vpFREE2 Forums

Strategy Comparison

#1

As for adjusting your strategy to go for more royals or quads, that

is generally a good way to decrease your bankroll rather than
increase it. However, you might want to look through the sections on
alternative strategies

http://members.cox.net/vpfree/FAQ_S.htm#WS

I am not questioning the current poster; only the "Strategies"
contained in the reference.

I considered the three strategies, 9/6 JoB Min-Cost-Royal Strategy,
9/6 JoB Min-Risk Strategy, and the 9/6 JoB Best-Shot-Royal Strategy.
Let me call these as Strategies #1, #2, and #3

As I suspect I am treading on Dogma, which is verboten, I will try to
provide "significant, persuasive documentation"; at least as far as I
understand it to be.

The preambles to each of these strategies make it appear to the
layman, as if these are significantly different, despite an admission
in the first case that #1 and #2 are very close but not identical.

I copied the strategies to an Excel spreadsheet, and compared them
side by side.

There are 49 rules in each strategy, starting from Royal Flush to
Garbage. #1 and #2 differ in steps 46 and 47 (of the 49 steps). If
you have suited AT and three junk cards, #1 recommends keeping the
suited AT, and #2 recommends keeping just the Ace. As the next best
choice, #1 recommends keeping the Ace, #2, keeping suited AT.

That's it. All the other rules from 1 to 49, except for 46 & 47 are
the same.

Is this minor, insignificant (remember the difference starts at the
46th of 49 rules) difference enough to warrant labeling these as two
major strategies? I have come across this situation before in some
routing programs. Let's say you ask for directions to go from point A
to point B, and ask for two choices. Minimum time, and minimum
distance. Out come a set of 49 steps for each, with the difference in
the 46th and 47th steps. In one you take a left and a right, in the
second you go straight for one more block and turn left. It is a
question of 99.8 miles and 125 minutes (minimum distance) for one and
99.9 miles and 124 minutes (minimum time) for another.

The difference between Strategies #2 and #3 is a little bit, but only
a little bit more.

The preferred choice in steps 28 and 29, for strategies #1 and #2 is
3 Cards to a Straight Flush with 0-gaps, and 0 high-cards, and then
suited QJ. For Strategy #3, this order is reversed.

Then it is the same until step 41. At 41/42, the choices are,
respectively, Jack and unsuited KT, for #1 and #2. For #3, this order
is reversed.

At 43/44, the choices are, respectively, unsuited AJ/AQ/AK and Queen
by itself, for #1 and #2. For #3, this order is reversed.
Then it becomes complicated! At 45/46/47, the choice in #1 is King by
itself, suited AT, and then Ace by itself. Let me abbreviate the
three choices as K/AT/A. For #2, the choice is K/A/AT. For #3, it is
AT/K/A. The difference in EV at these steps between the choices is in
the third decimal place.

I admit the theoretical difference between the strategies. One can
define a Risk Function, differentiate it to find the maxima and
minima, and declare the result as the optimal solution. Change the
Risk Function definition, you get a different maxima and minima. Fine
for theoretical purposes.

But these are strategies proposed to be adopted for the two/three
purposes defined. When they make no practical difference in the
outcome, which seems to me to be the case, what is their raison
d'être? These are like three normal distributions with a huge
overlap, with the means ever so slightly separated. It is impossible
to say which distribution a given sample belongs to.

In a practical situation, there is no difference in outcomes between
the two. It is as if I am arguing that it is possible to place 998
angels on the head of a pin, and you are saying, no, it should be
possible to place 998.5!

Optimal strategies, when derived mathematically, sometimes surprise
subject matter experts. But when they ponder over them, they see the
logic behind the strategy, in an aha moment. Thus, the strategies,
even if computer generated, must make intuitive sense; at least when
you try hard! Conceding that there must be a method to this madness,
I request anybody here who can look at the broad picture, and
recognize the forest from the trees, to explain how the little tweaks
present in the strategies take them along their chosen paths.

I am placing the Strategy Comparison (Excel Worksheet) in the files
section. I have very carefully formatted it so that bulk of the
information should be printed on two landscape pages. You can ignore
the 3rd and 4th pages.

PS: A question for the Ecel friendly and mathematically unchallenged.
Take the range of cells C6:C14 and D6:D14. Shouldn't the sumproduct
come out to the number given in C3? I am getting a considerably
higher number (in cell F15). Same thing for the other two cases.
Would this be due to round-off errors in the range of numbers?

···

--- In vpFREE@yahoogroups.com, "murphyfields" <jkludge@...> wrote:

#2

Having spent a lot of time with Excel, I'll tackle the easy one first...

You missed the note that says "ER (with 800 Unit Royal)"

Change the payout for a royal to 4000 and you get much closer. I
think the rest is just roundoff error.

Interesting doing a side-by-side comparison. In all honesty, I would
not expect huge differences, since a lot of the higher hands are
somewhat obvious. However, I did expect somewhat larger differences.

I always viewed alternative strategies as an interesting sidenote
saying "HEY, THERE IS NOTHING THAT SAYS YOU MUST MAXIMIZE EV." And in
the end it may be nice to have a reason to choose that suited AT over
just the ace, just to up your chances of hitting a royal.

As for why you might want to make the changes, I think I remember
someone pointing out that, in the long run, your chances of getting
significantly more royals than expected has a certain appeal. For my
level of play, it is probably not all that important which one I use.

- John

···

--- In vpFREE@yahoogroups.com, "Adams Myth" <Adams_Myth@...> wrote:

--- In vpFREE@yahoogroups.com, "murphyfields" <jkludge@> wrote:
> As for adjusting your strategy to go for more royals or quads, that
is generally a good way to decrease your bankroll rather than
increase it. However, you might want to look through the sections on
alternative strategies

> http://members.cox.net/vpfree/FAQ_S.htm#WS

I am not questioning the current poster; only the "Strategies"
contained in the reference.

I considered the three strategies, 9/6 JoB Min-Cost-Royal Strategy,
9/6 JoB Min-Risk Strategy, and the 9/6 JoB Best-Shot-Royal Strategy.
Let me call these as Strategies #1, #2, and #3

As I suspect I am treading on Dogma, which is verboten, I will try to
provide "significant, persuasive documentation"; at least as far as I
understand it to be.

The preambles to each of these strategies make it appear to the
layman, as if these are significantly different, despite an admission
in the first case that #1 and #2 are very close but not identical.

I copied the strategies to an Excel spreadsheet, and compared them
side by side.

There are 49 rules in each strategy, starting from Royal Flush to
Garbage. #1 and #2 differ in steps 46 and 47 (of the 49 steps). If
you have suited AT and three junk cards, #1 recommends keeping the
suited AT, and #2 recommends keeping just the Ace. As the next best
choice, #1 recommends keeping the Ace, #2, keeping suited AT.

That's it. All the other rules from 1 to 49, except for 46 & 47 are
the same.

Is this minor, insignificant (remember the difference starts at the
46th of 49 rules) difference enough to warrant labeling these as two
major strategies? I have come across this situation before in some
routing programs. Let's say you ask for directions to go from point A
to point B, and ask for two choices. Minimum time, and minimum
distance. Out come a set of 49 steps for each, with the difference in
the 46th and 47th steps. In one you take a left and a right, in the
second you go straight for one more block and turn left. It is a
question of 99.8 miles and 125 minutes (minimum distance) for one and
99.9 miles and 124 minutes (minimum time) for another.

The difference between Strategies #2 and #3 is a little bit, but only
a little bit more.

The preferred choice in steps 28 and 29, for strategies #1 and #2 is
3 Cards to a Straight Flush with 0-gaps, and 0 high-cards, and then
suited QJ. For Strategy #3, this order is reversed.

Then it is the same until step 41. At 41/42, the choices are,
respectively, Jack and unsuited KT, for #1 and #2. For #3, this order
is reversed.

At 43/44, the choices are, respectively, unsuited AJ/AQ/AK and Queen
by itself, for #1 and #2. For #3, this order is reversed.
Then it becomes complicated! At 45/46/47, the choice in #1 is King by
itself, suited AT, and then Ace by itself. Let me abbreviate the
three choices as K/AT/A. For #2, the choice is K/A/AT. For #3, it is
AT/K/A. The difference in EV at these steps between the choices is in
the third decimal place.

I admit the theoretical difference between the strategies. One can
define a Risk Function, differentiate it to find the maxima and
minima, and declare the result as the optimal solution. Change the
Risk Function definition, you get a different maxima and minima. Fine
for theoretical purposes.

But these are strategies proposed to be adopted for the two/three
purposes defined. When they make no practical difference in the
outcome, which seems to me to be the case, what is their raison
d'être? These are like three normal distributions with a huge
overlap, with the means ever so slightly separated. It is impossible
to say which distribution a given sample belongs to.

In a practical situation, there is no difference in outcomes between
the two. It is as if I am arguing that it is possible to place 998
angels on the head of a pin, and you are saying, no, it should be
possible to place 998.5!

Optimal strategies, when derived mathematically, sometimes surprise
subject matter experts. But when they ponder over them, they see the
logic behind the strategy, in an aha moment. Thus, the strategies,
even if computer generated, must make intuitive sense; at least when
you try hard! Conceding that there must be a method to this madness,
I request anybody here who can look at the broad picture, and
recognize the forest from the trees, to explain how the little tweaks
present in the strategies take them along their chosen paths.

I am placing the Strategy Comparison (Excel Worksheet) in the files
section. I have very carefully formatted it so that bulk of the
information should be printed on two landscape pages. You can ignore
the 3rd and 4th pages.

PS: A question for the Ecel friendly and mathematically unchallenged.
Take the range of cells C6:C14 and D6:D14. Shouldn't the sumproduct
come out to the number given in C3? I am getting a considerably
higher number (in cell F15). Same thing for the other two cases.
Would this be due to round-off errors in the range of numbers?

#3

Having spent a lot of time with Excel, I'll tackle the easy one

first...

You missed the note that says "ER (with 800 Unit Royal)"
Change the payout for a royal to 4000 and you get much closer. I
think the rest is just roundoff error.

Speaking of not seeing the forest for the trees, I didn't notice the
odd looking payoffs. Why do the numbers have fractional payoffs? I am
indeed missing something.

   RoyalF StFl Quad FH Fl Str 3Knd 2Pr HiPr
#1 4879.97 250 125 45 30 20 15 10 5
#2 4739.34 252.55 125.62 45.07 30.03 20.01 15.01 10 5
#3 5193.4 244.19 123.57 44.83 29.93 19.97 14.99 10 5

···

--- In vpFREE@yahoogroups.com, "murphyfields" <jkludge@...> wrote:

#4

> As for adjusting your strategy to go for more royals or quads, that

is generally a good way to decrease your bankroll rather than
increase it. However, you might want to look through the sections on
alternative strategies

> http://members.cox.net/vpfree/FAQ_S.htm#WS

I am not questioning the current poster; only the "Strategies"
contained in the reference.

I considered the three strategies, 9/6 JoB Min-Cost-Royal Strategy,
9/6 JoB Min-Risk Strategy, and the 9/6 JoB Best-Shot-Royal Strategy.
Let me call these as Strategies #1, #2, and #3

As I suspect I am treading on Dogma, which is verboten, I will try to
provide "significant, persuasive documentation"; at least as far as I
understand it to be.

I doubt that anyone here considers these strategies to be "dogma".
Tread all you like.

The preambles to each of these strategies make it appear to the
layman, as if these are significantly different, despite an admission
in the first case that #1 and #2 are very close but not identical.

I don't claim they are "significantly different," only that they are each
best at reaching the particular objective for which they were designed.

Comparing strategies by looking at the number of changes in the
strategy chart isn't a reliable way to assess the significance of the
changes, because the chart doesn't give any indication of how
frequently the player would have to play differently when using
the "other" strategy. It is better to look at numbers that measure
performance by the same standards that are used to optimize
the strategies.

I have some reason to believe that the fact that these came out so
close together is just an accident. If we change games to DB 10/7
then the following observations can be made about these same
strategies (I'll call them MR for "min risk", MCR for "min-cost royal"
and BSR for "best-shot royal"):

The MR and MCR strategies are still pretty close together. The
MR strategy lumps 4/flush hands together when they have
either no high cards or only one high card, and these are at
position 23 in the table. The MCR inserts "4/royal (1 high)"
at position 21, increasing the total number of rules to 64.
The only other strategy change between MR and MCR is to swap
the "suited KJ/KQ" and "3/str-flush (1 hole, 0 high)" at positions
40 and 41 in the MCR table.

The MCR strategy gives a cost of 715.71 units for a royal, while
MR strategy gives 716.32. By comparison, the BSR strategy has a cost
of 761.54 units. So, compared to MCR strategy, the cost for a royal is only
0.085% higher when using MR strategy, but BSR strategy raises the
cost of a royal by 6.4% above the minimum.

The bankroll needed for a 50% RoR is 5569 units when using MR strategy.
The MCR strategy needs 5609 units. The BSR strategy requires 10,602
units to give the same 50% RoR. The MCR strategy needs a bankroll that
is 0.72% larger to give the same level of risk, while the BSR strategy needs
a bankroll that is 90.38% larger than MR strategy! This is a result of the
tiny EV of the DB 10/7 game, and the fact that BSR strategy just happens
to require a larger EV tradeoff than the MR strategy. For the 9/6 JoB
game with a 1300 unit jackpot, the EV reduction is about the same for
MR/MCR/BSR strategies.

Comparing MR to BSR, the first strategy difference causes steps
7 and 8 to trade places. The BSR strategy prefers a full house
over trip aces, while the MR strategy breaks up the full house to
draw to trip aces. There are several other differences, and they
are mostly in the upper half of the priority list.

Another way to judge the "distance" between strategies is to look
at the royal cycle. For the DB 10/7 game, max-EV strategy has a
royal cycle of 48048 hands, compared to royal cycles of 48723
for MR strategy, 50010 for MCR strategy and 37139 for BSR strategy.
Once again the numbers for MR and MCR are much closer than
for BSR.

I copied the strategies to an Excel spreadsheet, and compared them
side by side.

There are 49 rules in each strategy, starting from Royal Flush to
Garbage. #1 and #2 differ in steps 46 and 47 (of the 49 steps). If
you have suited AT and three junk cards, #1 recommends keeping the
suited AT, and #2 recommends keeping just the Ace. As the next best
choice, #1 recommends keeping the Ace, #2, keeping suited AT.

The strategies are generated by my program. Different programs, given
the same payoff schedules, might generate a slightly different list of
priorities. Accounting for penalty cards would also be likely to bring
out additional differences in the strategies.

That's it. All the other rules from 1 to 49, except for 46 & 47 are
the same.

Is this minor, insignificant (remember the difference starts at the
46th of 49 rules) difference enough to warrant labeling these as two
major strategies?

I don't recall using the word "major" to describe any of these strategies.
I think such a distinction would be quite arbitrary. In fact, I don't feel
it is really meaningful to "rank" strategies. They strive to reach different
goals. But, if you compare any one of these strategies to the max-EV
strategy for the same game, then you'll see a much bigger difference
than what you describe here.

I have come across this situation before in some
routing programs. Let's say you ask for directions to go from point A
to point B, and ask for two choices. Minimum time, and minimum
distance. Out come a set of 49 steps for each, with the difference in
the 46th and 47th steps. In one you take a left and a right, in the
second you go straight for one more block and turn left. It is a
question of 99.8 miles and 125 minutes (minimum distance) for one and
99.9 miles and 124 minutes (minimum time) for another.

This is exactly analogous. Optimization problems such as these almost
always produce a tradeoff. Different paths are "best" in their own way,
and the differences often appear in only a few places.

Comparing strategies by looking at the number of changes in the
strategy chart isn't a reliable way to assess the significance of the
changes, because the chart doesn't give any indication of how
frequently the player would have to play differently when using
the "other" strategy. It is better to look at numbers that measure
performance by the same standards that are used to optimize
the strategies.

The difference between Strategies #2 and #3 is a little bit, but only
a little bit more.

The preferred choice in steps 28 and 29, for strategies #1 and #2 is
3 Cards to a Straight Flush with 0-gaps, and 0 high-cards, and then
suited QJ. For Strategy #3, this order is reversed.

Then it is the same until step 41. At 41/42, the choices are,
respectively, Jack and unsuited KT, for #1 and #2. For #3, this order
is reversed.

At 43/44, the choices are, respectively, unsuited AJ/AQ/AK and Queen
by itself, for #1 and #2. For #3, this order is reversed.
Then it becomes complicated! At 45/46/47, the choice in #1 is King by
itself, suited AT, and then Ace by itself. Let me abbreviate the
three choices as K/AT/A. For #2, the choice is K/A/AT. For #3, it is
AT/K/A. The difference in EV at these steps between the choices is in
the third decimal place.

I admit the theoretical difference between the strategies. One can
define a Risk Function, differentiate it to find the maxima and
minima, and declare the result as the optimal solution. Change the
Risk Function definition, you get a different maxima and minima. Fine
for theoretical purposes.

But these are strategies proposed to be adopted for the two/three
purposes defined. When they make no practical difference in the
outcome, which seems to me to be the case, what is their raison
d'être? These are like three normal distributions with a huge
overlap, with the means ever so slightly separated. It is impossible
to say which distribution a given sample belongs to.

How different would the strategies have to be before you would
conclude that they made a "practical difference?"

In a practical situation, there is no difference in outcomes between
the two. It is as if I am arguing that it is possible to place 998
angels on the head of a pin, and you are saying, no, it should be
possible to place 998.5!

It is true that the three particular strategies you've chosen are
extremely close to each other. Given a different game, the same
three goals may produce strategies that much bigger differences.

Optimal strategies, when derived mathematically, sometimes surprise
subject matter experts. But when they ponder over them, they see the
logic behind the strategy, in an aha moment. Thus, the strategies,
even if computer generated, must make intuitive sense; at least when
you try hard! Conceding that there must be a method to this madness,
I request anybody here who can look at the broad picture, and
recognize the forest from the trees, to explain how the little tweaks
present in the strategies take them along their chosen paths.

I've about given up hope for making any of this very intuitive.
The best suggestion I have is to read the section on "Virtual Payoffs"
and ask lots of questions. The strategy changes are mathematically
equivalent to playing as if the payoffs are changed to match the
objective. The correct virtual payoffs are listed in the table called
"9/6 JoB Virtual Payoffs." Here's an extra hint for the Min-RoR payoffs:
they are all proportional to the probability that the new bankroll will
allow you to play forever without going broke. So, a royal payoff of
1300 units will increase your probability of playing forever by a
factor of 1003.5 as compared to starting with a single unit. Similarly,
hitting quads for a payoff of 25 units will increase this probability by
a factor of 24.713.

···

On Wednesday 30 August 2006 8:57 am, Adams Myth wrote:

--- In vpFREE@yahoogroups.com, "murphyfields" <jkludge@...> wrote:

#5

These are the "virtual payoffs" that are used to generate the playing
strategy. They are rounded to two decimal places. The optimal strategy
is mathematically equivalent to playing as if the payoffs had these
values. You can think of this as a "weighting function" than shifts the
payoffs to values that are aligned to the specific objection that the
strategy is trying to optimize.

···

On Wednesday 30 August 2006 3:38 pm, Adams Myth wrote:

--- In vpFREE@yahoogroups.com, "murphyfields" <jkludge@...> wrote:
> Having spent a lot of time with Excel, I'll tackle the easy one

first...

> You missed the note that says "ER (with 800 Unit Royal)"
> Change the payout for a royal to 4000 and you get much closer. I
> think the rest is just roundoff error.

Speaking of not seeing the forest for the trees, I didn't notice the
odd looking payoffs. Why do the numbers have fractional payoffs? I am
indeed missing something.

   RoyalF StFl Quad FH Fl Str 3Knd 2Pr HiPr
#1 4879.97 250 125 45 30 20 15 10 5
#2 4739.34 252.55 125.62 45.07 30.03 20.01 15.01 10 5
#3 5193.4 244.19 123.57 44.83 29.93 19.97 14.99 10 5

#6

This is confusing. Given that a certain game has a certain payoff
table, say 4000/250/125/45/30/20/15/10/5, you go about generating
strategies #1, #2, and #3. Then what is the rationale for altering
the payoff number?

The strategy should adapt to the paytable; not the paytable to the
strategy - what is it that I am missing?

···

--- In vpFREE@yahoogroups.com, Steve Jacobs <jacobs@...> wrote:

   RoyalF StFl Quad FH Fl Str 3Knd 2Pr HiPr
#1 4879.97 250 125 45 30 20 15 10 5
#2 4739.34 252.55 125.62 45.07 30.03 20.01 15.01 10 5
#3 5193.4 244.19 123.57 44.83 29.93 19.97 14.99 10 5

These are the "virtual payoffs" that are used to generate the

playing strategy. They are rounded to two decimal places. The
optimal strategy is mathematically equivalent to playing as if the
payoffs had these values. You can think of this as a "weighting
function" than shifts the payoffs to values that are aligned to the
specific objection that the strategy is trying to optimize.

#7

Yes, it is confusing, and I am probably not the best person to explain
it. First I suggest checking

http://members.cox.net/vpfree/FAQ_S.htm#VIR

I think the idea is that there are some fairly simple relationships,
where optimizing over one criterion is methematically equivalent to
optimizing for max EV if the pay table is changed. The magic point is
where the EV is exactly 100%, which can be achieved by adjusting the
payout for just the royal (one of the strategies) or by scaling all of
the payouts.

I can see how this is possible, but I won't pretend to be able to
derive it myself.

- John

···

--- In vpFREE@yahoogroups.com, "Adams Myth" <Adams_Myth@...> wrote:

This is confusing. Given that a certain game has a certain payoff
table, say 4000/250/125/45/30/20/15/10/5, you go about generating
strategies #1, #2, and #3. Then what is the rationale for altering
the payoff number?

The strategy should adapt to the paytable; not the paytable to the
strategy - what is it that I am missing?

--- In vpFREE@yahoogroups.com, Steve Jacobs <jacobs@> wrote:
>> RoyalF StFl Quad FH Fl Str 3Knd 2Pr HiPr
>> #1 4879.97 250 125 45 30 20 15 10 5
>> #2 4739.34 252.55 125.62 45.07 30.03 20.01 15.01 10 5
>> #3 5193.4 244.19 123.57 44.83 29.93 19.97 14.99 10 5

> These are the "virtual payoffs" that are used to generate the
playing strategy. They are rounded to two decimal places. The
optimal strategy is mathematically equivalent to playing as if the
payoffs had these values. You can think of this as a "weighting
function" than shifts the payoffs to values that are aligned to the
specific objection that the strategy is trying to optimize.

#8

Adams Myth wrote:

This is confusing. Then what is the rationale for altering
the payoff number?

Building on John's answer ... the adjusted (virtual) paytable is
selected as a means by which to find a strategy that achieves a
desired goal. It represents a paytable where max-ER strategy and the
strategy for the desired goal are one and the same.

Having found it, the resulting strategy also achieves the desired goal
when used against our actual paytable.

- Harry

#9

Harry's response is a good summary. The virtual payoffs are used to
transform the problem from "maximize objective X" into a problem
that is easier to solve -- maximizing "virtual" EV of a slightly
different payoff schedule.

This is sort of like using polar coordinates or cylindrical coordinates
to make it easier to solve problems that are very difficult when using
Cartesian coordinates. I hope that explanation is helpful.

···

On Thursday 31 August 2006 1:56 pm, Harry Porter wrote:

Adams Myth wrote:
> This is confusing. Then what is the rationale for altering
> the payoff number?

Building on John's answer ... the adjusted (virtual) paytable is
selected as a means by which to find a strategy that achieves a
desired goal. It represents a paytable where max-ER strategy and the
strategy for the desired goal are one and the same.

Having found it, the resulting strategy also achieves the desired goal
when used against our actual paytable.

#10

This is confusing. Given that a certain game has a certain payoff
table, say 4000/250/125/45/30/20/15/10/5, you go about generating
strategies #1, #2, and #3. Then what is the rationale for altering
the payoff number?

The payoffs aren't altered after the strategy is generated, they are
altered as part of the process for generating the strategy. The tricky
part is figuring out how to "stretch" the payoffs in a way that matches
the goal. Then, the altered payoffs are used by a VP program (pick
your favorite VP program) to generate a max-EV strategy.

The strategy should adapt to the paytable; not the paytable to the
strategy - what is it that I am missing?

The strategy is derived from the original paytable, by stretching
the payoffs in just the right way, then finding a max-EV strategy for
the stretched payoffs (I call them "virtual" rather than "stretched"
but the concept is the same).

I realize that this may seem like a really bizarre thing to do, but
that is how I generate alternate strategies which are mathematically
precise.

···

On Thursday 31 August 2006 12:15 pm, Adams Myth wrote:

#11

It's frightening, but I am beginning to understand this. It is more
akin to a logaritmic transformation than a polar-cartesian conversion.

···

--- In vpFREE@yahoogroups.com, Steve Jacobs <jacobs@...> wrote:

The strategy is derived from the original paytable, by stretching the

payoffs in just the right way, then finding a max-EV strategy for the
stretched payoffs (I call them "virtual" rather than "stretched" but
the concept is the same).