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Bob Dancer's Multi-Strike Poker Analysis (long post plus math)

#1

Harry Porter posted this link -- http://www.multistrikepoker.com -- to a nice, informative article by Bob Dancer regarding Multi-Strike Poker. Thanks again, Harry.

Anyway, I really enjoyed playing the game on the above-linked site, so I decided to try to investigate strategy a bit in case I find a machine in the casinos I visit. During the process, I was stumped by Bob's results which are substantially different from my own. Now, ordinarily I would assume that I was employing faulty methods, but in this case I think I used a very simple straighforward method following the info which Bob mentioned in his article, and I'd like to discuss this with the math gurus on this site -- and Bob as well, if he sees this.

Bob's article contains a chart which indicates the difference that multistrike makes in ER for various games, including JOB-9/6; it indicated a slight improvement from 99.54% ER associated with standard 9/6, to 99.79% when playing JOB-9/6 on all four levels of a multistrike machine. He also provided instructions on how a person could use WinPoker to practice the necessary strategy changes for this game.

I made some initial calculations of my own which seemed logical to me, but I'll be the first to concede that they are probably faulty, based upon my limited knowledge and ability in math and statistics; they show that, even when using nearly perfect altered strategy at each level, the combined ER for the game still declines more than 1.5% from normal, to only 97.99%. That's what prompted me to then try to follow Bob Dancer's approach, as best that I could figure it out, but in doing that my results were substantially lower than my own -- to only 94.43 percent -- a drop of over 5 percent from standard JOB-9/6, as opposed to the .25% gain Bob had predicted. According to my figures, the net result of using Bob's strategy is even worse than not making any strategy changes at all; my calculations of the game played without employing any strategy changes -- just using the same perfect play for JOB-9/6 on all four levels -- cuts the ER down to only 97.80%, which is nearly as good as I was able to get with my own strategy changes.

Anyway, I'm sure that there is an explanation to show how I have erred in my calculations, as so often has occurred in the past, ... but then again, no one is above making errors, and I'd just like to make sure that Bob is correct before I start practicing this game and then playing it at the casinos and possibly losing money that I shouldn't. Will Bob Dancer or someone else please take the time to look my methods over; there is no need to check the math because I've already gone over everything three different times with a calculator.

I won't get into my particular methods for my own calculations unless and until it is determined that Bob's results are in error, because if he happens to be correct, it is pretty much pointless to get into my approach -- except for my own learning process which probably doesn't interest too many other folks.

Here are the methods I used to test Bob's results:

1. I followed Bob's instructions with WinPoker to determine perfect strategy -- according to Bob -- for each of the four levels. I did this by changing the paytables and then running an analysis using the adjustments Bob specified in his article; i.e., I added six coins to the default amount for each hand in the single-coin pay column for the first-level strategy; I added four coins to the default amount for each hand for second-level strategy; I added two coins to default amount for third-level strategy; and I kept the default values for fourth-level strategy.

2. I used WinPoker to run an analysis for each of the above changes to determine the frequency distribution of hands; I multiplied the frequency for each hand ... times the _ACTUAL_ amount paid for that hand (e.g., I always used 800 for the Royal, despite that the strategy was determined with a setting of 806, or 804, or 802), and then totalled those expected wins and divided by coins x total hands played to determine the actual ER for that strategy -- as opposed to what WinPoker was stating for the inflated paytables.

3. I calculated the ratio of winning to losing hands for each strategy.

4. I noted the exact percentage of FREE-RIDES for each level, as given by the Multistrike Game on its paytables page -- 7.7% in level one, 7.0% in level two, and 6.4% during level three. For any hand on any given level where a FREE-RIDE is received, the proper play is to revert to normal optimum play for JOB-9/6 in order to maximize ER.

5. I calculated the ER for each level as well as the percentage of hands at each level which will advance to the next level, whether by FREE-RIDE or a winning hand.

6. I totalled the ER for each of the four levels and then divided by 4 to arrive at the overall average ER for the entire game.

Here is the math I did:

···

************
Tier 1 � Using +6 strategy, i.e., 806, 256, etc.
Winpoker Analysis:
  Total Return: 96.32171075%
  Calculated Win/Lose Ratio = 0.46880103 vs. 0.53119897
Free-Rides = .077
ER for Tier 1 = (.077 x .9954390379) + (.923 x .9632171075)
    = 0.9656981961408
Advance to next level = .077 + (.923 x 0.46880103) = 0.50970335069
************
Tier 2 � Using +4 strategy, i.e., 804, 254, etc.
Winpoker Analysis:
  Total Return: 96.96938807%
  Calculated Win/Lose Ratio = 0.46749323 vs. 0.53250677
Free-Rides = .07
ER for Tier 2 =
=[(.07 x .9954390379)+(.93 x .9696938807)] x .50970335069 x 2 =
    = 0.99034957527720155435152
Advance to next level = .07 + (.93 x .46749323) = 0.5047687039
************
Tier 3 � Using +2 strategy, i.e., 802, 252, etc.
Winpoker Analysis:
  Total Return: 76.85301819
  Calculated Win/Lose Ratio = 0.46107790 vs. 0.53892210
Free-Rides = .064
ER for Tier 3 =
=[(.064 x .9954390379) + (.936 x .7685301819)] x ...
  ... x .50970335069 x .5047687039 x 4 =
    = 0.80586203622362759228844482568258
Advance to next level = .064 + (.936 x .46107790) = 0.4955689144
************
Tier 4 � Using +0 strategy, i.e., 800, 250, etc.
Winpoker Analysis:
  Total Return: 99.5439%
  Calculated Win/Lose Ratio = 0.45456533 vs. 0.54543467
Free-Rides = None at this level
ER for Tier 4 =
= .9954390379 x .50970335069 x .5047687039 x .4955689144 x 8 =
    = 1.0153566578165987771439455722388
************
Average ER for Multistrike Game total of 4 levels divided by 4:
= [0.965698196 + .99034957528 + 0.8058620362 + 1.0153566578]/4
= 3.77726646545822792378 / 4 = 0.944316616364556980946
************

I'd sure appreciate any assistance in helping me to understand where I may have gone wrong in the above.

Thanks.

Bill Velek

#2

Although I have nothing to add or comment on your analysis, I have been
concerned with Bob's numbers only because I have yet to see anyone "agree"
with the numbers Bob has shed upon us. I really hope you get some good
answers to your analysis either confirming your numbers, Bob's numbers, or
new numbers!

#3

Here is the math I did:
************
Tier 1 – Using +6 strategy, i.e., 806, 256, etc.

My guess is that this is your problem...

The schedule for 9/6 JoB is 800-50-25-9-6-4-3-2-1 for 1 coin or 4000-
250-125-45-30-20-15-10-5 for 5 coins. It looks like the schedule you
are typing into winpoker is an incorrect hybrid of those two tables,
where the payout for a royal flush is too low.

Also, if you are using the 5-coin payouts, you have to add 30, 20 and
10 coins to each number, not 6, 4, and 2.

#4

The first thing I noticed was that your EV figure (%101+) for the
top line seemed ludicrously low, given that everything pays 8X. Then
I think I found out what's wrong with your approach, and possibly
Dancer's as well.

On Level 1, you are betting 20 coins. Therefore, the return on Level
1 should be expressed as
Nothing -20
Jacks -15
Two Pair -10
Trips -5
Straight +0

And so forth, except that the actual return is (result)+value of
advancing.

Level 2: We are risking 15 coins (since 5 coins have already been
resolved at level 1)
Nothing -15
Jacks -5
Two Pair +5, etc.

Similarly, the paytable at level 3 is based on 10 coins-in, and
level 4 on 5 coins-in.

The "value of advancing" figure (from Level 1) should simply be a
product of (EV of Level 2 @ 15 coins) + (EV of Level 3 @ 10 coins
X "winning frequency" of level 2) + (EV of Level 4 @ 5 coins
X "winning frequency" of Level 2 X "winning frequency" of Level 3).

To put it in English, we divide the 20-coin bet into four segments.
The first segment is played at an EXTREMELY negative EV because of
the loss of the other three bets that accompanies failing to make
JOB on the bottom line. Conversely, making Jacks or another paying
hand is valuable not because of the line 1 payback but because it
enables us to play line 2.

The value of playing line 2 is the EV of the play, PLUS a 46% chance
(I believe that was the figure) of playing line 3 (multiply the EV
of line 3 by .46), PLUS a (.46)x(.46) chance of playing line 4.

The value of playing line 3 is the EV of a 10-coin bet against the
4X paytable, plus a 46% chance of advancing.

The value of playing line 4 is the EV of a 5-coin bet against the 8X
paytable.

The reason this game is a loser is that there is LESS than a
cumulative 50% chance of advancing on each line (even with strategy
adjustments) so that we get to play the top line, for example,
considerably less than 1/8 of the time. (This is (.46) cubed which I
think is about 10%.) In fact, the "Free Ride" card is needed just to
prevent this game from resembling a midnight mugging.

It is easy to factor in the "Free Ride" card by multiplying its
(stated) chance of occuring, and taking this as a fraction, changing
the (.46) to (1) in all calculations.

Somebody else can do the math---but I'll bet this method of
calculation shows the game to be a real stinkeroo.

#5

tooncesthecatwhocoulddriveacar wrote:

> Here is the math I did:
> ************
> Tier 1 � Using +6 strategy, i.e., 806, 256, etc.

My guess is that this is your problem...

The schedule for 9/6 JoB is 800-50-25-9-6-4-3-2-1 for 1 coin or 4000-
250-125-45-30-20-15-10-5 for 5 coins. ...

Thanks, but I'm well aware of that.

... It looks like the schedule you
are typing into winpoker is an incorrect hybrid of those two tables,
where the payout for a royal flush is too low.

First, those paytable figures are precisely the ones that Bob Dancer recommended using in his article, which is why I tried them and then mentioned them, in particular, in my post. Although I don't agree with using those particular values, Bob's _approach_ is nevertheless a valid way to cause WinPoker to generate perfect play strategy for purposes other than full-coin play in a VP game, and I then used them solely for the purpose of retrieving from WinPoker the information which I needed to determine the resulting ER's for those tiers when those tiers are played according to that strategy. In the interest of brevity, I will skip those details and assume that you know how that is done, but if you need me to explain that, then please let me know and I'll be glad to do so.

To really appreciate what I had said in my post, you need to be familiar with Multistrike, and better yet, with Bob Dancer's article. Here's the link: http://www.multistrikepoker.com -- and that site has a shockwave version of the game that you can play online; it's really fun and you ought to try it if you haven't done so already.

Also, if you are using the 5-coin payouts, you have to add 30, 20 and
10 coins to each number, not 6, 4, and 2.

Actually, I believe that those calculations were probably made in 'betting units', so as long as the Royal was adjusted proportionately from the single-coin value of 250 up to the full-coin value of 800 per coin, that part really doesn't make any difference; that is, so long as the same coin-in value is used consistently throughtout all calculations, it doesn't matter whether we use 800/1-coin or 4000/5. This last point can be seen with WinPoker by changing the Royal payout in the single-coin column to 800 and then running a comparison of the analysis for both short-coin and full-coin play; they are _identical_.

On another point, however, I just noticed as I reviewed my original post that I had made a mistake in _labeling_ the paytable used for each tier by referring to them as "806, _256_, etc.", "804, _254_, etc." and so forth; this scared me for a second because I thought I had made a major blunder, but when I checked my actual calculations, I confirmed that I actually used the correct figures of "806, _56_, 31, etc." (and NOT 256 for the STFL); the same is the case for "804, _54_" ... "802, _52_" ... and "800, _50_".

I will be answering other concerns in responses to other posts in this thread.

Thanks for your comments and interest.

Bill Velek

#6

Bill Velek wrote:

> Here is the math I did:
> ************
> Tier 1 � Using +6 strategy, i.e., 806, 256, etc.

snip

First, those paytable figures are precisely the ones that Bob Dancer recommended using in his article ...

Sorry, but I should have revised that statement before making my post, in light of what I later pointed out toward the end of that post -- that is ...

On another point, however, I just noticed as I reviewed my original post that I had made a mistake in _labeling_ the paytable used for each tier by referring to them as "806, _256_, etc.", "804, _254_, etc." and so forth; this scared me for a second because I thought I had made a major blunder, but when I checked my actual calculations, I confirmed that I actually used the correct figures of "806, _56_, 31, etc." (and NOT 256 for the STFL); the same is the case for "804, _54_" ... "802, _52_" ... and "800, _50_".

What I had actually meant to state was that Bob Dancer had recommended using the specific adjusted paytables that I actually used, in accordance with his article, which were:

Level 4 (8X pay) = 800, 50, 25, 9, 6, 4, 3, 2, 1
Level 3 (4X pay) = 802, 52, 27, 11, 8, 6, 5, 4, 3 -- +2 to all hands
Level 2 (2X pay) = 804, 54, 29, 13, 10, 8, 7, 6, 5 -- +4 to all hands
Level 1 (1X pay) = 806, 56, 31, 15, 12, 10, 9, 8, 7 -- +6 to all hands

SORRY FOR ANY CONFUSION I CREATED.

Bill Velek

#7

mkl54321 wrote:

The first thing I noticed was that your EV figure (%101+) for the
top line seemed ludicrously low, given that everything pays 8X. ...

The reason why the EV is so low despite an 8X payout is because you only reach that level _approximately_ once in every eight times that you pay for that level, which basically cancels out the 8X payout. The figures which I calculated using WinPoker, FREE-RIDE info from the Multistrike game/website, and Bob's method for _roughly_ establishing perfect-play strategy for each level, as shown at the bottom of my original post, are as follows:

You will advance from Level-1 to Level-2 50.970335069% of the time; ...

... then out of the number of times that you happen to reach Level-2 [which is approximately half of your bets], you will advance to Level-3 only 50.47687039% of the time [which is approximately a quarter of your bets]; ...

... and then, out of the number of times that you succeed in reaching Level-3 [about one-quarter of your bets], you will advance to Level-4 only 49.55689144% of the time [which is approx. an eighth of your bets].

The precise percentage of times that you should reach Level-4 is a product of the three percentages just quoted, and is shown in my calculation of Level-4 ER at the end of my original post, as follows:
ER for Tier 4 = ER for perfect 9/6 strategy (.9954390379) times the % of times reaching that level (.50970335069 x .5047687039 x .4955689144) times the 8X multiplier for the paytable at Level-4. The actual frequency of reaching Level-4 computes out to 12.75%, which is only slightly more often than once in every 8 games, and that explains why there is a jump -- for that level -- from the normal 99.5439% for 9/6 to the 101.5356 ER calculated for just Level-4.

... Then
I think I found out what's wrong with your approach, and possibly
Dancer's as well.

On Level 1, you are betting 20 coins. Therefore, the return on Level
1 should be expressed as
Nothing -20
Jacks -15
Two Pair -10
Trips -5
Straight +0

And so forth, except that the actual return is (result)+value of
advancing.

Level 2: We are risking 15 coins (since 5 coins have already been
resolved at level 1)
Nothing -15
Jacks -5
Two Pair +5, etc.

Similarly, the paytable at level 3 is based on 10 coins-in, and
level 4 on 5 coins-in.

Well, it might be possible to make calculations using negative values ["there's more than one way to skin a cat"], but since there is no way to enter values for "Nothing" in WinPoker as you suggested, I couldn't attempt that approach.

The "value of advancing" figure (from Level 1) should simply be a
product of (EV of Level 2 @ 15 coins) + (EV of Level 3 @ 10 coins
X "winning frequency" of level 2) + (EV of Level 4 @ 5 coins
X "winning frequency" of Level 2 X "winning frequency" of Level 3).

I'm not following what you mean there.

To put it in English, we divide the 20-coin bet into four segments.
The first segment is played at an EXTREMELY negative EV because of
the loss of the other three bets that accompanies failing to make
JOB on the bottom line. Conversely, making Jacks or another paying
hand is valuable not because of the line 1 payback but because it
enables us to play line 2.

Besides the inability to enter values for "Nothing" hands, WinPoker will not allow me to enter negative values either. But aside from that, I'm not sure that this would be the proper approach anyway, for this reason:
If you merely set the lowest hands to Zero and run the analysis, you will find that WinPoker adjusts the strategy so that you end up with an even higher percentage of losing hands when what you're actually attempting to accomplish is a lower percentage of losing hands so that you can advance to the higher levels more frequently.

The value of playing line 2 is the EV of the play, PLUS a 46% chance
(I believe that was the figure) of playing line 3 (multiply the EV
of line 3 by .46), PLUS a (.46)x(.46) chance of playing line 4.

snip

I will agree that this is basically correct; the only disagreement I'd have is that the chance of moving from one level to another varies between all levels, due in part to the various percentages of FREE RIDES that are set in the machine, as well as changing strategy (and therefore changing ratios of winning/losing hands) for different levels. When you factor all of that together, you should get the figures that I've already explained earlier. Incidentally, just so readers will understand the method I used to determining the "Advance to next level" figures, I had given the following calculation for moving from Level-1 to Level-2:
Advance to next level = .077 + (.923 x 0.46880103) = 0.50970335069

The .077 is to consider the FREE-RIDE percentage (7.7%) taken from info provided by the game itself. The .923 figure reflects the remaining percentage of games wherein the FREE-RIDE does _not_ occur, and when multiplied times the normal percentage of winning hands for that given strategy (0.46880103), it automatically prorates that figure to adjust for the times that you needlessly receive a FREE-RIDE on an already winning hand.

The reason this game is a loser is that there is LESS than a
cumulative 50% chance of advancing on each line (even with strategy
adjustments) so that we get to play the top line, for example,
considerably less than 1/8 of the time.

I think that we're heading down the same road, but with the adjusted strategy and the FREE-RIDE, I've demonstrated where the 4th-Tier is reached slightly more often than 1/8 of the time.

(This is (.46) cubed which I
think is about 10%.) In fact, the "Free Ride" card is needed just to
prevent this game from resembling a midnight mugging.

It is easy to factor in the "Free Ride" card by multiplying its
(stated) chance of occuring, and taking this as a fraction, changing
the (.46) to (1) in all calculations.

I'm not sure what you're suggesting, but it appears at the least that your figures above are very _rough_, whereas I'm using very precise figures, even to the point of not limiting my decimal places as I have in these posts.

Somebody else can do the math---but I'll bet this method of
calculation shows the game to be a real stinkeroo.

It is, based on my own calculations. As I stated previously, _MY_ calculations reveal an ER of 97.99% for Full-Coin JOB-9/6, but I was concerned that I might be wrong since Bob Dancer's article described this as actually increasing the ER to 99.79% Now that I look at those numbers near each other, do you think that somebody just transposed the '7' and a '9'?

Thanks for a very intriguing discussion.

Bill Velek

#8

Here is the math I did:
************
Tier 1 – Using +6 strategy, i.e., 806, 256, etc.
Winpoker Analysis:
  Total Return: 96.32171075%
************
Tier 2 – Using +4 strategy, i.e., 804, 254, etc.
Winpoker Analysis:
  Total Return: 96.96938807%
************
Tier 3 – Using +2 strategy, i.e., 802, 252, etc.
Winpoker Analysis:
  Total Return: 76.85301819
************
Tier 4 – Using +0 strategy, i.e., 800, 250, etc.
Winpoker Analysis:
  Total Return: 99.5439%
************

Bill, I think your problem is that you probably goofed up the Tier 3
result. As you move from Tier 1 to Tier 4, your strategy should get
closer and closer to normal, so the tier 3 strategy should yield a
return between 96.97% and 99.54%. A 76.85% return seems way too low,
and is probably just a miscalculation.

But it sounds like your method of solving the problem is correct.

I'd sure appreciate any assistance in helping me to understand

where I

···

--- In vpFREE@yahoogroups.com, Bill Velek <billvelek@a...> wrote:

may have gone wrong in the above.

Thanks.

Bill Velek

#9

tooncesthecatwhocoulddriveacar wrote:

> Here is the math I did:

snip

> Tier 3 � Using +2 strategy, i.e., 802, 252, etc.
> Winpoker Analysis:
> Total Return: 76.85301819

snip

Bill, I think your problem is that you probably goofed up the Tier 3
result. ...

I just posted a detailed description in the WinPoker yahoogroup explaining why this ER is, in fact, correct for this particular altered strategy. For those who are interested and don't subscribe to that group, I'll paste it in its entirety at the bottom of this post.

... As you move from Tier 1 to Tier 4, your strategy should get
closer and closer to normal, so the tier 3 strategy should yield a
return between 96.97% and 99.54%. ...

Normally that would appear to be true, but this is not the case because the altered paytable is not the REAL paytable; it is only used to dupe WinPoker into altering its strategy. Any alteration from the normally perfect strategy is going to cause a reduction in the REAL ER, and when we make subtle changes in strategy, the results are not always linear.

I've done a lot of experimenting with altered strategies, and if I were to graph the results we would see that the graph has many relative peaks and valleys as it generally moves in an upward trend while approaching perfect strategy. Arbitrarily taking three altered strategies, such as are produced by the +6, +4, and +2 values added per Bob Dancer's suggestion, can easily match up with peaks and valleys that would cause this to be non-linear.

... A 76.85% return seems way too low,
and is probably just a miscalculation.

_PLEASE_ look at the WinPoker post, either on that group or at the bottom of this post.

But it sounds like your method of solving the problem is correct.

That's what I've been trying to confirm. Thanks.

Cheers.

Bill Velek

···

--- In vpFREE@yahoogroups.com, Bill Velek <billvelek@a...> wrote:

*********************
Here is the WinPoker post referenced above:

Please don't lose patience with me, and bear with me on this. I'm not
trying to prove that I'm right, and I'll be delighted to confirm that
Bob is correct and that MultiStrike has a higher ER because I like
playing the game. However, let's please look at whether I've made the
mistake that you and Tom suspect.

Larry218 wrote:

>> Bill-
>>
>> Tom tried to point out that your Level 3 number derived from Winpoker
>> is clearly incorrect (the rest of your approach looks pretty good but
>> I haven't gone through it in detail).
>>
>> Look at the 4 numbers that you extracted from WinPoker:
>>
>> Tier 1 � Using +6 strategy, i.e., 806, 256, etc.
>> Winpoker Analysis:
>> Total Return: 96.32171075%
>> ************
>> Tier 2 � Using +4 strategy, i.e., 804, 254, etc.
>> Winpoker Analysis:
>> Total Return: 96.96938807%
>> ************
>> Tier 3 � Using +2 strategy, i.e., 802, 252, etc.
>> Winpoker Analysis:
>> Total Return: 76.85301819
>> ************
>> Tier 4 � Using +0 strategy, i.e., 800, 250, etc.
>> Winpoker Analysis:
>> Total Return: 99.5439%

First, just to clear the record on a point which doesn't affect the
calculations, and as I had explained in another post, the above
"labeling" of each tier paytable was in error; e.g., for Tier 3, it was
NOT "802, 252, etc.", but rather "802, 52, etc.", as shown below.

>> Your result for Tier 3 is terribly low, probably as a result of
>> incorrect paytable entry. This is the figure that has skewed your
>> result.

The way I did my calculations was to use 'cut and paste' to copy the
WinPoker analysis for each paytable adjustment into a document, and then
I used the copy function to transfer each number to my calculator to do
the math, and then I used the copy function to transfer the resulting
calculation back to my working document. This approach was used for two
reasons: first, it is much easier and quicker than trying to type the
numbers into my calculator, and second, it reduced the chance of an
error in entering the numbers into the calculator. Admittedly, it is
possible the cut and paste the wrong number, which is why I went through
the entire process _three_ times without finding any disagreement in my
figures.

Now, I've just run the WinPoker analysis again for the Tier-3 strategy,
and used the copy and paste function to move that data over to this
post. I have run the analysis for both full-coin and short-coin to show
that it makes no difference when the short-coin Royal has been properly
adjusted. Here are those analysis tables:

JACKS OR BETTER
Hand Name Payout Frequency % Prob. Occurs Every % Ret.
ROYAL FLUSH 4010 60.841077 0.002% 42717.19 1.88%
STRAIGHT FLUSH 260 197.50090 0.008% 13159.23 0.40%
4 OF A KIND 135 6158.7797 0.237% 421.9927 6.40%
FULL HOUSE 55 30039.482 1.156% 86.51814 12.71%
FLUSH 40 26692.694 1.027% 97.36597 8.22%
STRAIGHT 30 23057.326 0.887% 112.7173 5.32%
3 OF A KIND 25 194836.93 7.497% 13.33916 37.48%
TWO PAIR 20 338805.12 13.036% 7.670958 52.14%
JACKS OR BETTER 15 578474.37 22.258% 4.492783 66.77%
NOTHING 0 1400637.0 53.892% 1.855556 0.00%
Total Return 191.3265%
Variance 21.66811

JACKS OR BETTER
Hand Name Payout Frequency % Prob. Occurs Every % Ret.
ROYAL FLUSH 802 60.841077 0.002% 42717.19 1.88%
STRAIGHT FLUSH 52 197.50090 0.008% 13159.23 0.40%
4 OF A KIND 27 6158.7797 0.237% 421.9927 6.40%
FULL HOUSE 11 30039.482 1.156% 86.51814 12.71%
FLUSH 8 26692.694 1.027% 97.36597 8.22%
STRAIGHT 6 23057.326 0.887% 112.7173 5.32%
3 OF A KIND 5 194836.93 7.497% 13.33916 37.48%
TWO PAIR 4 338805.12 13.036% 7.670958 52.14%
JACKS OR BETTER 3 578474.37 22.258% 4.492783 66.77%
NOTHING 0 1400637.0 53.892% 1.855556 0.00%
Total Return 191.3265%
Variance 21.66811

As you can see by comparing them, the ER, Variance, and all frequency of
hands are identical. Now, I have just reviewed, once again, the
hard-copy that I had made of my working document, and the above figures
are what I had used.

Please note that the adjustments to the paytable are identical to the
instructions given in Dancer's article at http://tinyurl.com/6nbp --
i.e., that the +2 be added to each hand, as was just done, above.

Now, looking at that table, you will see that WinPoker gives this
strategy an ER of 191.3265%; this would be true if the game was actually
paying the amounts entered into the paytable. But this is not the case,
so we need to find out what the REAL ER is when using this particular
strategy. I did this by calculating the total proportional payouts for
each hand by multiplying the above 'Frequency' for each hand by the
actual amount the machine pays, i.e., 800, 50, 25, 9, 6, 4, 3, 2, 1.

I will now paste that portion of the info from my working document,
which merely replaces the "% Probable", "Occurs Every", and "% of Ret."
columns with a new column identified as "Earnings".

Hand Name Payout REAL Frequency Earnings x 5
ROYAL FLUSH 802 800 60.841077 243364.308
STRAIGHT FLUSH 52 50 197.50090 49375.225
4 OF A KIND 27 25 6158.7797 769847.4625
FULL HOUSE 11 9 30039.482 1351776.69
FLUSH 8 6 26692.694 800780.82
STRAIGHT 6 4 23057.326 461146.52
3 OF A KIND 5 3 194836.93 2922553.95
TWO PAIR 4 2 338805.12 3388051.2
JACKS OR BETTER 3 1 578474.37 2892371.85
  Sub-Total 1198323.043677 9986896.1755
NOTHING 0 1400637.0
  Total 2598960.043677 x 5 = 12994800.218385

You can compare the figures in the 'Frequency' column, above, and see
that they are identical to the WinPoker analysis pages pasted above.
To arrive at the figures in the "Earnings x 5" column, I multiplied the
'Frequency' of each hand x the respective "REAL" payout, e.g., for the
Royal, I multiplied 60.841077 x 800 = 48672.8616 ... and then multiplied
it again x 5 = 243364.308 ... and then pasted it into the right column.
The reason it was multiplied by 5 is because in later calculations I
am using '5' as the value of the betting units.

I then totaled the two columns to determine the total number of hands
played and the total amount earned; because I had multiplied winning for
each hand x 5, I also multiplied the Total number of hands played x 5 to
arrive at the total number of coins-in. I then divided the Total Amount
Earned by the Total Amount of Coin-in to arrive at the REAL ER, i.e.,
9986896.1755 / 12994800.218385 = 0.76853018189310618303275204069262, and
then converted it to a percentage = 76.853018189310618303275204069262

So, if you follow perfect-play strategy after the +2 adjustment is made
to each hand, as Bob suggested, then according to WinPoker's analysis
(which I trust) and my very simple straight-forward calculations which I
don't think can be disputed, you arrive at an ER of 76.85301819 which is
what I had indicated I had used in my further calculations. That figure
is the one which you referred to as probably being erroneous. Of
course, it is actually only part of the answer to what you can expect to
win on Tier-3 because you will only be using this altered strategy 93.6%
of the time because 6.4% of Tier-3 hands will include a FREE-RIDE, and
you should use normal 9/6 strategy at those times. That was all
included in my calculations.

Now, you will also note that I had inserted a Sub-Total line to show the
total number of 'winning' hands; multiplying that figure by 100 (to
express the result as a percentage) and then dividing that by the total
number of hands gives you the percentage of winning hands. Conversely,
dividing the number of "Nothing" hands by the total number of hands will
give you the percentage of losing hands. The math is as follows:

1198323.043677 x 100 / 2598960.043677 = 46.10779017524319% winning hands
1400637.0 x 100 / 2598960.043677 = 53.8922098247568071398% losing hands

Those figures, along with the appropriate FREE-RIDE figure of 6.4
percent for this tier, were used later to determine the probability of
advancing from Tier-3 to Tier-4.

I won't get into the rest of the calculations because they've been
discussed in other posts. This was just to show you that the ER figure
I arrived at for Tier-3 is correct, no matter how odd it looks.

>> I do find it interesting that you say that you don't trust IGT and
>> all of the regulation agencies yet have unwavering faith in a private
>> program that has no such scrutiny. This is not to take anything away
>> from WinPoker. I often use it as a tool in the same way that you are
>> and feel confident in the results it generates.

Even people at IGT and agencies can make mistakes. I'm just trying to
understand how I've gone astray in my calculations. And yes, I do have
unwaivering faith in WinPoker; isn't it blasphemy to say otherwise here?
<g> Seriously, while I have no idea whether there are programming
errors in WinPoker, or some inaccurate figure in some sort of a database
that it might draw from, I think I pretty well understand the methods it
uses and they seem very logical and reliable to me; in addition, I have
faith in it because it is so popular and has been used for years by so
many pros that I would think any errors would have been discovered by
now. Are there any inconsistencies between it's analysis and those from
FrugalVP?

>> It is the scrutiny of the regulation agencies that allows us all to
>> play in the regulated areas with pretty high confidence that we are
>> getting a fair game. I have heard and seen some amazing things going
>> on with video poker in areas that are not regulated.

Agreed.

>> In any case, I hope that you are able to work the math to your
>> satisfaction. I can verify that independent of Chris Brune's
>> analysis we have come up with the same 99.79% optimal return for 9-6
>> Jacks or Better.

Who are the "we" that you're referring to? ... uhhhh, to whom you are
referring, for the grammar freaks

Thanks for posting, Larry.

Cheers.

Bill Velek

#10

As I suspected earlier, you made a calculation error that resulted in
the ridiculously low return for Tier 3. The sum of numbers on the
far right column is 12,879,219 not 9,986,896 which most likely left
off the earnings on a pair of Jacks-Aces.

Dividing the actual subtotal into the total hands results in a 99.11%
return at tier 3, which I would suspect gets you in the vicinity of
Dancer's result.

···

> --- In vpFREE@yahoogroups.com, Bill Velek <billvelek@a...> wrote:
> > Here is the math I did:
Hand Name Payout REAL Frequency Earnings x 5
ROYAL FLUSH 802 800 60.841077 243364.308
STRAIGHT FLUSH 52 50 197.50090 49375.225
4 OF A KIND 27 25 6158.7797 769847.4625
FULL HOUSE 11 9 30039.482 1351776.69
FLUSH 8 6 26692.694 800780.82
STRAIGHT 6 4 23057.326 461146.52
3 OF A KIND 5 3 194836.93 2922553.95
TWO PAIR 4 2 338805.12 3388051.2
JACKS OR BETTER 3 1 578474.37 2892371.85
  Sub-Total 1198323.043677 9986896.1755
NOTHING 0 1400637.0
  Total 2598960.043677 x 5 = 12994800.218385

#11

tooncesthecatwhocoulddriveacar wrote:

snipped columns of data

As I suspected earlier, you made a calculation error that resulted in
the ridiculously low return for Tier 3. ...

Thanks. ... and man, do I have egg all over my face. I swear, I really thought that I had triple checked my math completely for each table ... but there were 4 separate strategy tables for Bob Dancer's method, and I obviously missed this one. Makes me wonder if I checked another table 6 times.

... The sum of numbers on the
far right column is 12,879,219 not 9,986,896 which most likely left
off the earnings on a pair of Jacks-Aces.

You're absolutely correct.

Dividing the actual subtotal into the total hands results in a 99.11%
return at tier 3, which I would suspect gets you in the vicinity of
Dancer's result.

It not only get me in that vicinity, it actually hits the nail right on the head at Bob's predicted ER. My calculations now reveal that Bob's strategy yields a 99.7917% ER, which is _exactly_ what he said it was.

Many thanks to all who participated in this discussion. I hope that the thread presented at least somewhat of a learning experience for many of us. And Bob Dancer, once again I want to say that I hope I didn't offend you in anyway by questioning this; I just wanted to see the math for myself, and I was having problems seeing it -- due mostly to my stupid math error, above.

I do have one last question about MultiStrike, although it is not really a "math" question:

Are the Casinos free to change the percentages of FREE-RIDES at whim if they suddenly experience too many players winning too much? I know they can't change the RNG or monkey with the deck because the game has to be as random as a regular deck of cards. But, the FREE-RIDE has nothing to do with a deck of cards; it's sort of like mixing just a slight amount of 'slots' with VP, and unless the gaming commissions in various states would prohibit it, it seems that this would be a good way for the casinos to lower the ER on any MultiStrike games when they want.

More importantly, will the casinos be required to display the percentages of FREE-RIDES anywhere on or within the game? If they are permitted to have differing FREE-RIDE percentages, and if they aren't required to reveal that vital information, then it seems to me that it will be completely impossible for anyone to know whether they are playing a positive or near positive machine, even if they are playing a game that _usually_ has a high ER.

Just to see how much of a difference it makes, I just used the same calc for Bob's Strategy, but I reduced the FREE-RIDE percentages by 1% at each level ... FROM: 7.7%, 7.0%, and 6.4% ... TO: 6.7%, 6.0%, and 5.4%.
Recomputation yields 98.1861% ER -- a reduction in the ER of slightly more than 1.6%

Once again, thanks to everyone for their help and patience with me; I hope I haven't lost _all_ my credibility after such stupid mistakes.

Cheers.

Bill Velek