vpFREE2 Forums

Multi Strike math

#1

I think I have the basics of multi-strike math down, although I've
probably screwed up something somewhere. I'm at work and they don't
have winpoker here so I made up numbers to use for inputs. I'll have
to run it for real tonight, but it looks reasonable with the made-up
numbers I used, based on what i remember. Corrections/further
explanations welcome, as well as suggestions for easier ways to
accomplish the same ends.

Also, I need to know the frequencies of free rides by level –
anybody know that number? I guess it would be on the LED Gaming
website somewhere, I'll try to find it tonight, but meanwhile if
somebody knows please post.

All right, first, here's how to figure the ER as I see it. You've
got four levels you need to average, each of which has a
different "Hand ER" (expected value of a hand played at that level,
not yet counting the value of advancing) due to different strategies
employed. The highest Hand ER is on the top level. Lower levels are
willing to sacrifice some Hand ER in order to improve the odds of
advancing.

So suppose you have the following chart of ERs by level:

level mult ER
1 1 98.50%
2 2 98.80%
3 4 99.10%
4 8 99.54%

Next you have to know how often you get to each level. So for each
strategy, you need to know the % of wins, which you can get when you
run the average win (ER). Assume the following numbers (I'll get to
how to run these strategies later). Also, you have to know the
percent of free rides and then combine that number with the win
percent, being careful not to double count cases where you both win
the hand and receive a free ride. This is done by adding the win
percent to the product of free rides and percent of hands lost. For
example, in level 2 below, the table says the odds of getting to
that level by winning the first level hand is 49.4%. With 4% of
hands having a free ride, the odds of advancing to level 2 are 49.4%
+ (4% times 50.6%) or 51.424%.

Now the table looks like this so far:

level mult Hand ER %adv-win %adv-FR %adv-cum
1 1 98.50% 100.0% 0.0% 100.000%
2 2 98.80% 49.4% 4.0% 51.424%
3 4 99.10% 47.7% 4.0% 25.605%
4 8 99.54% 46.0% 4.0% 12.331%

So to finish calculating the ER, we simply multiply the % Advance –
Cumulative for each level by the Multiplier for that level and then
multiply by the Hand ER, and divide the result by 4 (same as
multiplying each by 25% and summing):

lev mult Hand ER %win %FR %adv bet contrib
1 1 98.50% 100.0% 0.0% 100.00% 25% 24.63%
2 2 98.80% 49.4% 4.0% 51.424% 25% 25.40%
3 4 99.10% 47.7% 4.0% 25.605% 25% 25.37%
4 8 99.54% 46.0% 4.0% 12.331% 25% 24.55%
              
                                      total 99.95%

Now, last but not least, how to get real numbers and how to run
strategy?

To do this, we start at the top level, level 4, because here we know
the strategy is the same as the regular one-line game. So our ER is
99.54% and the advance percent doesn't matter because there is no
further level to advance to.

Next, we go to level 3 and do an iterative process. First, assuming
46% of level 3 winners will advance to level 4, we calculate the
additional value of all winning hands as 46% times the ER 99.54%
times 2 (because level 4's multiplier of 8 is 2 times the level 3
multiplier of 4). The answer, 1.991, is added to all levels of the
pay table for level 3, and the strategy is rerun. This will give a
new number for the percent of winning hands, which we then
substitute in the above formula and refigure the add-on per hand,
and rerun the ER. Repeat this as necessary until the % advance
doesn't change (which should only take two or three iterations at
most).

Next, go to level 2 and repeat the process. The only difference here
is that in addition to the level 3 ER, all winning hands receive in
addition two times the level 3 winning percentage times the level 3
add-on. For example, for level 2's additional value per winning
hand, you use 2 times the level 3 ER plus 2 times the level 3
advance percent % times the level 3 add-on, or 2 * .9910 + 2 *

Based on the made-up numbers in the table above, the results are as
follows:

Add-ons to pay table for each level
lev mult val adv
1 1 6.291
2 2 4.040
3 4 1.991
4 8 na

Again, these are all for my made-up numbers. I'll post the real
numbers when I finish running the winpoker calculations, hopefully
tonight or tomorrow morning.

#2

<<I'm at work and they don't
have winpoker here>>

What sort of a job is that - I'd quit!!!! :slight_smile:

···

________________________________________
Jean $¢ott - Go to http://www.FrugalGambler.biz
  for VP software and strategy cards and for the
  Frugal series of books.

[Non-text portions of this message have been removed]

#3

LOL, Jean - I wonder if maybe I could file a complaint with OSHA?

···

--- In vpFREE@yahoogroups.com, "Jean Scott" <QueenofComps@f...> wrote:

<<I'm at work and they don't
have winpoker here>>

What sort of a job is that - I'd quit!!!! :slight_smile:
________________________________________
Jean $¢ott - Go to http://www.FrugalGambler.biz
  for VP software and strategy cards and for the
  Frugal series of books.

[Non-text portions of this message have been removed]

#4

--- In vpFREE@yahoogroups.com, "blaw57" <blaw57@y...> wrote,
delusionally:

I think I have the basics of multi-strike math down....

Ok, math gurus, I need help. This Dancer article gives check figures
of 99.79% on the yield and add-on values of 5.94, 3.96 and 1.99. I
match the level 3 number, but I'm off increasing amounts on the other
two - I get 3.99 and 6.00. And I'm way off on ER, coming up with
99.43% versus 99.79%. Where did I go astray (other than the friends I
chose in high school, I mean)?

Here are my results:

level mult Hand ER %win %FR %advnc cum adv bet contrib
1 1 95.967% 100.00% 0.00% 100.00% 100.00% 25% 23.99%
2 2 96.615% 46.879% 7.70% 50.97% 50.969% 25% 24.62%
3 4 98.860% 46.749% 7.00% 50.48% 25.728% 25% 25.43%
4 8 99.544% 46.108% 6.40% 49.56% 12.750% 25% 25.38%
      45.457% (dancer 99.79%) 99.43%

level mult calc add-on 5 coin used dancer
1 1 5.996650294 29.983 2970 5.94
2 2 3.987055874 19.935 1980 3.96
3 4 1.99088 9.954 995 1.99
  
Note: the %win in the table refers to the previous level, thus level
1 wins 46.879% of hands, etc. Think of it and FR as the %'s required
to get to this level. Hand ER, on the other hand, refers to the level
where its listed. But I don't think I mixed those up in my formulas
because, since its so confusing anyway, I triple checked it.

Also, I got wierd results in winpoker when I tried to go to 3-decimal
accuracy. Running the basic JOB scale gives an ER of 99.47% not
99.54%. Conceivably something is wrong there, but 2 decimals comes
out apparently right on the basic schedule so I'm assuming its right
on the MS runs too. For checking, the winpoker results are below. To
get the hand ERs, I take the "% Prob" from these tables and multiply
by the actual paytable. To get the win %, I take the Nothing % Prob
and subtract it from 1.

level 3
JACKS OR BETTER
Hand Name Payout Frequency % Prob.
ROYAL FLUSH 400995 60.841077 0.002%
STRAIGHT FLUSH 25995 197.5009 0.008%
4 OF A KIND 13495 6158.7797 0.237%
FULL HOUSE 5495 30039.482 1.156%
FLUSH 3995 26692.694 1.027%
STRAIGHT 2995 23057.326 0.887%
3 OF A KIND 2495 194836.93 7.497%
TWO PAIR 1995 338805.12 13.036%
JACKS OR BETTER 1495 578474.37 22.258%
NOTHING 0 1400637 53.892%
Total Return
Variance
      
level 2
JACKS OR BETTER
Hand Name Payout Frequency % Prob.
ROYAL FLUSH 401980 63.458326 0.002%
STRAIGHT FLUSH 26980 190.05973 0.007%
4 OF A KIND 14480 5755.2704 0.221%
FULL HOUSE 6480 28671.85 1.103%
FLUSH 4980 25975.581 0.999%
STRAIGHT 3980 22079.188 0.850%
3 OF A KIND 3480 180311.65 6.938%
TWO PAIR 2980 321860.9 12.384%
JACKS OR BETTER 2480 629903.91 24.237%
NOTHING 0 1384148.1 53.258%
Total Return
Variance
      
level 1
JACKS OR BETTER
Hand Name Payout Frequency % Prob.
ROYAL FLUSH 402970 63.345149 0.002%
STRAIGHT FLUSH 27970 190.18938 0.007%
4 OF A KIND 15470 5735.8288 0.221%
FULL HOUSE 7470 28624.556 1.101%
FLUSH 5970 24007.482 0.924%
STRAIGHT 4970 18826.967 0.724%
3 OF A KIND 4470 179981.46 6.925%
TWO PAIR 3970 322153.48 12.395%
JACKS OR BETTER 3470 638775.22 24.578%
NOTHING 0 1380601.5 53.121%
Total Return
Variance

#5

blaw57 wrote, delusionally :wink:

>
> I think I have the basics of multi-strike math down....

I'll pour over this and your prior post when I'm able to give them
dedicated concentration.

What I will note initially is that you earlier spoke of an interative
method. Dancer takes a shortcut:

Refinement of game ER on each iteration isn't likely to lead to much
change in play strategy nor change in ER of a level. Thus, he
assumes, when determining the EV added to a given level by the
potential for advance to higher levels, that the ER of each upper
level is 100% and the probability of advancement from one level to the
next (including Free Play effect) is exactly 50%.

For example, he assumes a win on Level 3 adds an EV for Level 4
advance of: (1 cr EV for every credit wagered, reflecting a 100% ER) x
(2, the increase in multiplier with a one level advance) = 2.

He thus adds 2 to every basic pay in the paytable to determine Level 3
EV, (or 10 to the 5 credit pays) and calculates the EV of that level
with Winpoker.

Extending the logic above (I'll leave it to you to fill things in),
the addition for Level 2 is 4; the addition for Level 3 is 6. These
increments account for potential multiple level advancement and are
referred to short-hand as 6/4/2 in most discussions.

Having determined an ER for each level, the next step is to combine
them into an overall ER. It's been awhile since I've reviewed
Dancer's article, but I believe he sticks to the 50% advance
assumption. Thus, Level 2 EV is multiplies by 50%, Level 3 by 25%,
and Level 4 by 12.5%.

blaw, I'm flying through this just now and I suspect there are more
than a few incomplete thoughts expressed here. I hope it's clear
enough that it still proves of help. My first read through of your
initial post suggested that it was fully on track but that you were
going to stumble a bit when it came to determining the ultimate EV of
each level via an iterative method.

One thing to keep in mind is that you need a program that will
generate a strategy that's consistent with the math you use. At the
point that you start dealing with fractional credits in your
refinement, you may find yourself at a loss for a detailed strategy to
play if your strategy program won't accomodate partial credits.

- Harry

#6

I responded to blaw with:

Extending the logic above (I'll leave it to you to fill things in),
the addition for Level 2 is 4; the addition for Level 3 is 6. These
increments account for potential multiple level advancement and are
referred to short-hand as 6/4/2 in most discussions.

It should be clear from the previous paragraph that I slipped here and
that the "6" increment is for Level 1, not Level 3. Also, I've run
through this before a little more eloquently ... you may wish to check
the archives.

Having determined an ER for each level, the next step is to combine
them into an overall ER. It's been awhile since I've reviewed
Dancer's article, but I believe he sticks to the 50% advance
assumption. Thus, Level 2 EV is multiplies by 50%, Level 3 by 25%,
and Level 4 by 12.5%.

Actually, on second thought (and I haven't taken the time to review
the Dancer article), at this point in the calculation the advance
rates can be determined precisely from the win frequency for a given
level's strategy and the related Free Ride frequency. There's no need
to approximate and I'm sure he'd use actual expected advance rates for
each level.

- H.

#7

Thanks, Harry, I look forward to your comments. On your points here,
Dancer does mention the 6/4/2 method in the article I read, but he
also gives the exact values for JOB as an illustration. Those are
what I was trying to tie to as an indication that my math was right,
and as mentioned I'm a little off on levels 2 and 1 (you work
backwards down the levels, with each dependent on the next higher
level numbers). I agree the 6/4/2 method is probably close enough for
strategy determination, although I do have a custom strategy program
I wrote that will take fractional credits. For ER determination, I
just now tried using the weights 50/25/12.5 and that puts me even
further off - good thought, but that's apparently not the answer.

The iterative process I mentioned has very little impact - I doubt it
would change the ER on the second pass more than 0.01% from the first
pass. It's just a means to tie up loose ends and to guard my "Anal-
est of the Analysts" trophy, and maybe get my hundreths of a percent
to reconcile exactly to the 4th decimal. On the numbers I presented I
didn't bother to use the iterative process once I saw I was off
Dancer's numbers.

Looking over my numbers, I was kind of doubting the Hand ER/win %
numbers that I calculated in winpoker, since I monkeyed with the
input to add two decimal places. For example, the numbers say that
for level 1 versus 2, you should be willing to let ER drop by 0.65
for a 0.13 increase in win % - that didn't sound right. But I just
ran the same numbers in Cindy Liu's on-line Vp analyzer and seem to
get essentially the same numbers. And, if higher levels are worth 6
to level one, then 6 times 0.13% is 0.78%, which is indeed higher
than the 0.65% Hand ER giveup the higher win rate cost you. So I
guess those numbers are okay.

I bet I'm overlooking something basic like using the wrong weights in
the averages or something silly like that.

Anyhow, this is all just an academic exercise to me, as the best MS
in Tunica pays about 98.5% - most are even worse than that. Which is
really silly considering the massive strategy errors that are
possible (likely) in this game - they could offer 100% MS and get it
played 10 times as much and probably still get a 3% hold out of it.
The only MS I play here is an occasional $10 on short coin nickel
Super Double Bonus MS at Harrahs, with a short royal and resultant
97% ER, and on which I'm currently $100 ahead thanks to incredible
4OAK luck on the top line (er, excuse me, I didn't mean to
say "luck", but it has to happen to somebody and I always seem to be
standing there at the time).

--- In vpFREE@yahoogroups.com, "Harry Porter" <harry.porter@v...>
wrote:

blaw57 wrote, delusionally :wink:
> >
> > I think I have the basics of multi-strike math down....

I'll pour over this and your prior post when I'm able to give them
dedicated concentration.

What I will note initially is that you earlier spoke of an

interative

method. Dancer takes a shortcut:

Refinement of game ER on each iteration isn't likely to lead to much
change in play strategy nor change in ER of a level. Thus, he
assumes, when determining the EV added to a given level by the
potential for advance to higher levels, that the ER of each upper
level is 100% and the probability of advancement from one level to

the

next (including Free Play effect) is exactly 50%.

For example, he assumes a win on Level 3 adds an EV for Level 4
advance of: (1 cr EV for every credit wagered, reflecting a 100%

ER) x

(2, the increase in multiplier with a one level advance) = 2.

He thus adds 2 to every basic pay in the paytable to determine

Level 3

EV, (or 10 to the 5 credit pays) and calculates the EV of that level
with Winpoker.

Extending the logic above (I'll leave it to you to fill things in),
the addition for Level 2 is 4; the addition for Level 3 is 6. These
increments account for potential multiple level advancement and are
referred to short-hand as 6/4/2 in most discussions.

Having determined an ER for each level, the next step is to combine
them into an overall ER. It's been awhile since I've reviewed
Dancer's article, but I believe he sticks to the 50% advance
assumption. Thus, Level 2 EV is multiplies by 50%, Level 3 by 25%,
and Level 4 by 12.5%.

blaw, I'm flying through this just now and I suspect there are more
than a few incomplete thoughts expressed here. I hope it's clear
enough that it still proves of help. My first read through of your
initial post suggested that it was fully on track but that you were
going to stumble a bit when it came to determining the ultimate EV

of

each level via an iterative method.

One thing to keep in mind is that you need a program that will
generate a strategy that's consistent with the math you use. At the
point that you start dealing with fractional credits in your
refinement, you may find yourself at a loss for a detailed strategy

to

···

play if your strategy program won't accomodate partial credits.

- Harry

#8

blaw57 wrote:

Thanks, Harry, I look forward to your comments.

My interest is sufficiently piqued that I'll give this a much stronger
read-through/review in the next day or two.

In the mean time, you may wish to review the archives here. Bill
Velek ran though a work up of the numbers for which he put together a
lengthly discussion. Review his posts on the subject to find the link
to that discussion.

- H.

#9

Not a math comment - I try not to strain my brain. But a fun report. Yesterday I got a royal on the 3rd line of a 50-cent machine for $8,000. I was a real hero on the bank where I was playing - everyone else, but Brad, was play 1-cent! This is the first royal we have either one have had on a MS that wasn't on Line 1.

This would have been a better story if I had still been on drugs. But they have finally worn off - darn it!

···

________________________________________
Jean $¢ott - Go to http://www.FrugalGambler.biz
  for VP software and strategy cards and for the
  Frugal series of books.

[Non-text portions of this message have been removed]