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Card counters sometimes make big bets: find out why this is important.
This article continues our series on ways for card counters to fine tune their advantage over the casino in blackjack. We've looked at a number of different variables, with some surprising results:
Choosing the right bankroll: How playing without a sufficient bankroll can lead to a greater than 50% losing trip record, even with a positive expectation from card counting.
- Picking the right game: Which blackjack games actually yield a better hourly win rate for the card counter? 8-deck/S17, or 6-deck/H17?
- Number of players at the table: A lot has been written about the pros and cons of more or less players at the blackjack table playing with you. The actual statistical impact may surprise you.
- In this article, we examine the impact of your bet spread: the range from your minimum bet to your maximum bet, corresponding to the card count advantage.
Bet spreads and card counting
Almost all effective card counting systems generate a player advantage by having the player bet less on rounds where the edge is in the casino's favor, and by betting more when the player has the positive edge. Computer simulations prove that as the distribution of remaining cards changes from the average, the player advantage will vary enormously. With most count systems such as Stanford Wong's popular High-Low method, we look for a higher than average number of tens and aces remaining to be played. With High-Low, we keep track of the running count by summing the twos to sixes at a value of +1, and the tens and aces as a value of -1 (7, 8, and 9 have no value). We then divide the run count by the number of decks remaining to be played, to come up with the true count. Using Blackjack Audit, a blackjack simulator from DeepNet Technologies, we can compute the player expectation (sum of all winnings plus losses, divided by the sum of all bets) across a range of true counts. For example, Table 1 shows the results for the following game: eight deck, S17, DAS, 75% penetration, Full High-Low count system with indices, 1 to 10 bet spread, 1 billion blackjack simulation rounds in BJ Audit.
| <= -8 |
417328 |
0.042% |
-4.5% |
| -7 |
874905 |
0.087% |
-3.92% |
| -6 |
2368982 |
0.237% |
-3.51% |
| -5 |
5899566 |
0.590% |
-3.12% |
| -4 |
13761658 |
1.376% |
-2.52% |
| -3 |
30383946 |
3.038% |
-2.02% |
| -2 |
64551134 |
6.455% |
-1.48% |
| -1 |
134601851 |
13.460% |
-0.99% |
| 0 |
503301006 |
50.330% |
-0.31% |
| 1 |
129732306 |
12.973% |
0.41% |
| 2 |
62316119 |
6.232% |
0.98% |
| 3 |
29335819 |
2.934% |
1.55% |
| 4 |
13296903 |
1.330% |
2.12% |
| 5 |
5675311 |
0.568% |
2.80% |
| 6 |
2258306 |
0.226% |
3.41% |
| 7 |
829916 |
0.083% |
3.94% |
| >= 8 |
395607 |
0.040% |
4.6% |
Table 1: Player expectation at different true counts for High-Low
Not surprisingly, the player's advantage starts when the true count reaches 1, and continues to climb from there. This is why the High-Low count system uses a minimal one unit bet with any true count less than one (or zero, if you are 'wonging' by sitting out hands with negative counts), and starts increasing the bet for true counts greater than one. For the data above, the bet was increased to 3 units at TC = 1, 5 units at TC =2, and 10 units for TC >= 3.
The results in table 1 might make you wonder why players shouldn't just bet the maximum possible bet as soon at the true count is 1 or higher, rather than 'scale' the bets with the increasing true count as noted above. Although making the bet spread simply 10 units at TC >= 1 will increase your expectation and hourly win rate, it does so at a significant increase in your bankroll volatility as shown in Table 2. More importantly, few casinos would tolerate a player making such radical bet swings from hand to hand: bet spreading is the most common indicator of a card counter.
| Average wagers/round |
2.26 bet units |
3.65 bet units |
| Average profit/round |
.012 bet units |
.020 bet units |
| Standard deviation/round |
3.43 bet units |
5.83 bet units |
| 5% lifetime ROR bankroll |
1512 bet units |
2543 net units |
| Expectation |
0.5148% |
0.5486% |
Table 2: Comparison of a single 10-unit bet spread to a standard 1-10 bet spread
The lifetime bankroll numbers in Table 2 reflect the amount of money you need to play to have no more than a 5% chance of losing that initial bankroll, assuming an infinite number of games.
Lowering your bet spread
Your hourly earning rate is directly related to your minimum bet, given a fixed bet spread. If you double your betting spread from $5/$50 to $10/$100, you will double your hourly earning rate. The drawback is that you must double your bankroll as well to maintain the same risk of ruin. This subtle point is easy to disregard in practice, and can have large consequences on your earnings potential.
Suppose, for example, that you normally play blackjack with a $5 to $50 bet spread and a $5000 session bankroll with the following standard game: eight deck, DAS, S17, full High-Low count system, 75% penetration. On arrival at the casino, the tables are busy and you can only play at a $15 table, but it does have the added benefit of using six decks. Although you intend to play a 1 to 10 bet spread, you actually limit your highest bet to $75 as you nervously watch your bank fluctuate radically in front you. The following chart compares these playing environments:
| Expectation |
0.52% |
0.37% |
0.71% |
| Average bet/round |
$11.35 |
$26.50 |
$37.19 |
| Lifetime ROR ($5000 bankroll) |
11% |
28% |
30% |
| Avg. rounds to double bankroll |
64,387 |
21,560 |
7455 |
| Avg. rounds to lose bank |
67992 |
21,603 |
7676 |
| Expected hourly win rate |
$5.90 |
$9.80 |
$26.40 |
Table 3: Affects of adjusting the bet spread
Compare the middle six deck game to the eight deck game:
- Bad: About 1 in 3 times you play, you will lose all $5000 and walk out a grand loser.
- Good: When you do win, you will win almost twice as much even though the expectation is worse (caused by the smaller bet spread).
- Bad: You will lose (or double) your $5000 bankroll about three times faster.
- Summary: The loss caused by the decreased bet spread is not met by the gain from two less decks.
Now, consider the game in the third column, which uses the same bankroll, but has you stick to the proper $15 to $150 bet range. The expectation increases almost 50% over the eight deck game, simply by removing two decks (the increased bet does not affect the expectation). Not surprisingly, the win rate jumps up to $27 per hour. Yet, the risk of ruin is almost unchanged from the middle game. This amazing result can be understood by recognizing that the significantly improved expectation tempers the fluctuations caused by the larger bets, the exact opposite of what we saw in the middle game! The only down side is that your big win or loss is going to happen 3 times faster still compared to the middle game.
Conclusions
This revealing exercise reinforces an often stated but frequently disregarded observation: most of the earnings potential in multi-deck blackjack is delivered by using a large bet spread. Hopefully this analysis shows you the huge risks of intentionally or unintentionally lowering your bet spread.
A common mistake by novice card counters is to start lowering their maximum bet as their bankroll gets low, hence reducing the amount of profit they can make when they have the rare advantage over the casino. The solution is to play with a sufficient bankroll to reflect a reasonable risk of ruin for the amount of time you intend to play, lowering the occurrence of trip-ending sessions and the need to temper your maximum bets. Note that the bankroll numbers in this article are lifetime values, rather than trip risk of ruin results. Trip ROR differs in that it assesses the probability of losing a bankroll amount only within a fixed quantity of blackjack rounds. Both lifetime and trip ROR can be computed in Blackjack Audit, both empirically with simulations and with one-step mathematical equations programmed into the product for quick approximations that are generally accurate within 5% or better.
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