = 312 cards dealt to 75% pen) = 1/2*312*(.75) = 117 cards. This would need to be roughly estimated. If simming this could be determined exactly.
Any deviations from basic strategy are made according to true count (TC) similar to HiLo. Utilizing TC would require it to be roughly computed. If simming TC could be computed exactly. If only basic strategy is used for playing then computation of a true count (TC) wouldn't at all be necessary with this system.
Betting and Playing Strategy
6 decks, h17, spna3, spa1, DAS, nohsa pen = 75% ~= 234 cards to cut card
count tags {2,-2,-2,-2,-2,-2,-2,1,1,2}
Running count (RC) = 2*{A,T} present in shoe + {8,9} present in shoe - 2*{2,3,4,5,6,7} present in shoe
initial running count (IRC) = 0 for any number of decks
True count (TC) = 52*RC / (cards remaining) (rounded to 1 decimal place if simming)
betting system: (flat bet either 1 or 3 units and leave (wong out) according to RC below)
flat bet 1 unit at start of shoe;
leave (wong out) if RC <= -4 at end of any round (including first round)
if < 117 cards have been dealt (> ~195 cards remaining):
if RC >= +20 at start of a round flat bet 3 units; continue to bet 3 units thereafter
leave (wong out) if RC <= +8 at end of any round
when >= 117 cards have been dealt (<= ~195 cards remaining):
at end of round where 117th card is dealt:
if RC <= +8 leave (wong out) regardless if bet is 1 or 3 units
if RC > +8 change wong out to RC <= +3 regardless if bet is 1 or 3 units
if RC >= +15 at start of a round flat bet 3 units; continue to bet 3 units thereafter
leave (wong out) if RC <= +3 at end of any round regardless if bet is 1 or 3 units
TC INDEXES
always play according to TC indexes (if simmed round to 1 decimal place)
take insurance if TC >= 7 or 8 (insurance correlation is not so good for this count system)
2 3 4 5 6 7 8 9 T A CD Insurance
________________________________________________________________ ________________
16 s s s s s 20 19 15 s>0 7 LL ins TC >= 7
15 s s s s s 24 26 25 9 11 LM ins TC >= 7
14 s s s s s 27 33 35 17 15 LT ins TC >= 8
13 -2 -5 s s s h h h h 28 LA ins TC >= 7
12 7 3 -1 s s h h h h 58
MM ins TC >= 6
11 d d d d d d d d -10 -2 MT ins TC >= 7
10 d d d d d d d -4 8 6 MA ins TC >= 6
9 2 -2 -6 d d 8 h h h h
8 h h 12 8 5 h h h h h TT ins TC >= 8
TA ins TC >= 7
TT s s 13 11 9 s s s s s
99 -8 p p p p 6 s s s 4 AA ins TC >= 6
88 p p p p p p p p p<14 -4
77 p p p p p p p>=5 s s>=14 s>=14 Generic Insurance
66 -5 -10 -13 p p h h h h h ________________
55 d d d d d d d d d>=8 d>=6 xx ins TC >= 7
44 h 17 5 p p h h h h h
33 -1* -13 p p p p h h h h
22 p p p p p p p>=8 h h h
AA p p p p p p p p p p
A9 s s 13 11 9 s s s s s
A8 s s 6 3 -1 s s s s s
A7 1 -5 -13 d d s s h h h
A6 4 -7 d d d h h h h h
A5 h 7 -5 d d h h h h h
A4 h 13 -1 d d h h h h h
A3 h 16 3 d d h h h h h
A2 h 15 6 -1 d h h h h h
The above are mid-shoe running count indexes divided by 3 to convert to true count, since in this case mid-shoe is 156 cards remaining = 3 decks. Mid-shoe indexes are the statistically best ones to use for all shoe compositions. I have included those which may only have a remote chance of being encountered for reference if nothing else. You could employ as many indexes as can be mastered but the main ones in no particular order are: 16,15,14 v T, 16,15,14 v A, 16 v 9, 12 v 2, 12 v 3, 10 v T, 10 v A, 9 v 2, 9 v 7, 8 v 6, 8 v 5, 99 v 7, 44 v 4, A8 v 6, A8 v 5, A8 v 4, A6 v 2, A5 v 3, A3 v 4, A2 v 4, insure at TC >= 8. (In anecdotal evaluation with my own software I never split tens or double on A-9 even though there are indexes for these plays.) A simplified option is to always use basic strategy for playing and the betting strategy for betting.
Simming this system
If you are playing in a casino you will necessarily need to make estimates of cards/decks remaining in order to arrive at a true count given the current running count (RC). However simulation software always knows the exact number of cards remaining to be dealt as well as the current running count and can compute true count (TC) to the card using the following formula:
***** TC = 52*RC/(unseen cards remaining) *****
This TC can be rounded to 1 decimal place and compared to the relevant index in the above table to determine if a basic strategy deviation is in order. Also the betting strategy calls for leaving (wonging out) at the end of a round if RC <= -4 and for continuing to bet 3 units until wonging out at a running count determined by cards remaining. Therefore simulation software can determine exactly when to leave the game (wong out.) It is possible for simulation software to make a virtually perfect evaluation of the system based upon the outlined system.
Minimal Betting
The main advantage of the above system is that it employs minimal betting. I have investigated using it for a game with 1 player versus dealer using my own software and it seems to have a possibility of succeeding long term. My software does not allow playing in a game with multiple players. Whether or not it could be expected to succeed long term would have to be determined by exhaustive simulation of enough rounds/conditions to make that determination. To me it is not so important that it succeeds but that it is an effort to eliminate betting too much while maintaining some hope of success.
The system is loosely based on the following model of eliminating house edge.
Simple Model of Eliminating House Edge
Suppose we had a game with a pool of 3 black marbles and 2 white marbles. Player can bet any amount and wins even money on his bet. He wins if a white marble is randomly drawn and loses if a black is drawn. Whether player wins or loses he can bet any amount on the next random draw made with the marble previously drawn removed for up to 3 draws. The house edge for this game is a massive 20% if player plays every round and never varies his bet. However, player has the option of leaving any time he chooses so he adopts the following betting strategy. Round 1: bet 1 unit; if player wins he is lucky since this happens only 40% of the time and he leaves with 1 unit profit; if he loses he goes on to round 2. Round 2: bet 1 unit; if player wins, which now happens 50% of the time, he is now even and again leaves since the next round would be very unfavorable to him; if he again loses he goes on to round 3. Round 3: after losing first 2 rounds player has a 2/3 chance of winning the third; if he bets 1 unit he is always an overall loser; if he bets 2 units he exactly eliminates the house edge in the long run; if he bets 3 units he is an overall winner in the long run.
This model can be very loosely representative of a blackjack game. Fewer initial number of marbles loosely represents fewer decks. Fewer number of initial marbles makes removal of marbles more greatly effect probabilities as does the removal of cards in blackjack. More initial number of marbles loosely represents more decks. Increased number of initial marbles causes probabilties to change more slowly as marbles are removed as does the removal of cards in blackjack for a larger number of decks. The main difference is that in blackjack the same shoe composition has many possible outcomes whereas in the marble model each draw has only 1 possible outcome.