223 lines
8.4 KiB
Plaintext
223 lines
8.4 KiB
Plaintext
Optimal Wagering
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Copyright 1991, Michael Hall
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Permission to repost, print for own use.
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I think I've got some good discoveries here... even if you don't
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follow the math, you can get some useful blackjack information here.
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The question of optimal wagering has been brewing on rec.gambling
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for a while. I rephrase this question as the following:
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* What's the optimal win per hand as a portion of bankroll and
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what is the betting pattern necessary for this?
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That is, we want to maximize E/a' where E is the win per hand
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and a' is the required bankroll.
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E is simply defined by:
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E=sum{WiPiEi}
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where i is the situation
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Wi is the wager for that situation
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Pi is the probability of that situation
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Ei is the expected value of that situation
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I defined a' in previous articles. Unfortunately, I made a
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slight error, in that I left out a couple of sqrt's. I
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hope the following is correct...
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log((1/R) - 1)
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a'= ----------------------------(sqrt(s^2 + E^2))
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/sqrt(s^2 + E^2) + E\
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log| ------------------- |
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\sqrt(s^2 + E^2) - E/
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where R is the risk of ruin
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E is the win per hand
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s^2 is the variance of E
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a' is the necessary units of blackjack bankroll
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[Incidentally, the Kelly criterion leads to a bankroll formula
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proportional to the one above, and so Kelly betting produces the
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same optimal wagering schemes as the ones shown below.]
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I tried to maximize E/a' by taking the derivatives wrt Wi and setting
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them to 0. That got really ugly. Then I tried to maximize E or minimize
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R using various formulations of Lagrange multipliers. That got really ugly
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too. I did come up with the partial derivatives, which are ugly themselves,
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but solving for the Wi's is where it gets *really* ugly.
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So I gave up and just wrote a program to evaluate the function given
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Wi's as input, and then I wrote a program to do a simple hill-climbing
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on this function in the space of integers between 1 and some maximum
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bet like 4 or 8. My intuition is that hill-climbing should converge to
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the global maximum and not a local maximum of this function, but I don't
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have any proof of this. BTW: my program does adjust for the basic
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variance of blackjack, increasing the effective bet size by 1.1 and other
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such things.
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For a downtown Vegas single deck 75% penetration (Snyder's tables in
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"Fundamentals of Blackjack" by Chambliss and Rogenski), here is the
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optimal betting patterns I found for spreads of 1-2, 1-4 and 1-8:
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SINGLE DECK
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DOWNTOWN VEGAS
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1-2 1-4 1-8
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ADV FREQ HI-LO BET BET BET
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Ei Pi Wi Wi Wi
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-.026 .065 -5 1 1 1
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-.021 .030 -4 1 1 1
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-.016 .055 -3 1 1 1
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-.011 .070 -2 1 1 1
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-.006 .100 -1 1 1 1
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-.001 .200 0 1 1 1
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+.004 .095 +1 1 1 1
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+.009 .075 +2 1 1 2
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+.014 .050 +3 2 2 3
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+.019 .045 +4 2 3 5
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+.024 .040 +5 2 4 6
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+.029 .035 +6 2 4 7
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+.034 .030 +7 2 4 8
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+.039 .030 +8 2 4 8
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+.044 .080 +9 2 4 8
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The Hi-Lo column shows the approximate High-Low (or Hi-Opt I) count for
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each advantage, though you should adjust for the extra advantage from
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strategy deeper into the deck. Note that the bet should not be raised
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until a true count of 3, unless you are using a very wide spread.
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You might fool a few pit critters by your low bet at a true count of 2.
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(Or at least you won't get nailed when you increase your bet at a true
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count of 2, like I did once.) For the 1-2 and 1-4 spreads, the betting
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pattern is easy to remember - true count minus 1 (minimum of 1, maximum
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of 2 or 4.) [More exact results using simulations for the input data
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showed that the optimal spread for Hi-Lo here is actually to bet equal
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to the true count.]
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Here's the same stuff, but for 2 decks:
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DOUBLE DECK
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(BSE of -0.2% assumed)
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1-4 1-8 1-16
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ADV FREQ HI-LO BET BET BET
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Ei Pi Wi Wi Wi
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-.027 .060 -5 1 1 1
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-.022 .040 -4 1 1 1
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-.017 .060 -3 1 1 1
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-.012 .080 -2 1 1 1
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-.007 .110 -1 1 1 1
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-.002 .200 0 1 1 1
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+.003 .110 +1 1 1 2
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+.008 .085 +2 3 3 5
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+.013 .055 +3 4 5 8
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+.018 .045 +4 4 7 11
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+.023 .040 +5 4 8 14
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+.028 .030 +6 4 8 16
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+.033 .025 +7 4 8 16
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+.038 .020 +8 4 8 16
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+.043 .040 +9 4 8 16
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Here's the same stuff, but for 8 decks:
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EIGHT DECKS
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(NEGATIVE COUNTS PLAYED)
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1-8 1-16 1-32
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ADV FREQ HI-LO BET BET BET
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Ei Pi Wi Wi Wi
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-.030 .010 -5 1 1 1
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-.025 .010 -4 1 1 1
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-.020 .020 -3 1 1 1
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-.015 .060 -2 1 1 1
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-.010 .130 -1 1 1 1
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-.005 .510 0 1 1 1
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.000 .130 +1 1 1 1
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+.005 .060 +2 8 8 10
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+.010 .030 +3 8 15 20
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+.015 .015 +4 8 16 30
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+.020 .010 +5 8 16 32
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+.025 .010 +6 8 16 32
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+.030 .005 +7 8 16 32
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EIGHT DECKS
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(NEGATIVE COUNTS NOT PLAYED)
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0-8 0-16 0-32
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ADV FREQ HI-LO BET BET BET
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Ei Pi Wi Wi Wi
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-.030 .010 -5 0 0 0
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-.025 .010 -4 0 0 0
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-.020 .020 -3 0 0 0
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-.015 .060 -2 0 0 0
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-.010 .130 -1 0 0 0
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-.005 .510 0 1 1 1
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.000 .130 +1 1 1 1
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+.005 .060 +2 4 5 8
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+.010 .030 +3 8 10 16
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+.015 .015 +4 8 15 24
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+.020 .010 +5 8 16 31
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+.025 .010 +6 8 16 32
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+.030 .005 +7 8 16 32
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What follows are statistics on all these different optimal spreads.
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The bankroll requirements assume we want to have a 20% chance of
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losing *half* the bankroll before winning *half* the bankroll.
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One you lose half the bankroll, I'd advise cutting the bet size
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in half. (Note that the desired risk of ruin has absolutely no effect
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on the optimal betting pattern - it just changes the bankroll
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by a constant amount.)
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UNIT^2 UNITS
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% BANK GAIN UNIT GAIN VARIANCE REQUIRED
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PER HAND PER HAND PER HAND BANKROLL
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DECKS SPREAD| E/(2a') E s^2 2*a'
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-------------*--------------------------------------------
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1-Deck FLAT |.001420% .0050? 1.27 352
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1-Deck 1-2 |.008027% .0165 2.47 206
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1-Deck 1-4 |.014170% .0348 6.16 245
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1-Deck 1-8 |.018132% .0695 19.19 383
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2-Deck 1-4 |.002765% .0170 7.55 615
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2-Deck 1-8 |.006787% .0433 19.92 638
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2-Deck 1-16 |.009916% .0946 65.16 955
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8-Deck 1-8 |.000251% .0064 11.77 2550
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8-Deck 1-16 |.000673% .0162 28.00 2401
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8-Deck 1-32 |.001033% .0328 75.24 3177
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8-Deck 0-8 |.000675% .0086 7.82 1263
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8-Deck 0-16 |.001047% .0169 19.33 1600
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8-Deck 0-32 |.001288% .0326 59.57 2532
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Some things to conclude, given the above table:
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* A 1-2 spread on a single deck is more than 6 times more profitable
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than a 0-32 spread on 8 decks! Even flat betting a single deck
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is probably better. 8 decks stink!
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* It takes a 1-16 spread on double decks to beat a 1-2 spread on single
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decks! (Can this be true?)
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* A 1-8 spread buys you 29% more income over a 1-4 spread on
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a single deck, but you'll probably lose more than that from
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the extra countermeasures.
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* Given a $6,125 bankroll, you could spread $25-$100 on a single
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deck, making $86.8/hour (.014170%*6125*100). This is probably
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overly optimistic, since it rare that you can freely spread
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1-4 on a 75% penetration downtown Vegas game.
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* You need about a 1-32 spread on 8 decks before you can get away
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with playing through negative counts. A 1-8 spread gets killed
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sitting through negative counts, as the high bankroll requirement
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shows.
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One thing that might be fun is playing around with the above
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betting spreads. They are optimal, but how weird can you get
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without sacrificing much of the E/a'?
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I'd like to acknowledge Blair for getting me to think in terms of
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percent bankroll win.
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