One of the most "amusing" stories I heard about a team betting full-Kelly occured back in the mid-1980s. A small group of investors put together a $200,000 bankroll, with a group of a dozen known and trusted players to take on the casinos of Nevada. All players were to bet full-Kelly, and the team would distribute the profits when they had doubled the bank. They estimated that it would take three to six weeks with the talent they had, and the number of hours the players should be able to play per week. The investors would get half the profit—$100,000—and the players would divvy up the other $100K, or about $8,500 per player. Sounded like a good plan…

After numerous ups and downs, with more downs than ups, the team hit a major downswing about three months into play. They had already gone twice as long as they were expected to go, but now the total bank remaining was only about $60,000. They had lost $140,000. The investors, all veteran players, took it in stride. They contacted the players and told them that $60,000 was simply not a big enough bankroll to front a dozen players. The investors were going to have to bite the bullet and put together a new bankroll; then they'd start over. So, they called in the remaining funds from the players.

But a funny thing happened…there were no remaining funds! Or, at least, there wasn't nearly the $60,000 there was supposed to be. All funds combined, the players had only about $11,000 in their possession! Where did the money go?

As it turned out, the players had all been taking informal "advances" on their expected $8,500 payday. Twelve players. Three months. They all had to live, eat, pay rent, buy gas…$49,000 had been dwindled away in the players' day-to-day living expenses. All the missing money, in fact, was accounted for. Each player knew how much he had advanced himself. An average draw of about $4,000 per player was not really that much money for three months living expenses. None of the players had been living high on the hog.

In fact, if this team had been betting quarter-Kelly, that $140,000 loss would have been a $35,000 loss. No big deal on a $200,000 bank. And the initial estimated time for doubling the bank would have been five to ten weeks, instead of three to six. Obviously, with this initial downswing, the team would not have made the estimated time schedule for doubling, but the fact is that even with the players taking their living-expense advances, they would have still had a substantial bankroll after three months, and a good chance of eventually hitting their goal.

If you bet half-Kelly, you cut your flux in half, but your win rate is still 75% of what it would be with full-Kelly betting. Better yet, with quarter-Kelly betting, you cut your flux to one-quarter of what it would be with full-Kelly betting, but your win rate is still 56% of the full-Kelly haul. If you've got a bankroll of $50K or better,

quarter-Kelly betting makes sense, but the problem with betting this way on a $10K bank is that the initial hourly expectation is so low you might consider getting a job that pays better to increase your bank before beginning your blackjack career. But on any size bank, I'd never advise betting greater than half-Kelly.

• Blackjack – How To Read The 6 Deck/75% Dealt Chart Part5 (0)
• Blackjack – How To Read The 6 Deck/75% Dealt Chart Part4 (0)
• Blackjack – How To Read The 6 Deck/75% Dealt Chart Part3 (0)

Do not be lured by the "magic" of Kelly betting. Although, mathematically, full-Kelly betting appears to be the fastest way to increase a bankroll with no possibility of going broke, if you do not have a replenishable bankroll, full-Kelly betting causes HUGE fluctuations in the process of winning at the "fastest" rate. These fluctuations are absolutely intolerable in real-world games. And they're not tolerable for practical reasons, whether you think you have the stomach for them or not.

Unless you have a very large bank, a full-Kelly betting strategy will often dip your bankroll down to the brink of extinction. If you run a full-Kelly betting pattern on a computer simulation, no problem—your computer couldn't care less if your $10,000 bankroll goes down to $700 before turning around and climbing back up. Nor does your computer mind when your high bets go from $200 to $20, and then back up to $600. But for real-world players, full-Kelly betting is a bankroll roller coaster you'd rather not ride. It's the betting system that performs best on paper, not in real-world casinos.

For a player trying to make it as a pro on a small bankroll (say $10,000), full-Kelly betting is suicide. Players like the thought that with Kelly betting, "it's impossible to go broke." It's true that you can't lose your whole bank if you never bet more than a small fraction of it. But how can you continue playing when your high bets have to be reduced from $100 to $15 because your bankroll took a negative swing from $10,000 to $1500? Where are you going to find a casino that you can beat with $15 high bets?

Negative flux on full-Kelly betting has been the death of more would-be blackjack pros, and has spelled doom for more blackjack teams, than anything else. Talk to some of the old-timers who were playing back in the '70s and '80s when everyone was Kelly crazy. It was a common occurrence for a team to go broke, with one and all chanting: "But we were Kelly betting!" Betting full-Kelly, your winning streaks are way bigger, but your losing streaks are way grimmer. All of the successful big teams today have their players betting very small fractions of the "ideal" Kelly bets.

• Blackjack – How To Read The 6 Deck/75% Dealt Chart Part5 (0)
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• Blackjack – How To Read The 6 Deck/75% Dealt Chart Part3 (0)

In any case, our immediate interest here is in using a bigger advantage over the house to cut your fluctuations. Employing S-l, in the game described above, the counter using this strategy would win only 16 units for every 10,000 hands played, but one standard deviation on these hands would be about 110 units. That win rate of 0.11% is so small that even with 100,000 hands played, and an expectation of 156 units, the standard deviation, 348 units, is still more than twice the expectation.

With this small of an edge, whether or not you win or lose is a crapshoot, even over a period of years. One of the main goals of the pro player is to raise his expectation of winning so that it's higher than the standard deviation as soon as possible. If your expected unit win is still less than one standard deviation after 100,000 hands, you may have the edge on the house, but you're still just gambling.

If you use S-3, however, with a win rate of 1.26%, your unit expectation is 126 after 10,000 hands played, with a standard deviation of only 110 units. (Note: the standard deviation numbers are not in the chart above, which has been condensed from the charts in the 6-deck report.) So, with S-3, you're more likely to be in the black after 10,000 hands played than you would be after 100,000 hands played with the S-l betting strategy. That higher win rate really pays off if you're trying to get past those initial short-term losses.

But there are other problems with the S-3 betting strategy, some of which we've mentioned already. Since we're only playing 26 hands per 100 seen, it's going to take a hell of a lot longer to play 10,000 hands in real-world hours, with most of our time spent circling around the pit like a buzzard just watching for a decent betting opportunity. It's not a practical solo approach to beating the game.

So, let's look at S-8, playing the same number of hands per hour (69) as S-l, but spreading from 1 to 8 units as the count increases. Let's say that we spread from $25 to $200 in order to get this 1 to 8 spread. The average bet is 2.23 units, which—with a $25 unit—translates to $55.75. It's a significantly smaller average bet than the one used by the S-l player, who is flat betting $100, even though we are playing the same number of hands, and bet twice as much ($200) on the best situations.

So, playing S-8 our Hourly Unit expectation is 1.62, and with a $25 unit, this comes to $40.50, significantly higher than any of the flat-betting strategies, despite the smaller unit size and smaller average bet. Finally, even though using S-8 we play the same number of hands as the S-1 bettor, and are sometimes betting twice as much as the S-l bettor, after 10,000 hands the S-l bettor's standard deviation (with a $100 betting unit) is $11,000, while ours (with a $25-$200 betting spread) is only $8,950. So, S-8 outperforms S-l by winning almost 4 times as much per hour, with an average bet almost half the size of S-l, and has significantly reduced fluctuations.

Although casual players who have outside income can seriously consider playing with very small edges in order to obtain comps and other amenities, pros must always do whatever possible to get as big of an edge over the house as possible in order to cut the flux. Table-hopping to avoid betting when the house has any significant advantage is one way to do this. Let's look at some other methods:

• Blackjack – How To Read The 6 Deck/75% Dealt Chart Part4 (0)
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• Blackjack – How To Read The 6 Deck/75% Dealt Chart Part2 (0)

Moving on, we get to Win Rate %: This is the actual advantage over the house that you have with this betting spread. With S-l, your Win Rate is 0.16%, which means you're barely squeaking by. The highest Win Rate is from S-4, which is a whopping 1.74%. So, let's look at that betting strategy. The S-4 player is waiting for a full 1 % advantage over the house before he puts any money on the table. Should we all just use S-4, and wait until a 1% advantage occurs before we bet? Not necessarily. Look at the data in the next row:

Hourly Units: This data tells how many units we would expect to win per hour, based on 100 hands per hour, with each betting system. Why does S-4 slightly under-perform 5-2 and S-3, which have notably lower overall Win Rates'? The answer is found in the first row we looked at, Hands Bet/100. With S-4, we're only betting on 14 hands for every hundred hands seen. So we've got a big win rate on a very small amount of money. Also, this 5-4 betting strategy would be extremely impractical in a real-world casino. Back-counting 86% of the hands you see, before making a bet, is not easy to pull off. You'll look like a back counter and you may find your action unwanted when you do try to put a bet down. You'll be the classic pit buzzard.

Practically, it's far easier, and less obvious, to come off the top of the shoe with a 1-unit bet, and play through all those zero-count situations when the house has a small edge, but the flat-bet approach (5-7) is simply too weak. So, let's look at our results if we bet one unit off the top, but raise our bet when we get the advantage to our side.
5-5,5-6,5-7, and S-8 show our results assuming we spread to 2 units, 3 units, 4 units, and 8 units respectively. Using any one of these approaches, we again play 69 hands per hundred, but significantly raise both the win rate and hourly unit expectation over the 5-7 results. This is the power of betting more in proportion to your advantage.

Look closely at the Hourly Units data: Assuming a $100 betting unit, the 5-7 strategy would provide 0.11 units, or $11 per hour. The most we can expect with any of the flat bet approaches (5-2 and S-3) is 0.28 units, or $28 per hour. But using a spread to 2, 3, 4, or 8 units, our hourly expectation goes up to $39, $62, $86, and $162 respectively. So, although you can beat many shoe games with a flat bet, a small betting spread at higher counts has a lot of value.

The data in the chart above is all taken from a report I initially published in 1987 titled "Beat the 6-Deck Game." If you are a serious player, investing any substantial amount of money, I urge you to get the complete report, which is inexpensive. There are 44 full pages of charts, showing 6-deck results based on frequency distributions with varying levels of penetration, different strategies ranging from flat-bet table-hopping styles, to table-hopping with spreads up to 16 units, to playing through all hands of the shoe without table-hopping or back-counting.

These charts also display the unit win compared to standard deviation, showing various numbers of hands played, to give you a handle on what sort of fluctuation to expect at the tables. Finally, for those who are not scared of the math, and who want data not contained in any of the charts I provide, there is a more complete explanation of how you can use frequency distributions to perform your own analyses of any betting strategies you care to test. It's not really difficult if you've got a pocket calculator and you follow the instructions. There are also similar reports available for 1-deck, 2-deck, 4-deck and 8-deck games. You'll find information in the back section of this book on obtaining these reports.

• Blackjack – How To Read The 6 Deck/75% Dealt Chart Part5 (0)
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• Blackjack – How To Read The 6 Deck/75% Dealt Chart Part2 (0)

Now, let's look at the five rows of data beneath the betting systems, labeled Hands Bet I100, Average Bet, Gain/Hand, Win Rate %, and Hourly Units. The data in these rows is derived from the columns. I'm not going to go into the precise formulas for obtaining all of these numbers, but I want you to understand the basic logic of how a mathematician figures out the value of a game and a betting strategy. None of this is rocket science.

Hands Bet/100 is a pretty easy number to come up with. Remember that the table-hopping/back-counting player is not betting on every hand; rather, he watches a lot of hands without betting, trying to avoid betting into any significant house edge. If you add up all of the numbers in the Hands column that show a bet greater than 0 in the respective 5 columns, you come up with how many hands (out of 100 hands seen), the player using each betting system would bet on. In 5-7, for instance, the numbers in the Hands column where we see player bets are:

34 + 13 + 8.5 + 4.5 + 3.5 + 2 + 2 + 1 + .5 = 69
Notice that with the betting systems where the player waits for a higher advantage before placing a bet, the entry in the Hands Bet/100 goes down.

Look at the next row, Average Bet. This is also pretty easy to figure out. Looking closely, you'll see that with 5-7 through 5-4, the Average Bet is always 1.0, because the player always bets one unit. But when the player starts spreading his bets upward, the average amount bet per hand goes up, as expected.

The next row is Gain/Hand. These entries tell us how many units, per hand played, the player would win with each betting system. The 5-7 entry is 0.002—if you multiply this number by your betting unit, you get the dollar amount you would win for every hand you played with this betting system.

Let's say your betting unit is $100.
$100 x 0.002 = .20, or 200 per hand
If that strikes you as an awfully small gain on a $100 bet, you're starting to think like a pro! (In fact, the actual gain is even smaller, only 160 per hand. The only reason we came up with 200 per hand is because the table entries are rounded off to three decimal places. The true gain per hand is 0.0016. Yuck!)

• Blackjack – How To Read The 6 Deck/75% Dealt Chart Part5 (0)
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• Blackjack – How To Read The 6 Deck/75% Dealt Chart Part2 (0)

Now we come to the various betting strategies we might try in this game. For our analysis, we'll look at eight different betting systems, which I've labeled S-l through S-8. Each of those columns represents a different way that card counters might bet in this game. The numbers in each column represent the number of units bet at each of the various advantages that occur. Note that zeros fill the whole upper portion for all of the betting systems. This is because all of the betting systems shown here are table-hopping approaches. The player is usually not betting anything when the advantage is negative.

In S-7, for instance, the player is using a 1 to 4 betting spread. Note that when the house has a 1/2% edge over him (-0.5%), such as when dealing right off the top of a fresh shoe, he bets 1 unit, and he also bets 1 unit when his edge is even with the house (0.0%). When his advantage is 0.5% over the house, he increases his bet to 2 units; and when it is 1.0% or more, he bets 4 units. The S-7 player is using a typical 1 to 4 betting spread in this game, but leaves the table when the count goes negative. Look at all eight betting systems, and make sure you understand how each player attacks this game.

6 Decks 75% Dealt
Adv. Hands S-1 S-2 S-3 S-4 S-5 S-6 S-7 S-8
-4.0% 0.0 0 0 0 0 0 0 0 0
-3.5% 1.0 0 0 0 0 0 0 0 0
-3.0% 2.0 0 0 0 0 0 0 0 0
-2.5% 3.0 0 0 0 0 0 0 0 0
-2.0% 4.0 0 0 0 0 0 0 0 0
-1.5% 8.0 0 0 0 0 0 0 0 0
-1.0% 13.0 0 0 0 0 0 0 0 0
-0.5% 34.0 1 0 0 0 1 1 1 1
0.0% 13.0 1 1 0 0 1 1 1 1
0.5% 8.5 1 1 1 0 2 2 2 2
1.0% 4.5 1 1 1 1 2 3 4 4
1.5% 3.5 1 1 1 1 2 3 4 8
2.0% 2.0 1 1 1 1 2 3 4 8
2.5% 2.0 1 1 1 1 2 3 4 8
3.0% 1.0 1 1 1 1 2 3 4 8
3.5% 0.5 1 1 1 1 2 3 4 8
4.0% 0.0 1 1 1 1 2 3 4 8
Hands Bet/100: 69 35 22 14 69 69 69 69
Average Bet: 1.00 1.00 1.00 1.00 1.32 1.51 1.71 2.23
Gain/Hand: .002 0.008 .013 .017 .006 .009 .012 .023
Win Rate %: .16% .79% 1.26% 1.74% .42% .59% 0.72% 1.05%
Hourly Units: .11 .28 .28 .24 .39 .62 .86 1.62

• Blackjack – How To Read The 6 Deck/75% Dealt Chart Part5 (0)
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• Blackjack – How To Read The 6 Deck/75% Dealt Chart Part3 (0)

The chart title, "6 Decks/75% Dealt (pg 177)," tells us the specific game we're looking at—with a different number of decks, or a different percentage dealt out, the frequency distribution would look different. Advantages change radically with different numbers of decks and greater or lesser penetration. This freak chart is specifically for a 6-deck game where the dealer is dealing out about 4 1/2 decks (75% of the cards) between shuffles—a fairly typical shoe game.

The first column, labeled Adv. (an abbreviation for "advantage"), shows the percent of occurance for various player advantages, positive and negative. Note that the entries in this column run from -4.0% to +4.0%. In a 6-deck game with 75% penetration, house/player advantages greater than this occur very rarely. If this chart was for a 6-deck game where 5 1/2 decks were being dealt out between shuffles (92% penetration), or for any single-deck game, we would need a freak chart that showed the occurrence of much greater house/player advantages.

The second column, labeled Hands, is the frequency distribution. That's the data that shows how many hands—per 100—will occur with the percent advantage shown in the first column. Example: In this six-deck game with 75% (4 1/2 decks) dealt, a player advantage of 1% will occur about 4.5 times per hundred hands. A house advantage of 1% will occur about 13 times per 100 hands. Note that the house advantage of 1% occurs almost 3 times as often as the player advantage of 1 % This is because the house has a 1/2% edge off the top. In a sense, the house gets a running start on you! Before you continue reading this explanation, find these numbers in the Hands column. Be sure you understand how to read the data in the first two columns before you go on to the rest of the chart.

• Blackjack – How To Read The 6 Deck/75% Dealt Chart Part5 (0)
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