# Thread: Prize Car: playing 2 spots vs 1

1. ## Prize Car: playing 2 spots vs 1

I posted this on a different web site as well. I am looking for answers to this question. Any input is appreciated.

Does playing 2 spots heads-up make more money than just playing 1 spot?

Every book I've read says to play one spot heads-up, unless you hit table max or other people joining in. For example, SW's book PBJ suggests that in a heads-up situation you should play one spot, and that it really doesn't matter much if you play one spot or two.

I think it does matter. Somebody please tell me I have delusions of grandeur and I am wrong (if applicable). Here's my reasoning:

If you're a seasoned card counter you're probably able to play quickly, and your speed at the bj table is mainly limited by the dealer's experience and speed, rather than your lack thereof.

In PBJ, the table on p. 235 shows the number of seconds per dealing round are 12 and 20, going from 1 hand to 2, respectively. However, those numbers are averages that include bj decisions that are sometimes made on "ploppy time."

If it doesn't matter much whether you play 1 spot or 2, then why not play 2 spots, since you can wave 2 pairs of 20's with one long wave and be done with them, and since you're able to put the money out there more quickly in DAS and DD situations (if that's your cover, change your cover). I suspect that playing two (or more*) hands makes more money in all situations, except where the TC's negative.

So if doubling the spots you're playing doesn't double the dealing time, but only a fraction of the dealing time, then isn't it worth it to always play 2 spots?

I attempted to work out the math on this. Assuming 240 cards are used from a 6-deck shoe, here's the paste from my excel sheet:

sec/hand 12.00
hands/min 5.00
hands/hr 300.00

cards/hand 2.70
players/round 2.00
cards/round 5.40
cards/shoe 240.00
rounds/shoe 44.44

shuffle time (sec) 100.00

Total shoe time
sec/shoe= 533.33
sec/shuffle 100.00
sec/tot shoe 633.33
min/tot shoe 10.56
shoes/min 0.09
shoes/hr 5.68
hands/hr 252.63

For comparison, my figures agree well with PBJ's chart on p. 237.

If you play two spots and take as much time as 2 different people (20 sec vs 12 sec), the hands would not take double the time, but only 67% more time (20/12 = 1.67). Moreover, couldn't this number really be more like 1.4 to 1.5, since we APs make decisions faster, even irrespective of dealer speed?

First, let's assume conservatively that all card counters have ploppy-like speed and we just can't make up our minds about anything. Furthermore, we have 2 different personalities and as a result it takes us the same time to make a decision playing 2 spots as it does 2 separate ploppies playing 1 spot each. Of course, we're betting proportionately lesson both spots to achieve our desired RoR levels. Notice I changed the number of players to 3 (you, you again, and the dealer).

sec/hand 20.00
hands/min 3.00
hands/hr 180.00

cards/hand 2.70
players/round 3.00
cards/round 8.10
cards/shoe 240.00
rounds/shoe 29.63

shuffle time (sec) 100.00

Total shoe time
sec/shoe= 592.59
sec/shuffle 100.00
sec/tot shoe 692.59
min/tot shoe 11.54
shoes/min 0.09
shoes/hr 5.20
hands/hr 154.01

If you multiply this number by 2, you will get 314. This number is 24.6% greater than 252. So my math now shows that playing 2 spots vs 1 heads-up makes 25% more money.

Now let's assume you're the average seasoned veteran card counter and can play reasonably quickly, and it doesn't take your brain forever to spit out index-related decisions. Then that 1.67 number drops a bit. So on the 2nd hand you take only half the time to make the decision. Using the "7n + 5" number from PBJ, where n is the number of players at the table, we see that each extra person increases the seconds/round by 7 more seconds. So let's halve that to 3.5. So the average number of seconds per round is now 20 - 3.5, or 16.5 seconds. That gives us a new data set:

sec/hand 16.50
hands/min 3.64
hands/hr 218.18

cards/hand 2.70
players/round 3.00
cards/round 8.10
cards/shoe 240.00
rounds/shoe 29.63

shuffle time (sec) 100.00

Total shoe time
sec/shoe= 488.89
sec/shuffle 100.00
sec/tot shoe 588.89
min/tot shoe 9.81
shoes/min 0.10
shoes/hr 6.11
hands/hr 181.13

This new number, 181.13, multiplied by your 2 spots, yields 362.26. This new figure is 43.75% greater than our original 252 hands/hr.

So from my analysis, it seems that playing 2 spots instead of 1 will earn the average AP anywhere between 30% and 40% more money. If some finds any significant errors in my math or analysis, please let me know!

Prize Car

2. ## Prize Car: Re: playing 2 spots vs 1, plus some ideas

Additionally, here's the analysis of the advantages of playing 3 spots instead of 2 when you're playing with one other player. Also, near the end of this post, you'll find the 3 spot analysis of heads-up play.

Playing 2 spots vs one other player (3 players total + dealer = 4):

sec/hand 27.00
hands/min 2.22
hands/hr 133.33

cards/hand 2.70
players/round 4.00 (you, you, the other player, and the dealer)
cards/round 10.80
cards/shoe 240.00
rounds/shoe 22.22

shuffle time (sec) 100.00

Total shoe time
sec/shoe= 600.00
sec/shuffle 100.00
sec/tot shoe 700.00
min/tot shoe 11.67
shoes/min 0.09
shoes/hr 5.14
hands/hr 114.29

This number times 2 spots equals 228.58 hands/hr.

If we play 3 spots, you get.....

sec/hand 35.00
hands/min 1.71
hands/hr 102.86

cards/hand 2.70
players/round 5.00
cards/round 13.50
cards/shoe 240.00
rounds/shoe 17.78

shuffle time (sec) 100.00

Total shoe time
sec/shoe= 622.22
sec/shuffle 100.00
sec/tot shoe 722.22
min/tot shoe 12.04
shoes/min 0.08
shoes/hr 4.98
hands/hr 88.62

88.62 times 3 equals 265.86. This is a 16.3% increase in win rate vs playing just 2 spots. This is using the standard table set that was created by Stanford as he was watching a few card counters, a few very slow people, but mostly just average-speed ploppies, I assume. So let's run everything again assuming any subsequent hands you have take you only half the time. So we subtract 3.5 seconds/round from the first data set above, and we subtract 7 seconds/round from the second data set. Then we have the following:

sec/hand 23.50
hands/min 2.55
hands/hr 153.19

cards/hand 2.70
players/round 4.00
cards/round 10.80
cards/shoe 240.00
rounds/shoe 22.22

shuffle time (sec) 100.00

Total shoe time
sec/shoe= 522.22
sec/shuffle 100.00
sec/tot shoe 622.22
min/tot shoe 10.37
shoes/min 0.10
shoes/hr 5.79
hands/hr 128.57

128.57 x 2 = 257.14, which is actually slightly more profitable than a single-spot heads up play (252 hands/hr).

and......

sec/hand 27.00
hands/min 2.22
hands/hr 133.33

cards/hand 2.70
players/round 5.00
cards/round 13.50
cards/shoe 240.00
rounds/shoe 17.78

shuffle time (sec) 100.00

Total shoe time
sec/shoe= 480.00
sec/shuffle 100.00
sec/tot shoe 580.00
min/tot shoe 9.67
shoes/min 0.10
shoes/hr 6.21
hands/hr 110.34

This means that playing 3 spots quickly (110.34 x 3 = 331.02 hands/hr) will earn 28.7% more money (331.02 / 257.14 = 1.287) than just playing 2 spots quickly.

Just for kicks:
3 spot analysis of a heads-up game, slow and fast:

sec/hand 27.00
hands/min 2.22
hands/hr 133.33

cards/hand 2.70
players/round 4.00 (you, yourself, you, and the dealer)
cards/round 10.80
cards/shoe 240.00
rounds/shoe 22.22

shuffle time (sec) 100.00

Total shoe time
sec/shoe= 600.00
sec/shuffle 100.00
sec/tot shoe 700.00
min/tot shoe 11.67
shoes/min 0.09
shoes/hr 5.14
hands/hr 114.29

This times 3 equals 342.87 hands/hr, which is 36% more profitable than 1-spot heads-up play, and 11.3% more profitable than playing 2 spots heads-up.

Given the increased heat from playing 3 spots, I think I'll start playing just 2 spots at all positive counts from now on, unless anyone has found any errors with these calculations.

Also, here are some ideas floating around in my head:

1. In a single-deck where the count can skyrocket or plummet on consecutive hands (or even the same hand), you want to play fewer spots, so you can have more time to gradually raise your bet. In a shoe game, this probably doesn't matter nearly as much. This goes against my previous post, but I'm not sure to what extent.

2. In a single deck game you want to play fewer spots becuase more hands means more variance. Playing 1 spot, you get cards #1 and #3. But playing 3 spots, you get cards #1, 2, 3, 5, 6 and 7. By that time, who knows how bad the count got! SW himself said pretty much the same thing in PBJ. This also goes against what I said earlier, but I doubt the change in variance is so huge as to change the RoR to such an extent to lower your bets enough to negate the advantage of playing more spots. But alas, I've not done that math, and by golly I'm not going to! (gotta watch a movie with my wife).

Cheers,

Prize Car

3. ## Don Schlesinger: Re: playing 2 spots vs 1

> I posted this on a different web site as well. I am
> looking for answers to this question. Any input is
> appreciated.

> Does playing 2 spots heads-up make more money than
> just playing 1 spot?

I guess I've answered this question 100 times in the past 30 years, maybe more. And, of course, it is answered in BJA3, as well. But, I can't comment on your numbers tonight -- it's too late -- so I'll read what you've written tomorrow. But, the short answer is that, if you're betting optimally, you should win about the same, with heads-up play, whether you play one hand or two.

Curiously, I don't see anywhere in your analysis how you're going to bet the two hands. Don't you think that should have appeared somewhere in your discussion??

Don

4. ## Prize Car: Re: playing 2 spots vs 1

> I guess I've answered this question 100 times in the
> past 30 years, maybe more. And, of course, it is
> answered in BJA3, as well.

I agree, in BJA3 it says to play one hand heads-up, and with others, play 2 hands at high counts.

> But, I can't comment on
> your numbers tonight -- it's too late -- so I'll read
> what you've written tomorrow. But, the short answer is
> that, if you're betting optimally, you should win
> about the same, with heads-up play, whether you play
> one hand or two.

I am claiming that you can win significantly more with 2 hands, whether or not you're heads-up.

> Curiously, I don't see anywhere in your analysis how
> you're going to bet the two hands. Don't you think
> that should have appeared somewhere in your
> discussion??

Yes, you're right, I didn't mention how to play two hands. To be explicit, the amounts would be proportional to maintain the desired fractional kelly/RoR. So each of the two bets would be somewhere around 70-75% of the original 1-spot bet. Also, I am conservatively saying "play all", although of course you'll win more money if you cut back to one hand at all non-positive counts.

I have done my own excel calcs, and I am also doing CVChapterX calcs to further support my conclusions.

HEADS-UP
ploppy speed

1 spot 2 spots
sec/round 12.00 20.00
rounds/min 5.00 3.00

cards/hand 2.70 2.70
players/round 2.00 3.00
cards/round 5.40 8.10
cards/shoe 240.00 240.00
rounds/shoe 44.44 29.63

shuffle t(sec) 100.00 100.00

sec/shoe= 533.33 592.59
sec/shuffle 100.00 100.00
sec/tot shoe 633.33 692.59
min/tot shoe 10.56 11.54
shoes/min 0.09 0.09
shoes/hr 5.68 5.20
hands/hr 252.63 154.01

earnings/hr 252.63 308.02
profit 100.00% 121.93%

HEADS-UP
normal speed

1 spot 2 spots
sec/round 10.00 16.50
rounds/min 6.00 3.64

cards/hand 2.70 2.70
players/round 2.00 3.00
cards/round 5.40 8.10
cards/shoe 240.00 240.00
rounds/shoe 44.44 29.63

shuffle t(sec) 100.00 100.00

sec/shoe= 444.44 488.89
sec/shuffle 100.00 100.00
sec/tot shoe 544.44 588.89
min/tot shoe 9.07 9.81
shoes/min 0.11 0.10
shoes/hr 6.61 6.11
hands/hr 293.88 181.13

earnings/hr 293.88 362.26
profit 100.00% 123.27%

One other plyr
ploppy speed

1 spot 2 spots
sec/round 20.00 27.00
rounds/min 3.00 2.22

cards/hand 2.70 2.70
players/round 3.00 4.00
cards/round 8.10 10.80
cards/shoe 240.00 240.00
rounds/shoe 29.63 22.22

shuffle (sec) 100.00 100.00

sec/shoe= 592.59 600.00
sec/shuffle 100.00 100.00
sec/tot shoe 692.59 700.00
min/tot shoe 11.54 11.67
shoes/min 0.09 0.09
shoes/hr 5.20 5.14
hands/hr 154.01 114.29

earnings/hr 154.01 228.57
profit 100.00% 148.41%

One other player
normal speed

1 spot 2 spots
sec/round 16.50 23.50
rounds/min 3.64 2.55

cards/hand 2.70 2.70
players/round 3.00 4.00
cards/round 8.10 10.80
cards/shoe 240.00 240.00
rounds/shoe 29.63 22.22

shuffle (sec) 100.00 100.00

sec/shoe= 488.89 522.22
sec/shuffle 100.00 100.00
sec/tot shoe 588.89 622.22
min/tot shoe 9.81 10.37
shoes/min 0.10 0.10
shoes/hr 6.11 5.79
hands/hr 181.13 128.57

earnings/hr 181.13 257.14
profit 100.00% 141.96%

5. ## Don Schlesinger: Re: playing 2 spots vs 1

> I agree, in BJA3 it says to play one hand heads-up,
> and with others, play 2 hands at high counts.

Alternatively, you could play two hands all the time. In later years, I've come to prefer that method, but again, heads-up, it hardly matters. It helps much more when there are others at the table.

> I am claiming that you can win significantly more with
> 2 hands, whether or not you're heads-up.

No, you can't.

> Yes, you're right, I didn't mention how to play two
> hands. To be explicit, the amounts would be
> proportional to maintain the desired fractional
> kelly/RoR. So each of the two bets would be somewhere
> around 70-75% of the original 1-spot bet. Also, I am
> conservatively saying "play all", although
> of course you'll win more money if you cut back to one
> hand at all non-positive counts.

OK, fine.

> I have done my own excel calcs, and I am also doing
> CVChapterX calcs to further support my conclusions.

But, I asked you about the wagering, because I suspected that that's where you went wrong ... and you did. See below.

> HEADS-UP
> ploppy speed

> 1 spot 2 spots

> hands/hr 252.63 154.01

You don't mean "hands"; you mean "rounds." And, for two spots, the number of hands would double, to 308.02.

> earnings/hr 252.63 308.02
> profit 100.00% 121.93%

No, sorry. This is your mistake. You're ascribing the same earnings to the \$100 hands (one spot, \$1 per hand)) that you are to the \$75 hands (two spots). But, the two spots earn only 75 cents each, per hand. So, the earnings per hour are 75% of your number, or \$231, and you're back where you started from ... or even a little worse.

So, as I've said many times, when you play two hands by yourself, in an effort to improve on one-hand heads-up play, you're not only playing BY yourself, you're also playing WITH yourself!! :-)

> HEADS-UP
> normal speed

> 1 spot 2 spots
> sec/round 10.00 16.50
> rounds/min 6.00 3.64

> cards/hand 2.70 2.70
> players/round 2.00 3.00
> cards/round 5.40 8.10
> cards/shoe 240.00 240.00
> rounds/shoe 44.44 29.63

> shuffle t(sec) 100.00 100.00

> sec/shoe= 444.44 488.89
> sec/shuffle 100.00 100.00
> sec/tot shoe 544.44 588.89
> min/tot shoe 9.07 9.81
> shoes/min 0.11 0.10
> shoes/hr 6.61 6.11
> hands/hr 293.88 181.13

> earnings/hr 293.88 362.26
> profit 100.00% 123.27%

Same comments as above.

> One other plyr
> ploppy speed

> 1 spot 2 spots
> sec/round 20.00 27.00
> rounds/min 3.00 2.22

> cards/hand 2.70 2.70
> players/round 3.00 4.00
> cards/round 8.10 10.80
> cards/shoe 240.00 240.00
> rounds/shoe 29.63 22.22

> shuffle (sec) 100.00 100.00

> sec/shoe= 592.59 600.00
> sec/shuffle 100.00 100.00
> sec/tot shoe 692.59 700.00
> min/tot shoe 11.54 11.67
> shoes/min 0.09 0.09
> shoes/hr 5.20 5.14
> hands/hr 154.01 114.29

> earnings/hr 154.01 228.57
> profit 100.00% 148.41%

Here, even at 75%, the two spots win more, as I've always stated.

> One other player
> normal speed

> 1 spot 2 spots
> sec/round 16.50 23.50
> rounds/min 3.64 2.55

> cards/hand 2.70 2.70
> players/round 3.00 4.00
> cards/round 8.10 10.80
> cards/shoe 240.00 240.00
> rounds/shoe 29.63 22.22

> shuffle (sec) 100.00 100.00

> sec/shoe= 488.89 522.22
> sec/shuffle 100.00 100.00
> sec/tot shoe 588.89 622.22
> min/tot shoe 9.81 10.37
> shoes/min 0.10 0.10
> shoes/hr 6.11 5.79
> hands/hr 181.13 128.57

> earnings/hr 181.13 257.14
> profit 100.00% 141.96%

Ditto.

Clear??

Don

6. ## Prize Car: my CV sim results say that I am wrong

Hi Don,

Here are the CV ChapterX results for one hand vs two, with both heads-up and with one other player playing. I will do this for 6, 2 and single deck, \$1 dollar chip denominations (for accuracy)

System: Hi-low, sweet 16
Hands run: system sim, 2x10^9
Decks: 6
Rules: pen 4.62/6, 77% (72 cards cut off)
S17, DAS, no surrender, no resplit aces
1 hands play-all, or 2 hands play-all
Spread: 1:20
Bankroll: 50k
Min. chip size: \$1. I reduced the chip size so we can get an accurate representation of a general idea, rather than the data being skewed because of some artifact like a chip denomination. Later, we'll do the sim with customary chip sizes as well.

Heads-up, one hand, slow, \$1 chips
TC<1 = 16
TC 1 = 56
TC 2 = 135
TC 3 = 213
TC 4 = 302
TC 5 = 320

Ave. bet: 53.31
Win/hr: 147.07
SD/hr: 1733.71
RoR: 0.7%
SCORE: 28.56
N0: 35,019

Now let's see what happens if we double up on our hands, heads-up, while decreasing our bet amounts proportionately:

Heads-up, two hands, slow, \$1 chips
TC<1 = 2x11
TC 1 = 2x41
TC 2 = 2x100
TC 3 = 2x157
TC 4 = 3x220

Ave. bet: 76.35
Win/hr: 128.92
SD/hr: 1611.42
RoR: 0.7%
SCORE: 41.56
N0: 24.061

So you win less.

But what if it doesn't take you 20 seconds/round to decide. Let's say instead of 20 sec/round for 2 spots, it only takes you 15 sec/round. Not the craziest idea I've had. So that's 196 hands/hr. That's a win rate of 164.08/hr (with the same 0.7% RoR) with the same chip ramp as immediately above. That's a 12% improvement. But wait, you say, because you have to compare apples to apples. So instead of 12 sec/hand playing 1 spot, you're faster and can do it in 10. That's 294 hands/hr, which CVCX says gives a win rate of 171.58/hr. So playing 1 spot heads-up wins more, and 2 spots wins less, according to CVCX.

So according to CVCX, I am wrong.

If I leave the table when the count drops, my win rates for 1 vs 2 hands are:

1 hand:
321.54/hr

2 hands:
307.48/hr

Well, maybe that answers it all.

Prize

7. ## Prize Car: Re: my CV sim results say that I am wrong

I reran for double deck. Same as for 6D.

Prize

8. ## Don Schlesinger: Re: my CV sim results say that I am wrong

> I reran for double deck. Same as for 6D.

But, you didn't even comment on the fact that you originally calculated the per-hand bets incorrectly. That's the crux of the problem.

I'm not saying that there can't be an error in BJA3, but, after so many years, rather than reinvent the wheel, don't you think that, if there had been an error, someone would have found it by now??

Don

9. ## Prize Car: the answer

I understand what you mean now. I didn't multiply the new win rate by 0.73. That was the problem, and thanks for finding it!

So the way I had my sim set up previously, it was placing 2 bets where before there was only 1, but for the full amount, not the 0.73x. Thank you for that. It took you and CVCX to slap me a couple of times in the face before I figured things out.

I do have one other point. That is, if the dealer is fast or slow. I will simulate this in CVCX (which precisely agrees with the chart on Page 24 of BJA3, by the way), and let you know....

Prize

P.S. Even if I sim this, I don't think it would be accurate. We'd have to actually go to casinos and categorize slow and fast dealers to get some real data, rather than me loosely approximate dealer speeds.

10. ## Prize Car: results of 1 spot vs 2 spots with slow vs fast dealer

Here are the results of the sims. This is also with a 6 Deck, same rules as before. It appears that the 7n + 5 figure from PBJ can be altered to reflect dealer speed. If the dealer is fast, he would be able to reduce the initial 12 seconds (heads-up) from maybe 12 to 9 or even faster, but what component of that is dealer, and what component is due to our average AP's speed? If you assume that a fast dealer will reduce the 7 to maybe 5, and the 5 to maybe a 3, then that leaves us with 5n + 3, which is 8 seconds. This even underscores SW's comment that some fast dealer can reach 500 hands/hr, as 6 seconds plugged into my equation is 436 hands/hr, but if you reduce the shuffle time from 100 sec to less, then there's your 500 hands/hr figure.

But if you assume you don't take long to make decisions, and the dealer's fast, that can subtract an additional 1 from the player component. So now we have:

4n + 3, or 7 seconds per round for a fast dealer, is for one spot. I'm going to use 7 seconds for this equation. For two spots, you can subtract another seconds since we're the same person. That would be 4(2) + 3 - 1, or 10 seconds. Let's plug that into our *correct* equation using the 0.73 multiplier:

earnings/hr 389.19 393.97
profit 100.00% 101.20%

This is corroborated by CVCX:

earnings/hr 311.88 307.21
profit 100.00% 98.48%

So both my excel sim and CVCX are close to each other (within 3%). So with a fast dealer, it's a wash. Now let's see what happens with a slow dealer....

Let's take the 7n + 5 number and modify it accordingly. If the dealer's slow, the "5" will definitely be slower, but he'll also react slower thereby increasing the "7" as well. So let's assume 7 for the 5, and for the 7n, let's add one second, and only reduce the total by half a second because we're the same player playing 2 spots. So that gives us

8n + 7 for 1 spot, and
8n + 7 - 0.5 for 2 spots

1 spot = 15 sec
2 spots = 22.5 sec

earnings/hr 208.70 203.13
profit 100.00% 97.33%

CVCX gives us:

167.57
158.74

difference: 94.4%

So very similar.

In this case, it's slightly better to play 1 hand.

Now let's break up the components of that equation and maximize for the 7n, then later maximize for the 5, and see what is better.

for the 7n + 5, let's say it changes to 5n + 7. So the dealer takes more time to do stuff. That would be 12 seconds. For 2 spots, 5n + 7 is 17, minus 1 is 16.

1 spot, 2 spots: 252.63 271.28

100.00% 107.38%

So it's over 7% more profitable if the dealer is slow with everything except you make a quick decision.

For the opposite, 7n + 5 turning into 9n + 3, you get 12 for 1 spot of course, and for 2 spots you get 21 sec.

1 spot, 2 spots: 252.63 215.63

100.00% 85.35%

This means playing 1 spot is much (almost 15%) more valuable.

So the bottom line is, when playing solo (heads-up):

1. If the dealer's fast, it's a wash.
2. If the dealer's slow, play 2 spots.
3. If the dealer's fast but takes more time with your hand, like confirming things like hard 19, etc., then play 1 spot.
4. If the dealer's fast with your decisions but regular to slow with his hand or whatever else, play 2 spots.

Any comments on this are appreciated.

Prize

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