There
was a popular
article in the Wall Street Journal on a hedge fund manager that racked up
$710,000 in profit at the blackjack table.
According to the report, he started with a $10,000 line of credit and
turned in his big profit over a couple of sessions at the tables in the
Bellagio. He summed up his strategy as
follows:
[He] follow[s]
the instructions on a blackjack strategy card he lays out on the table. “The
only thing I do is I vary my bet sizes based on how well I’m doing,” he said.
Each time he wins, he increases the size of his wager; similarly, he decreases
the size of his wager each time he loses. “The concept is, if I lose 10 hands
in a row, I’m going to lose my minimum bet at least nine of those times…but if
I win 10 hands in a row by pressing my bet every time—not doubling it, just
making it bigger—I’m going to win a lot more than when I lose 10 hands in a
row,” he said.
Setting
aside the ins and outs of “proper” blackjack strategy for a second (do you
always double down on 10?), I thought it would be interesting to simulate his
risk management and position sizing strategy.
Based
on the above quote, I extrapolated the following rules:
- Break down your starting bankroll into
units. For instance, if you have a
$10,000 bankroll to start with, divide your bankroll into 100 units for a basic
bet of $100 per hand.
- Your first bet will always be 1 unit.
- If the last hand was a win, the next bet size
will be increased by some multiplier. As
the article stated, you don’t double your bet but increase by some set amount.
- If the last hand was a loss, the next bet
size will be one unit times some multiplier.
- Expectancy and edge play a large role in
blackjack strategy. In general, the
house or casino has about a 5% edge over blackjack players. Using perfect blackjack strategy
and a 6 card deck, the house advantage can be reduced to 0.44%. According to one resource, a
card counter who performs flawlessly can expect a 1% edge over the casino.
So
in the best case scenario and assuming the hedge fund manager was not counting
cards, his edge would be -0.44%. This is
the definition of a negative expectancy outcome – there is no way that he could
win over the long run. Your winning
percentage would then be defined as 49.56%.
Building the
Spreadsheet
If
you have read any of the
recent posts, you know the ins and outs of a Monte Carlo simulation. For this simulation, I again assumed we would
run 1000 simulations of 500 hands and determine the statistics on the entire
run.
I
assumed we would start with a bankroll of $10,000 and that a unit would be $100
giving us 100 units to start with.
If
the previous hand was a loss, our Losing Multiplier would be 1. Stated otherwise, if we lose a hand then our
next bet will be equal to 1 unit times the Losing Multiplier. This is the “cut your losses” mentality.
If
the previous hand was a win, our Multiplier would be 1.1. If we win a hand then our next bet will be
equal to our profit on the previous hand times the multiplier. This is the “let your winners ride” reasoning.
Here
is a screenshot of the spreadsheet:
For
Hand No. 0, we reference $F$2 as our starting bankroll.
For
Hand No. 1, the formula is:
=IF(RAND()<=Winning
Percentage, Previous Account Balance + Unit, Previous Account Balance – Unit)
Or
=IF(RAND()<=$I$3,E7+$F$1,E7-$F$1)
This
tells Excel to generate a random number.
If the random number is less than or equal to our winning percentage,
then you add 1 unit to the starting bankroll.
If the random number is greater than our winning percentage, then you
subtract 1 unit from the starting bankroll.
We don’t worry about the Losing Multiplier or Winning Multiplier because
we don’t have a previous hand history.
For
Hand No. 2, we apply the “let winners ride” and “cut our losses”
mentality. The formula is:
=IF(RAND()<=$I$3,IF((F7-E7)>0,F7+((F7-E7)*$I$2),F7+$F$1*$I$1),IF((F7-E7)>0,F7-((F7-E7)*$I$2),F7-$F$1*$I$1))
Again,
we tell Excel to compute a random number to determine a “win” or a “loss” for
the current hand.
Then
you have to look at the bankroll for the previous hand to determine our bet
size. Remember, if we had a winner last
hand we would increase our bet size to an amount equal to the profit times the winning
multiplier. If we had a loser last hand,
our bet size is equal to one unit times the losing multiplier.
So
if Hand No. 2 is a win AND your bankroll from the last hand increased (last
hand was a winner), then take the profit from the last hand times winning multiplier
and add it to your bankroll. Conversely,
if Hand No. 2 is a loss AND your bankroll from the last hand decreased (last
hand was a loser), then subtract one unit times the losing multiplier to the
previous bankroll.
The
process described above is reversed if Hand No. 2 is a loser.
Applying
these rules to Simulation 1, here is a chart of our bankroll over 500 hands:
Running the
Statistics
Next,
we can look at the statistics after 1000 runs of 500 hands of blackjack to
determine our expectations for what this system gives us as far as profit and
loss.
|
Profit/Loss
|
||
|
Minimum
|
(8737)
|
|
|
Maximum
|
8763
|
|
|
Average
|
(527)
|
|
|
Median
|
(508)
|
|
|
Standard Deviation
|
2516
|
|
|
Low
|
High
|
|
|
1-SD (68% Confidence)
|
(3043)
|
1988
|
|
2-SD (95% Confidence)
|
(5559)
|
4504
|
Here
is the histogram for our system:
|
Profit/Loss
|
|||
|
Bins
|
Occurrences
|
Ind %
|
Cum %
|
|
(8,737)
|
1
|
0.10%
|
0.10%
|
|
(6,987)
|
3
|
0.30%
|
0.40%
|
|
(5,237)
|
22
|
2.21%
|
2.61%
|
|
(3,487)
|
92
|
9.24%
|
11.85%
|
|
(1,737)
|
194
|
19.48%
|
31.33%
|
|
13
|
277
|
27.81%
|
59.14%
|
|
1,763
|
225
|
22.59%
|
81.73%
|
|
3,513
|
129
|
12.95%
|
94.68%
|
|
5,263
|
38
|
3.82%
|
98.49%
|
|
7,013
|
12
|
1.20%
|
99.70%
|
|
8,763
|
3
|
0.30%
|
100.00%
|
Yikes,
we didn’t get anywhere close to $710,000!
In fact, this wasn’t even in the realm of possibility! As expected, you can easily expect to be down
$527 on average if you bet 110% of your profits after every winning hand. At best, you would be up more than $8,700
bucks 3 times out of 1000!
Changing Our
Parameters to Make $710,000
If
we are to accept that the house edge is .44% even playing perfectly, then what
winning multiplier would we have to use to even have a chance of making
$710,000?
By
using a Winning Multiplier of 1.6 (betting 160% of your profits after a winning
hand), then we do have an outside chance of making the type of money that we
want. Check out the stats below:
|
Profit/Loss
|
||
|
Minimum
|
(15375)
|
|
|
Maximum
|
819518
|
|
|
Average
|
(490)
|
|
|
Median
|
(3788)
|
|
|
Standard Deviation
|
32294
|
|
|
Low
|
High
|
|
|
1-SD (68% Confidence)
|
(32785)
|
31804
|
|
2-SD (95% Confidence)
|
(65079)
|
64098
|
|
Profit/Loss
|
|||
|
Bins
|
Occurrences
|
Ind %
|
Cum %
|
|
(15,375)
|
1
|
0.10%
|
0.10%
|
|
68,115
|
992
|
99.20%
|
99.30%
|
|
151,604
|
5
|
0.50%
|
99.80%
|
|
235,093
|
0
|
0.00%
|
99.80%
|
|
318,583
|
0
|
0.00%
|
99.80%
|
|
402,072
|
0
|
0.00%
|
99.80%
|
|
485,561
|
0
|
0.00%
|
99.80%
|
|
569,050
|
1
|
0.10%
|
99.90%
|
|
652,540
|
0
|
0.00%
|
99.90%
|
|
736,029
|
0
|
0.00%
|
99.90%
|
|
819,518
|
1
|
0.10%
|
100.00%
|
This
says that 1 time out of 1000, we will go on a heater and win $736,029 - $820,000! However, it is way more likely that we will
lose money with this system.
Limitations
on the Spreadsheet
1. During all of the runs, we don’t simulate
getting a blackjack and winning 3:2 or 2:1.
We always win what we bet. I
would imagine that this would skew the numbers up a bit.
2. It allows us to bet an unlimited amount at
anytime. Again, most casinos wouldn’t
let you bet $100,000 on one hand! This
happened more than once on our “heater.”
3. We don’t take into account surrendering,
insurance and all of the other fun stuff that goes into playing blackjack.
4. It assumes we play “perfectly” at all
times.
5. We assume that you can lose more than your
starting bankroll. Can you say “marker”?
Changing the
Multipliers and Winning Percentage
Again,
just for fun, let’s change some of the parameters around so that we can see
what happens.
What if we
were expert card counters and could have an Edge of 1% and a Winning Multiplier
of 1.1?
|
Profit/Loss
|
||
|
Minimum
|
(5683)
|
|
|
Maximum
|
8856
|
|
|
Average
|
1114
|
|
|
Median
|
1007
|
|
|
Standard Deviation
|
2536
|
|
|
Low
|
High
|
|
|
1-SD (68% Confidence)
|
(1423)
|
3650
|
|
2-SD (95% Confidence)
|
(3959)
|
6186
|
|
Profit/Loss
|
|||
|
Bins
|
Occurrences
|
Ind %
|
Cum %
|
|
(5,683)
|
1
|
0.10%
|
0.10%
|
|
(4,230)
|
13
|
1.31%
|
1.41%
|
|
(2,776)
|
44
|
4.42%
|
5.82%
|
|
(1,322)
|
97
|
9.74%
|
15.56%
|
|
132
|
208
|
20.88%
|
36.45%
|
|
1,586
|
227
|
22.79%
|
59.24%
|
|
3,040
|
191
|
19.18%
|
78.41%
|
|
4,494
|
113
|
11.35%
|
89.76%
|
|
5,948
|
66
|
6.63%
|
96.39%
|
|
7,402
|
22
|
2.21%
|
98.59%
|
|
8,856
|
14
|
1.41%
|
100.00%
|
What if we
were expert card counters and decided to increase our bets when we lost and
when we won – a losing multiplier of 1.1 and a winning multiplier of 1.1?
|
Profit/Loss
|
||
|
Minimum
|
(7347)
|
|
|
Maximum
|
9026
|
|
|
Average
|
1276
|
|
|
Median
|
1101
|
|
|
Standard Deviation
|
2749
|
|
|
Low
|
High
|
|
|
1-SD (68% Confidence)
|
(1473)
|
4024
|
|
2-SD (95% Confidence)
|
(4222)
|
6773
|
|
Profit/Loss
|
|||
|
Bins
|
Occurrences
|
Ind %
|
Cum %
|
|
(7,347)
|
1
|
0.10%
|
0.10%
|
|
(5,709)
|
6
|
0.60%
|
0.70%
|
|
(4,072)
|
23
|
2.31%
|
3.01%
|
|
(2,435)
|
54
|
5.42%
|
8.43%
|
|
(797)
|
132
|
13.25%
|
21.69%
|
|
840
|
234
|
23.49%
|
45.18%
|
|
2,477
|
212
|
21.29%
|
66.47%
|
|
4,114
|
189
|
18.98%
|
85.44%
|
|
5,752
|
89
|
8.94%
|
94.38%
|
|
7,389
|
43
|
4.32%
|
98.69%
|
|
9,026
|
13
|
1.31%
|
100.00%
|
