Simulation reveals shocking conclusion (1 vs 2 hands)

[ArcticInferno]

I'm trying to analyze the merits of playing two hands all the time.
I used CVData 5 to run the simulation.

Simulation parameters:
Decks: 6
Number of players: 2
Primary Strategy: Basic High-Low for both players
Number of rounds: 1 billion

First Seat: 1 unit = $50
Bet 1 unit for TC <= 1
Bet 3 units for TC = 2
Bet 6 units for TC >= 3

Second Seat: 1 unit = $25
Bet 1 unit on two hands for TC <=1 (yes, two hands)
Bet 3 units on two hands for TC = 2
Bet 6 units on two hands for TC >= 3

The Player 2 puts out the same amount of money as the Player 1,
in terms of absolute dollars. $50 = $25 x 2 hands

Player 2 had twice the number of hands played, as expected.
Player 1 win rate: $ 38.77
Player 2 win rate: $ 42.77
Player 1 Standard Deviation: 25.28 per 100 rounds
Player 2 Standard Deviation: 46.99 per 100 rounds
Player 1 SCORE: 9.41
Player 2 SCORE: 13.26

The higher win rate and SCORE for Player 2 was a pleasant discovery.
However, the shocking conclusion of Player 2 having a higher standard deviation
was not expected. Why???!!!
I was expecting (hoping) that playing multiple hands all the time would
actually lower the standard deviation.
Did I not run the simulation properly?
Can someone please repeat my simulation and confirm the findings?

 

[kewljason]

I'm trying to analyze the merits of playing two hands all the time.
I used CVData 5 to run the simulation.

Simulation parameters:
Decks: 6
Number of players: 2
Primary Strategy: Basic High-Low for both players
Number of rounds: 1 billion

First Seat: 1 unit = $50
Bet 1 unit for TC <= 1
Bet 3 units for TC = 2
Bet 6 units for TC >= 3

Second Seat: 1 unit = $25
Bet 1 unit on two hands for TC <=1 (yes, two hands)
Bet 3 units on two hands for TC = 2
Bet 6 units on two hands for TC >= 3

The Player 2 puts out the same amount of money as the Player 1,
in terms of absolute dollars. $50 = $25 x 2 hands

Player 2 had twice the number of hands played, as expected.
Player 1 win rate: $ 38.77
Player 2 win rate: $ 42.77
Player 1 Standard Deviation: 25.28 per 100 rounds
Player 2 Standard Deviation: 46.99 per 100 rounds
Player 1 SCORE: 9.41
Player 2 SCORE: 13.26

The higher win rate and SCORE for Player 2 was a pleasant discovery.
However, the shocking conclusion of Player 2 having a higher standard deviation
was not expected. Why???!!!
I was expecting (hoping) that playing multiple hands all the time would
actually lower the standard deviation.
Did I not run the simulation properly?
Can someone please repeat my simulation and confirm the findings?

I can't answer your question about the SD. Off hand it doesn't seem right to me. But I really wanted to comment on win rate comparison $38.77/100 hands vs $42.77/100 hands. Per 100 hands is a fair comparison simulation-wise, but in the real world which involves time, the results would be flipped, making the single hand the better play.

You didn't give specifics of the game, but lets assume 6 decks, 75% pen playing heads up. Thats 234 cards seen before the cut card. Playing 1 hand @ 2.7 cards per dealer and player (standard average, although in heads up play it as actually slightly less because dealer won't finish hand after player breaks), so 5.4 cards per round equals 43.44 rounds per shoe (rounded up as once the round is started it is of course completed) 44 rounds per shoe. Playing 2 hands using the same math would come to 28.88 rounds per shoe, rounded up to 29. Lets say the dealer is capable of dealing 4 shoes an hour, playing 1 hand would result in 176 rounds per hour x 38.77 cents per round ($38.77/100 round) equals $68.23 per hour. Playing two hands would using the same math would result in 116 rounds per hour (same 4 shoes) x 42.77 cents per round equals $49.61 per hour. So while playing 2 hands would result in a slightly higher win rate per 100 rounds, it would earn far less per hour.

 

[SleightOfHand]

I can't answer your question about the SD. Off hand it doesn't seem right to me. But I really wanted to comment on win rate comparison $38.77/100 hands vs $42.77/100 hands. Per 100 hands is a fair comparison simulation-wise, but in the real world which involves time, the results would be flipped, making the single hand the better play.

You didn't give specifics of the game, but lets assume 6 decks, 75% pen playing heads up. Thats 234 cards seen before the cut card. Playing 1 hand @ 2.7 cards per dealer and player (standard average, although in heads up play it as actually slightly less because dealer won't finish hand after player breaks), so 5.4 cards per round equals 43.44 rounds per shoe (rounded up as once the round is started it is of course completed) 44 rounds per shoe. Playing 2 hands using the same math would come to 28.88 rounds per shoe, rounded up to 29. Lets say the dealer is capable of dealing 4 shoes an hour, playing 1 hand would result in 176 rounds per hour x 38.77 cents per round ($38.77/100 round) equals $68.23 per hour. Playing two hands would using the same math would result in 116 rounds per hour (same 4 shoes) x 42.77 cents per round equals $49.61 per hour. So while playing 2 hands would result in a slightly higher win rate per 100 rounds, it would earn far less per hour.

You also have to consider that you would be betting more when you are playing 2 hands. Following the rule of 150%, playing 2 hands would yield $74.41 per hour.

 

[kewljason]

You also have to consider that you would be betting more when you are playing 2 hands. Following the rule of 150%, playing 2 hands would yield $74.41 per hour.

Well, I would yes! But that was not the terms of the original simulation. He specifically broke the wager in half $50 vs 2 x $25. I used his simulation for my thoughts.

 

[Nynefingers]

more details?

Can you post more details of the sim? Rules, pen, etc. Also, please clarify the standard deviation numbers, as they don't look even close to right. Your standard deviation per hand should be greater than what you quoted, and per hundred hands it should be 10x greater than that.

Using CVCX, S17 DAS RSA 75% pen, HiLo with Sweet 16 & Fab 4, using your betting ramps, and assuming the $25 and $50 unit sizes you used, I get the following numbers:

Heads up:
#hands          1            2
$/100        $42.29       $43.29
SD/100     $1246.83     $1032.95
SCORE         12.06        17.57

4 players:
#hands          1            2
$/100        $42.61       $42.61
SD/100     $1237.18     $1024.96
SCORE         11.86        17.28

I think Norm has said that CVCX's handling of playing two hands is an approximation, but still the point is the winrates will be similar and the SD will decrease when going to 2 hands with the same total amount of money wagered.

[edit]

Aha! I think I figured it out. You are reporting SD in units. Your SD increased from 25.28 units to 46.99 units when you went to 2 hands, but your units are half as big. I think you mean your SD went from $1264 to $1174.75. That is still a 7% decrease in SD. Not as big as I expected, but you also got a bigger increase in winrate than I did. These two factors combined to produce a 41% improvement in your SCORE.

 

[SleightOfHand]

Can you post more details of the sim? Rules, pen, etc. Also, please clarify the standard deviation numbers, as they don't look even close to right. Your standard deviation per hand should be greater than what you quoted, and per hundred hands it should be 10x greater than that.

Using CVCX, S17 DAS RSA 75% pen, HiLo with Sweet 16 & Fab 4, using your betting ramps, and assuming the $25 and $50 unit sizes you used, I get the following numbers:

Heads up:
#hands          1            2
$/100        $42.29       $43.29
SD/100     $1246.83     $1032.95
SCORE         12.06        17.57

4 players:
#hands          1            2
$/100        $42.61       $42.61
SD/100     $1237.18     $1024.96
SCORE         11.86        17.28

I think Norm has said that CVCX's handling of playing two hands is an approximation, but still the point is the winrates will be similar and the SD will decrease when going to 2 hands with the same total amount of money wagered.

[edit]

Aha! I think I figured it out. You are reporting SD in units. Your SD increased from 25.28 units to 46.99 units when you went to 2 hands, but your units are half as big. I think you mean your SD went from $1264 to $1174.75. That is still a 7% decrease in SD. Not as big as I expected, but you also got a bigger increase in winrate than I did. These two factors combined to produce a 41% improvement in your SCORE.

I was also a bit confused when I saw different win rates, as they theoretically should be the same. However, since his sim uses 2 players at the same table, if the player playing 2 hands went after the 1 handed player, he would be able to make marginally better playing decisions which would affect his win rate, although 10% seems a little much.

 

[Nynefingers]

I was also a bit confused when I saw different win rates, as they theoretically should be the same. However, since his sim uses 2 players at the same table, if the player playing 2 hands went after the 1 handed player, he would be able to make marginally better playing decisions which would affect his win rate, although 10% seems a little much.

I felt the same way, but I didn't bother running a sim since I didn't have the details to properly match up to ArcticInferno's sim. AI, did you check the "remove seat effect" box? Even that won't really give you a fair comparison. You really need to run 2 sims each with the same number of additional hands in play.

 

[QFIT]

(Dead link: http://www.blackjackinfo.com/bb/member.php?u=4511) _Nynefingers_ had it right. Click on the Customize button. Turn on "Display Std. Devs. in $" and Exit. The std dev will make more sense.

It's been long known that always playing two hands is superior.

 

[ArcticInferno]

I re-ran the simulation using the same settings and parameters.
Total number of players: 2
Remove seat effect: Check
6 decks, 75% penetration
Rounds per Hour: 100
Strategy: Basic High-Low for both players and all hands.

First Seat: 1 unit = $50
Bet 1 unit for TC <= 1
Bet 3 units for TC = 2
Bet 6 units for TC >= 3

Second Seat: 1 unit = $25
Bet 1 unit on two hands for TC <=1 (yes, two hands)
Bet 3 units on two hands for TC = 2
Bet 6 units on two hands for TC >= 3

The Player 2 puts out the same amount of money as the Player 1,
in terms of absolute dollars. $50 = $25 x 2 hands
There're two players at the table, and total of three hands are always played.

I used the default set of rules, such S17, DAS, insurance, etc. However,
the rules (or the penetration) should be irrelevant for the purposes of my
experiment, because I want to compare the two players who're playing
under the same set of conditions.

Player 1 win rate: $39.09/hr
Player 2 win rate: $42.71/hr
Player 1 Standard Deviation: $1263.71/100 rounds (25.27)
Player 2 Standard Deviation: $1174.38/100 rounds (46.98)
46.98/2 = 23.49 which is less than 25.27
Player 1 SCORE: 9.57
Player 2 SCORE: 13.23

Nynefingers, you're right. I misinterpreted the original data.
QFIT, I read many books, and none of them says that playing two hands is superior.
As a matter of fact, most authors advocate playing one hand unless there're other
players at high counts, or near the cut card. They often make statements like,
"... better to spread vertically than horizontally,..." etc.
Despite what's stated in the books, my experiment confirmed my theory.
A high limit such as $50 would require a smaller spread to yield the same amount
of money as a low limit such as $15 or $25 with a larger spread.
Also, the smaller spread would result in smaller standard deviation.
Now, if the high limit such as $50 is cut into two hands of $25, then the
standard deviation would be reduced further, while slightly increasing the yield.
Smaller spread would also draw less heat.
To increase yield (in terms of absolute dollars, not units), I would just play at higher
limits and maintain the small spread (and of course continue to play two hands).
I'll publish this data in "Blackjack Attack II, What Don didn't know". LOL!
By the way, I didn't really study http://www.blackjackincolor.com/penetration11.htm
prior to running my simulation. I read that link in depth only after I had confirmed
my theory.

 

[QFIT]

QFIT, I read many books, and none of them says that playing two hands is superior.

My book does. Blackjack Attack may also. You're right, I don't remember seeing it anywhere else. Come to think of it, it may not have been known until the last several years.

You are also right that it is an interesting find that surprised most of us.

 

[assume_R]

Is it really a fair comparison though? I suppose if you want the spread to be equal and the exact $$ bet at each TC to be equal, then even playing 2 hands at -counts would be advantageous. But shouldn't you also consider the fact that given a minimum bet allowed at a table, you can put a much lower bet out (and thus a higher spread) if you drop to 1 hand (1x1units versus 2x1units) at -counts??

So if you play at a $25 min table, then why would you bet 2x$25 versus 1x$25 at -counts? It would probably draw less heat to have a lower spread and to not switch between # of hands, but that could also be a subjective call.

 

[Renzey]

I am trying to determine the value of playing 2 hands all the time. Ran this sim;
Total number of players: 2

First Seat: 1 unit = $50
Bet 1 unit for TC <= 1
Bet 3 units for TC = 2
Bet 6 units for TC >= 3

Second Seat: 1 unit = $25
Bet 1 unit on two hands for TC <=1 (yes, two hands)
Bet 3 units on two hands for TC = 2
Bet 6 units on two hands for TC >= 3

Player 1 win rate: $39.09/hr
Player 2 win rate: $42.71/hr
Player 1 Standard Deviation: $1263.71/100 rounds (25.27)
Player 2 Standard Deviation: $1174.38/100 rounds (46.98)

I read many books, and none of them says that playing two hands is superior.
.

I apologize for not taking the trouble to read the entire thread thoroughly, but I believe your sim did include 2 simultaneous players at the table. But playing 2 hands all the time would also include heads up play. That's the sim I think you'd want to compare to playing 1 hand at a time. Did I miss something?

 

[Renzey]

I apologize for not taking the trouble to read the entire thread thoroughly, but I believe your sim did include 2 simultaneous players at the table. But playing 2 hands all the time would also include heads up play. That's the sim I think you'd want to compare to playing 1 hand at a time. Did I miss something?

Thinking about it a little further: Suppose you have 3 players, Joe, Bill and Bob at the table. All play Hi/Lo with identical betting ramps. All play their hands according to basic strategy to eliminate any advantage of seat position. Joe uses $50 units while Bill and Bob each use $25 units. At the end of eternity, Bill and Bob must each win half of what Joe wins -- no???

Now let's say that seats number 2 and 3 just happened to be BillyBob -- one person playing both hands! How could BillyBob's results be any different than Bill's and Bob's added together??? Can someone help me here???

 

[1357111317]

So just to make sure I understood this correctly. If you are paying HU vs a dealer and arent bumping into the table max you should always play 1 hand right? Compared to 1 hand in neutral/neg counts to 2 hands in positive counts?

 

[ArcticInferno]

My original simulation had two players at the same table.
The first player played one hand, while the second player played two hands all the time.
So, total of three hands were played against the dealer's same hand.
I assumed that the number of players at the table would have negligible impact in
6 decks.

For those who're curious about the "heads-up" results, I re-ran the simulation except
this time for 10 billion rounds.

First Table: 1 unit = $50
Number of players: 1
Bet 1 unit for TC <= 1
Bet 3 units for TC = 2
Bet 6 units for TC >= 3

Second Table: 1 unit = $25
Number of players: 1
Bet 1 unit on two hands for TC <=1
Bet 3 units on two hands for TC = 2
Bet 6 units on two hands for TC >= 3
Reset after shuffle: Unchecked (This is important to maintain two hands at all times.)
The second table has only one player who's playing two hands.

Player at Table 1 win rate: $37.77/hr
Player at Table 2 win rate: $38.55/hr
Player at Table 1 Standard Deviation: $1258.13/100 rounds
Player at Table 2 Standard Deviation: $1047.47/100 rounds
Player at Table 1 SCORE: 9.01
Player at Table 2 SCORE: 13.54

My theory is still correct. Even if you're playing heads-up, it's still better to play
two hands.

Personally, I would sacrifice yield for low variance. By lowering the standard deviation,
I'm basically buying piece of mind so I wouldn't get stomach ulcers and lose sleep at
night.
The slight increase in win rate was a pleasant bonus. But why is there an increase in
win rate at all? Maybe because more cards are dealt after the cut card,... ?
Which translates to slightly deeper penetration,... ?
But what about when both players are at the same table? Don't they get the same penetration?

 

[ArcticInferno]

I understand what you guys are saying about switching between one hand and two
hands based on the count. Some here have also commented about the 150% rule when
spreading to two hands at high counts, etc. Constantly switching between one hand
and two hands is like having the scarlet letter C on your chest. Do you split tens?
At Borgata in Atlantic City NJ, a dealer told me that he was told to look for players
who spread to two hands while raising the bets. It's extremely uncommon for a
"normal" player to constantly switch between one hand and two hands.

 

[QFIT]

The studies have been done. My fragile memory suggests that David D'Aquin suggested it first. But it is clear from the sims that two hands all the time is better. It's difficult to "logic it out" as you must deal with true count frequency changes caused by changing the number of hands during a shoe and variance correlation resulting from multiple hands against the same dealer upcard.

 

[kewljason]

My original simulation had two players at the same table.
The first player played one hand, while the second player played two hands all the time.
So, total of three hands were played against the dealer's same hand.
I assumed that the number of players at the table would have negligible impact in
6 decks.

For those who're curious about the "heads-up" results, I re-ran the simulation except
this time for 10 billion rounds.

First Table: 1 unit = $50
Number of players: 1
Bet 1 unit for TC <= 1
Bet 3 units for TC = 2
Bet 6 units for TC >= 3

Second Table: 1 unit = $25
Number of players: 1
Bet 1 unit on two hands for TC <=1
Bet 3 units on two hands for TC = 2
Bet 6 units on two hands for TC >= 3
Reset after shuffle: Unchecked (This is important to maintain two hands at all times.)
The second table has only one player who's playing two hands.

Player at Table 1 win rate: $37.77/hr
Player at Table 2 win rate: $38.55/hr
Player at Table 1 Standard Deviation: $1258.13/100 rounds
Player at Table 2 Standard Deviation: $1047.47/100 rounds
Player at Table 1 SCORE: 9.01
Player at Table 2 SCORE: 13.54

My theory is still correct. Even if you're playing heads-up, it's still better to play
two hands.

Personally, I would sacrifice yield for low variance. By lowering the standard deviation,
I'm basically buying piece of mind so I wouldn't get stomach ulcers and lose sleep at
night.
The slight increase in win rate was a pleasant bonus. But why is there an increase in
win rate at all? Maybe because more cards are dealt after the cut card,... ?
Which translates to slightly deeper penetration,... ?
But what about when both players are at the same table? Don't they get the same penetration?

I still say you are comparing by per 100 rounds which does not take the element of time in to the equation. It will take longer time-wise to play 100 rounds of two hands per round than 100 rounds of 1 hand per round, making your hourly rate much lower.

 

[ArcticInferno]

The number of hands wouldn't change during a shoe.
As a matter of fact, one of the parameters is to maintain
constant number of hands.

 

[ArcticInferno]

Hey kewljason, I agree with you about the time and stuff.
However, in practical viewpoint, in Atlantic City, I've
never been at a table alone. How much longer does it
take to deal me an extra hand when there're 2 or 3 other
random players at the table?
Also, I'm trying to lower the standard deviation, not increase yield.