Risk of Ruin and Required Return/Bankroll Requirement

[Harry_Porter]

In past months I've entered a couple of posts to the various vp groups
concerning the risk associated with video poker and the statistics that
are commonly presented in video poker software.

I'm going to discuss further evaluation of this and, frankly, if you
couldn't give a damn it's time to tune out :). (That's for your
benefit, Marsha!)

The most common statistic displayed is the variance of a game. I've
maintained that standard deviation, the square root of variance, is a
more accurate measure of a game's risk. However, at least one source
who I greatly respect has maintained that variance is the more pertinent
measure.

In the wee hours of a sleepless night I've had a chance to examine this
question more closely. A key aid in my review has been a Risk of Ruin
Spreadsheet that was developed by Dunbar (first name a mystery). The
source of his work is a short-term risk of ruin formula detailed in Don
Schlesinger's BJ Attack (1997), p. 162. The spreadsheet allows you to
determine the ROR for any specified bankroll given the standard
deviation and effective return of a game. While the formula was devised
for the specific purpose of evaluating blackjack, it is readily put to
use in looking at video poker.

My review revealed that both variance and standard deviation are
relevant measures. If you're interested in the bankroll to support play
of a game, then variance is the direct measure that is proportional to
bankroll requirements. However, if you're starting from a fixed
bankroll, standard deviation (the square root of variance) is the value
that will allow you to directly determine how the ROR of one game
compares to another as a function of return. These comparisons are most
readily made when either std. deviation or return is a common value.
Variation of both variables presents a more complex exercise.

The reason I've maintained that standard deviation is the more relevant
value to look at stems from my studies and work in finance. In
evaluating alternate investments, the return that is expected and
potential risk to that return is a direct function of standard deviation
(expressed in an extrapolated value known as a "beta"). However, my
position that this was applicable to video poker was an intuitive
proposition and I haven't been prepared to discuss the specific
application until now.

The explanation is fairly straightforward. As in finance, if you
evaluate the risk associated with two alternate games the same principal
applies and standard deviation will give you a relative measure of risk
of ruin as a function of return. However, consider how the return of a
game translates to potential swings in the results you might
experience. Any variation in actual return of play from the theoretical
ER will produce a compounding effect with each hand played. Therefore,
the potential downside that you face grows proportionally with the
number of hands played. This suggests that standard deviation, a
measure that pertains to return, won't reflect that compounding. On the
other hand, variance, the square of standard deviation, will measure the
potential growth of the absolute shortfall in cash return over time.
Thus bankroll, the cushion against that shortfall, is more directly
reflected by the variance statistic.

The application of this finding is clear cut. If you're interested in
how the bankroll requirements fluctuate for one game vs. another, then
variance is the creature that you're after. Given a percent increase in
variance, the bankroll requirements will increase accordingly (provided
returns are comparable). Determination of actual bankroll requirements
can be determined through a number of means. Application of the
blackjack formula will be a bit of a stretch for the general
quantitative inclined (and posed a bit of a challenge for me). Note
that the input to this formula is standard deviation, however it
effectively translates that value to variance in the formula. Other
means of measurement are available. (Dan Paymar has detailed one
formula that will pose a challenge of a different nature in a VP Times
article that can be found on his website.)

However, if what you're looking for is how alternate games will strain
your existing bankroll (relatively speaking), then standard deviation
will give you that indication (although alternate game returns do
complicate that calculation). The decision of what measure to apply in
looking at a game will depend on your intent.

I hope this discussion clarifies more than it obscures. I'm interested
in any feedback, either in support of my assertions or in argument, from
any knowledgeable sources.

- Harry

[Steve_Jacobs]

The most common statistic displayed is the variance of a game. I've
maintained that standard deviation, the square root of variance, is a
more accurate measure of a game's risk. However, at least one source
who I greatly respect has maintained that variance is the more pertinent
measure.

I fail to see how either can be "more pertinent" than the other. They each
encompass the same information. Taking sqrt(V) to get S.D. just "stretches"
the axes a bit, but in terms of accuracy they must inherently each be just
as accurate as the other.

In the wee hours of a sleepless night I've had a chance to examine this
question more closely. A key aid in my review has been a Risk of Ruin
Spreadsheet that was developed by Dunbar (first name a mystery). The
source of his work is a short-term risk of ruin formula detailed in Don
Schlesinger's BJ Attack (1997), p. 162. The spreadsheet allows you to
determine the ROR for any specified bankroll given the standard
deviation and effective return of a game. While the formula was devised
for the specific purpose of evaluating blackjack, it is readily put to
use in looking at video poker.

Don's formula is also an approximation based on EV and variance. This
is essentially equivalent to using a normal approximation in place of the
games true probability distribution. I seriously question the accuracy
of such approximations in the short term. They are fine for looking at
long term results as the number of plays approach infinity.

The exact ROR for VP games can be computed in a more direct fashion,
using the "characteristic equation" which represents the game. I don't
have time right now to go into detail (and I'll be at BARGE next week,
and thus offline) but I feel this is a vastly superior approach for answering
questions that involve risk of ruin. In addition, this approach allows any
game to be modelled with a "risk-equivalent coin".

My review revealed that both variance and standard deviation are
relevant measures. If you're interested in the bankroll to support play
of a game, then variance is the direct measure that is proportional to
bankroll requirements.

I disagree. I'd say that variance gives a "good approximation" but is
not a "direct measure". The true, exact ROR value (I'll call it XROR to
distinguish it from the other forms of ROR such as Don's approximate
formula,) computed from the characteristic equation, is a direct measure.
Anything else is not direct except perhaps for special cases.

However, if you're starting from a fixed
bankroll, standard deviation (the square root of variance) is the value
that will allow you to directly determine how the ROR of one game
compares to another as a function of return. These comparisons are most
readily made when either std. deviation or return is a common value.
Variation of both variables presents a more complex exercise.

XROR and EV are somewhat independent measures. Higher EV doesn't
imply lower XROR, nor vice-versa. You can compares games on one
basis or the other, but it is important to understand that these are like
apples and oranges. An analogy is height and weight. They measure
different things, and greater height doesn't necessarily imply greater
weight, even though they often track closely.

The explanation is fairly straightforward. As in finance, if you
evaluate the risk associated with two alternate games the same principal
applies and standard deviation will give you a relative measure of risk
of ruin as a function of return.

The problem with this is that ROR isn't a function of return. It is easy to
construct biased-coin games where game A has 10X EV of game B, but
game B has much lower risk of ruin.

However, consider how the return of a
game translates to potential swings in the results you might
experience. Any variation in actual return of play from the theoretical
ER will produce a compounding effect with each hand played. Therefore,
the potential downside that you face grows proportionally with the
number of hands played. This suggests that standard deviation, a
measure that pertains to return, won't reflect that compounding. On the
other hand, variance, the square of standard deviation, will measure the
potential growth of the absolute shortfall in cash return over time.
Thus bankroll, the cushion against that shortfall, is more directly
reflected by the variance statistic.

To throw another curve in here, consider the fact that for most VP games,
the playing strategy which is optimal from an XROR perspective is
significantly (read "measureably") different than the playing strategy which
maximizes EV. The goals "minimize ROR" and "maximize EV" are simply
different, and require different tactics.

However, if what you're looking for is how alternate games will strain
your existing bankroll (relatively speaking), then standard deviation
will give you that indication (although alternate game returns do
complicate that calculation). The decision of what measure to apply in
looking at a game will depend on your intent.

Since variance/std_dev are approximate, I prefer working directly
with XROR. A big advantage of this approach is that it applies directly
to problems of the form "given bankroll B and target T, what is the
probability of reaching target T before losing the entire bankroll". The
XROR approach allows these problems to be framed in terms of a repeated
coin flip using a risk-equivalent coin that "matches" the games exact
probability distribution. This can give some interesting insights in what
XROR "means" and the qualitative differences between XROR and EV.
The difference might be summarized by asking "do you want to reach
the target most quickly (max-EV) or do you want the highest probability
of reaching the target before going broke (min-ROR)". The difference
in these goals is subtle, and some may ask "what's the difference?"

I hope this discussion clarifies more than it obscures. I'm interested
in any feedback, either in support of my assertions or in argument, from
any knowledgeable sources.

This is an area that I've been spending a lot of time thinking about
in recent months. I'd be very interested in discussing/exploring this
further. I think this area is poorly understood, even by experts, and
there are some surprising conclusions that pop out of the math. One
example of this: gambling experts will tell you that progressions
cannot alter EV. They go on to conclude "progressions are worthless,"
but this is simply false. Progressions can have an impact on ROR.
In fact, when playing an unfavorable coin-flip game with the goal of
maximizing the probability of reaching a target bankroll, the optimal
betting strategy takes the form of a Martingale progression. You bet
just enough that winning reaches the goal. If you lose, you're now
twice as far from the goal and thus bet twice as much. Repeat until
you either reach the goal or lose your bankroll. This is the optimal
strategy from a ROR perspective, and any betting strategy which
deviates from this will result in an increase in ROR.

So, the fact that progressions cannot alter EV does NOT imply that
progressions are worthless. I have yet to see a single "expert" point
this out. In fact, experts are often smug about this to the point that
they won't respond to any article that asks about progressions, and
they belittle those who ask. I think this ultimately stems from the fact
that gambling experts are so focused on EV that the entire topic of
ROR has received much less study, and is poorly understood.

···

On Saturday 26 July 2003 06:04 am, Harry D. Porter wrote:

[Michael_Boutot]

In the real world a Martingale progression IS the road to ruin.
People bump into table limits after 5 or 6 losses which can occur
often even on 50/50 probabilities. They then can't bet enough to get
even on 1 win. Progressions are dangerous to an extreme if not
worthless in general.

> The most common statistic displayed is the variance of a game.

I've

> maintained that standard deviation, the square root of variance,

is a

> more accurate measure of a game's risk. However, at least one

source

> who I greatly respect has maintained that variance is the more

pertinent

> measure.

I fail to see how either can be "more pertinent" than the other.

They each

encompass the same information. Taking sqrt(V) to get S.D.

just "stretches"

the axes a bit, but in terms of accuracy they must inherently each

be just

as accurate as the other.

> In the wee hours of a sleepless night I've had a chance to

examine this

> question more closely. A key aid in my review has been a Risk of

Ruin

> Spreadsheet that was developed by Dunbar (first name a mystery).

The

> source of his work is a short-term risk of ruin formula detailed

in Don

> Schlesinger's BJ Attack (1997), p. 162. The spreadsheet allows

you to

> determine the ROR for any specified bankroll given the standard
> deviation and effective return of a game. While the formula was

devised

> for the specific purpose of evaluating blackjack, it is readily

put to

> use in looking at video poker.

Don's formula is also an approximation based on EV and variance.

This

is essentially equivalent to using a normal approximation in place

of the

games true probability distribution. I seriously question the

accuracy

of such approximations in the short term. They are fine for

looking at

long term results as the number of plays approach infinity.

The exact ROR for VP games can be computed in a more direct fashion,
using the "characteristic equation" which represents the game. I

don't

have time right now to go into detail (and I'll be at BARGE next

week,

and thus offline) but I feel this is a vastly superior approach for

answering

questions that involve risk of ruin. In addition, this approach

allows any

game to be modelled with a "risk-equivalent coin".

> My review revealed that both variance and standard deviation are
> relevant measures. If you're interested in the bankroll to

support play

> of a game, then variance is the direct measure that is

proportional to

> bankroll requirements.

I disagree. I'd say that variance gives a "good approximation" but

is

not a "direct measure". The true, exact ROR value (I'll call it

XROR to

distinguish it from the other forms of ROR such as Don's approximate
formula,) computed from the characteristic equation, is a direct

measure.

Anything else is not direct except perhaps for special cases.

> However, if you're starting from a fixed
> bankroll, standard deviation (the square root of variance) is the

value

> that will allow you to directly determine how the ROR of one game
> compares to another as a function of return. These comparisons

are most

> readily made when either std. deviation or return is a common

value.

> Variation of both variables presents a more complex exercise.

XROR and EV are somewhat independent measures. Higher EV doesn't
imply lower XROR, nor vice-versa. You can compares games on one
basis or the other, but it is important to understand that these

are like

apples and oranges. An analogy is height and weight. They measure
different things, and greater height doesn't necessarily imply

greater

weight, even though they often track closely.

> The explanation is fairly straightforward. As in finance, if you
> evaluate the risk associated with two alternate games the same

principal

> applies and standard deviation will give you a relative measure

of risk

> of ruin as a function of return.

The problem with this is that ROR isn't a function of return. It

is easy to

construct biased-coin games where game A has 10X EV of game B, but
game B has much lower risk of ruin.

> However, consider how the return of a
> game translates to potential swings in the results you might
> experience. Any variation in actual return of play from the

theoretical

> ER will produce a compounding effect with each hand played.

Therefore,

> the potential downside that you face grows proportionally with the
> number of hands played. This suggests that standard deviation, a
> measure that pertains to return, won't reflect that compounding.

On the

> other hand, variance, the square of standard deviation, will

measure the

> potential growth of the absolute shortfall in cash return over

time.

> Thus bankroll, the cushion against that shortfall, is more

directly

> reflected by the variance statistic.

To throw another curve in here, consider the fact that for most VP

games,

the playing strategy which is optimal from an XROR perspective is
significantly (read "measureably") different than the playing

strategy which

maximizes EV. The goals "minimize ROR" and "maximize EV" are simply
different, and require different tactics.

> However, if what you're looking for is how alternate games will

strain

> your existing bankroll (relatively speaking), then standard

deviation

> will give you that indication (although alternate game returns do
> complicate that calculation). The decision of what measure to

apply in

> looking at a game will depend on your intent.

Since variance/std_dev are approximate, I prefer working directly
with XROR. A big advantage of this approach is that it applies

directly

to problems of the form "given bankroll B and target T, what is the
probability of reaching target T before losing the entire

bankroll". The

XROR approach allows these problems to be framed in terms of a

repeated

coin flip using a risk-equivalent coin that "matches" the games

exact

probability distribution. This can give some interesting insights

in what

XROR "means" and the qualitative differences between XROR and EV.
The difference might be summarized by asking "do you want to reach
the target most quickly (max-EV) or do you want the highest

probability

of reaching the target before going broke (min-ROR)". The

difference

in these goals is subtle, and some may ask "what's the difference?"

> I hope this discussion clarifies more than it obscures. I'm

interested

> in any feedback, either in support of my assertions or in

argument, from

> any knowledgeable sources.

This is an area that I've been spending a lot of time thinking about
in recent months. I'd be very interested in discussing/exploring

this

further. I think this area is poorly understood, even by experts,

and

there are some surprising conclusions that pop out of the math. One
example of this: gambling experts will tell you that progressions
cannot alter EV. They go on to conclude "progressions are

worthless,"

but this is simply false. Progressions can have an impact on ROR.
In fact, when playing an unfavorable coin-flip game with the goal of
maximizing the probability of reaching a target bankroll, the

optimal

betting strategy takes the form of a Martingale progression. You

bet

just enough that winning reaches the goal. If you lose, you're now
twice as far from the goal and thus bet twice as much. Repeat until
you either reach the goal or lose your bankroll. This is the

optimal

strategy from a ROR perspective, and any betting strategy which
deviates from this will result in an increase in ROR.

So, the fact that progressions cannot alter EV does NOT imply that
progressions are worthless. I have yet to see a single "expert"

point

this out. In fact, experts are often smug about this to the point

that

they won't respond to any article that asks about progressions, and
they belittle those who ask. I think this ultimately stems from

the fact

···

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

On Saturday 26 July 2003 06:04 am, Harry D. Porter wrote:
that gambling experts are so focused on EV that the entire topic of
ROR has received much less study, and is poorly understood.

[Harry_Porter]

Steve Jacobs wrote:

I fail to see how either can be "more pertinent" than the other.
They each encompass the same information. Taking sqrt(V) to get
S.D. just "stretches" the axes a bit, but in terms of accuracy they
must inherently each be just as accurate as the other.

Steve, I appreciate the input. It may be that "pertinent" was a poor
choice of wording. What I'm saying is the in the case of ROR, sd is a
more direct indication of the relative ROR's between to games inasmuch
as there's a direction proportion between the two. Ditto between
bankroll requirement and variance.

Obviously there's a well-defined relationship between variance and
standard deviation; the former is the square of the latter.

However, let me pose an analogy that should clarify the reason for the
distinction I make. If I define a 2-D object to have a specific
dimension and I ask how many of these objects can fit in a square of a
given size, I can define that square both in terms of length of one
size in fixed units, or the area of that square in terms of area.
Now, obviously given one measurement you can determine the other, and
therefore you can find the answer to this problem using either
definition of the square. However, it's the case that the solution
will be directly proportional to the area of the square, not the
length of one size.

The problem players are faced with when presented with the variance
statistic of a game by any one of the popular video poker programs is
interpreting the statistic in some practical manner. Let's say we
take the hypothetical choice between play of 9/6 JB in one casino with
1% in cb and/or other cash incentives, and 10/7 DB with .37% cb.
(Clearly I've chosen these values to place total ER on par at 100.54%.)

In this circumstance, I'm saying that the relative bankroll
requirements of these two games is directly proportional to the
relative variance of the two games. Also that play of the two games
with the same bankroll availability will pose a ROR that's
proportional to the relative standard deviations of the two games.

That to me poses a more valuable statement than simply suggesting that
the relative variance of two games presents an ideal of the risk of a
game which is the general grasp under which most players play.

(And I want to take a moment to say that while I have demonstrated
these conclusions for myself with a reasonable degree of satisfaction,
I would be much more comfortable in making these assertions publicly
had I tested them thoroughly under a number of alternate scenarios.
Of course, it's the fortunate nature of some members of this group
that they'll question my statements where appropriate.)

Don's formula is also an approximation based on EV and variance.
This is essentially equivalent to using a normal approximation in
place of the games true probability distribution. I seriously
question the accuracy of such approximations in the short term.

I've discussed the fact that I've used the Dunbar formula published in
1997 in lieu of the update in 1999, which I understand to take into
account the short to medium term skewed distribution of vp. In that
discussion I've noted that there's a basic argument which would cause
one to anticipate the alternate relationships between s.d. and
variance which should hold up under either formula. The 1997 formula
was used as a matter of expedience.

I disagree. I'd say that variance gives a "good approximation" but
is not a "direct measure". The true, exact ROR value (I'll call it
XROR to distinguish it from the other forms of ROR such as Don's
approximate formula,) computed from the characteristic equation, is
a direct measure. Anything else is not direct except perhaps for
special cases.

Again I'll have to own up to using a general description (direct
measure) when it's not entirely accurate. For example, in my own
review I found that there were modest variations in the ratio of
standard deviation and ROR. However, plotted on an X-Y axis the
resulting regression line would likely show a very high linear
correlation between the two variable -- strong enough to suggest that
one would give you a strong basis on which to form an expected
relationship between two games.

You can compares games on one basis or the other, but it is
important to understand that these are like apples and oranges. An
analogy is height and weight. They measure different things, and
greater height doesn't necessarily imply greater weight, even though
they often track closely.

In this case the varables I've correlated (e.g. s.d. and ROR) reflect
a very strong direct linear relationship. That would not be the case
beween the height and weight of various people, given a number of
other significant variables. However, even in that case, if you're
able to define the other variables well enough and look at cases in
which they're held constant that a strong direct relationship will
likely reveal itself.

Now, that limitation may appear to weaken the value of the
relationships I've drawn. However, the key alternative variable is
the ER at which each game is played. That's obviously apparent when
examing the variance of two games. However, an understanding of how
the variance effects the bankroll requirements of a game clearly is
beneficial. In addition, presumably using a two-step approach would
be of use in determining the absolute relationship between two games.
One would first make the assumption that variance were a constant and
determine effect on return and then adjust that effect by the relative
actual variance of two games.

Now, I'm simplifying that suggested means tremendously. Understand
that it's by way of demonstrating the mechanics at play here. The
point I really am attempting to drive home is a means by which the
player can apply to values of variance and standard deviation to
develop a reasonably concrete grasp of their impact on bankroll
requirements and ROR. In absence of that, these statistics are of
little use to the typical player. Interpreted with a understanding of
the limitations, these relative aspects of two games are revealed to a
modestly useful extent. The alternative is to say to the player that
they're totally meaningless without a complete grasp of the
relationships involved.

The problem with this is that ROR isn't a function of return. It is
easy to construct biased-coin games where game A has 10X EV of game
B, but game B has much lower risk of ruin.

Clearly, and as noted both variance and return on critical components
of the ROR of a game. The intent of my exercise has been noted above.

To throw another curve in here, consider the fact that for most VP
games, the playing strategy which is optimal from an XROR
perspective is significantly (read "measureably") different than the
playing strategy which maximizes EV. The goals "minimize ROR" and
"maximize EV" are simply different, and require different tactics.

Again, I'll concede this point. However, in seeking a general
practical use of game statistics the relationships lend considerable
insight, even if not precise.

Since variance/std_dev are approximate, I prefer working directly
with XROR.

There's no argument if one has access to the XROR of a game. The
purpose of my review was to find a means by which the recreational
player might find a valuable interpretation of s.d. and variance in
their play without simply shrugging their shoulders. It's clearly
inadequate for anyone seeking to strongly/precisely quantify aspects
of a game.

Ultimately, I expect from your observations that you'll express the
opinion that my observations aren't sufficiently solid to serve as
such guidance to even the recreational player. That would leave them
without any means by which to generally apply these statistics, which
would be unfortunate.

This is an area that I've been spending a lot of time thinking about
in recent months. I'd be very interested in discussing/exploring
this further. I think this area is poorly understood, even by
experts, and there are some surprising conclusions that pop out of
the math.

As noted, my purpose has been to serve the recreational player. I
clearly have an interest in the more precise quantitative aspects of
vp, but I admit to being inadequately prepared to explore them at any
length and willing sit on the sidelines.

My principal concern with this discussion is that in absence of any
clear consensus, the average player is left without any clear
indication of the reliability of what I've put forth.

Again, Steve, I appreciate the earnest manner in which you've explored
what I've posted. I'll admit to some disappointment that you've found
what I've posted largely wanting, but certainly I anticipated that
might well be the reception I received.

- Harry

[Steve_Jacobs]

In the real world a Martingale progression IS the road to ruin.

Granted, those who are attracted to using progressions are almost
invariably least able to understand their effects. But, my point was
that "experts" are making certain mathematical claims in regards to
progressions which are unfounded.

People bump into table limits after 5 or 6 losses which can occur
often even on 50/50 probabilities.

Surely you aren't going to use what CAN happen to justify anything.
With 50/50 probabilities, 5 losses occur with probability 1/32, and
6 losses occur with probability 1/64.

They then can't bet enough to get even on 1 win. Progressions
are dangerous to an extreme if not worthless in general.

Yes, yes, this is the standard party line, "progressions are EVIL and
FATTENING and cause HAIR LOSS in adult males." You tow the line
very well. I suspect you either didn't read my post carefully, or didn't
understand what I said. For CERTAIN problems, such as the specific
case that I tried to describe carefully, a progression can be the
OPTIMAL solution. Those cases involve neg-EV games where the
goal is to maximize the probability of hitting the target. In such a
situation, if you bet minimum wagers, you can certainly expect to play
for a longer time, but you dramatically INCREASE the probability
that you will EVENTUALLY go broke.

Extreme example: You have $255 left from the $10,000 that you took
to Podunk Nebraska. You need $256 to buy a ticket to fly home, you're
just one lousy dollar away from your goal. The only game available is a
dishonest coin-flip that loses 55% of the time and wins 45% of the time.
If you "flat bet" one dollar at a time, hoping to win quickly and go home,
then your probability of reaching $256 is about 82%. If you use the
OPTIMAL martingale instead, you only lose if you lose 8 bets in a row,
which has a probability of (0.55)^8 = 0.84%, so that your probability of
winning increases to 99.16%.

Of course, in those 18% of cases where the flat-bettor does lose,
they get to play an average of 2550 trials, while the progression
bettor faces at most 8 wagers. But, when you are stranded in
Podunk Nebraska, why would you want to take a LONG TIME
while REDUCING the probability of getting on the plane?

Mathematical fact: progressions can be useful, even beneficial,
depending on the task at hand. While often a recipe for disaster,
an honest disclosure of mathematical truths calls for this fact to
be acknowledged. Those who consistently/vehemonently paint
progressions as evil/misguided/wrong in all situations, are either
lacking in facts or intentionally misleading.

···

On Saturday 26 July 2003 11:49 am, Michael Boutot wrote:

[Terrence_Murphy]

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

In past months I've entered a couple of posts to the various vp

groups

concerning the risk associated with video poker and the statistics

that

are commonly presented in video poker software.

I'm going to discuss further evaluation of this............

- Harry

   I love it when you talk dirty like that Harry!

   But all kidding aside, that was a great thought provoking article.
It's going to take awhile for my little brain to absorb it all.

   "I always gamble with a positive attitude. I'm positive I'm going
to lose."
                   --VP Pappy

[VP1]

Podunk:
an actual village in MA (and CT, MI, NY, VT) that has become figurative for
some small unimportant backwater hick town. Other unusual American towns
names (many from Native American), some of which are used like Podunk,
include Boring, Drain, Bend, and even Wanker's Corner OR, Bat Cave NC,
Bugtussle (TX, OK) and Bug Tussle (KY, TN), Cool, Weed CA, Hoboken (AL, CA,
GA, NJ), Hohokus NJ, Kalamazoo (AR, FL, MI, WV), Kankatee, Keokuk (IA, TX),
Kokomo (AR, CO, HI, IN, MS, TX), One blink, two blink TX, Oshkosh (NE, WI),
Punxsutawney PA, Skaneateles NY

[Harry_Porter]

vppappy wrote:

   I love it when you talk dirty like that Harry!

hmmm ... sounds like you have some odd ideas about my bedroom banter.

- H.

[Harry_Porter]

VP wrote:

Podunk:
an actual village in MA (and CT, MI, NY, VT) that has become
figurative for some small unimportant backwater hick town.

Other podunk towns are even more figuratively named, particularly some
quaint towns in the Amish country outside of Philly: Bird-in-Hand,
Blue Ball, and Intercourse.

Of course, to finish that tour you have to jump over to my former turf
and visit Climax, MI (not too far outside of Kalamazoo, of course).

(Hey VP Pappy, there must be another choice 'ville or two in Mich.
that I've forgotten about in the 20+ yrs. I've been away, no? Those
UP'ers have some fairly whimsical notions as I recall.)

- Harry

[Lawrence_Boxer]

Harry,

I was thinking of Dan Paymer's formula as I read your analysis and
I'm glad you mentioned it. Thank you for an engrossing analysis. I
do have a thought or two.

As related functions, don't both variance and SD pertain to longterm
ER, while ROR limits itself to a function of a fixed amount? My
argument a few months ago was that ROR assumes an increase in time
also, as a function of increased bankroll, but that's off topic.

My main question is, just how valuable are either ER, variance, or to
some extent ROR, to a short term player, defined as one who plays
perhaps several one-hour sessions a day for a few days, three or four
times a year? That discription would probably include the majority of
this group.

We've discussed the huge role of variance in short term play and it
seems that there is "general" agreement that it is a major factor in
results. Why, then, should a short term player be very concerned
with variance as it compares to several different games, or SD as it
applies to his or her wide swings in a given game during short term
play?

This is in no way an argument against optimal play, just a question
about the value of ER, variance, and SD to any but the most longterm
players.

I happen to believe that ER is a result of variance (chaos + time x X
=order), but don't hold that against me if you reply to this. <g>

lb

[Michael_Boutot]

Yes I will use what can happen - that is the very heart of risk of
ruin calaculation. The worst case scenario happens. 1 in 32 is very
likely compared to the 1 in 40,000 we all pray for. There is no
fact "lacking" or "intentional misleading" involved.

> In the real world a Martingale progression IS the road to ruin.

Granted, those who are attracted to using progressions are almost
invariably least able to understand their effects. But, my point

was

that "experts" are making certain mathematical claims in regards to
progressions which are unfounded.

> People bump into table limits after 5 or 6 losses which can occur
> often even on 50/50 probabilities.

Surely you aren't going to use what CAN happen to justify anything.
With 50/50 probabilities, 5 losses occur with probability 1/32, and
6 losses occur with probability 1/64.

> They then can't bet enough to get even on 1 win. Progressions
> are dangerous to an extreme if not worthless in general.

Yes, yes, this is the standard party line, "progressions are EVIL

and

FATTENING and cause HAIR LOSS in adult males." You tow the line
very well. I suspect you either didn't read my post carefully, or

didn't

understand what I said. For CERTAIN problems, such as the specific
case that I tried to describe carefully, a progression can be the
OPTIMAL solution. Those cases involve neg-EV games where the
goal is to maximize the probability of hitting the target. In such

a

situation, if you bet minimum wagers, you can certainly expect to

play

for a longer time, but you dramatically INCREASE the probability
that you will EVENTUALLY go broke.

Extreme example: You have $255 left from the $10,000 that you took
to Podunk Nebraska. You need $256 to buy a ticket to fly home,

you're

just one lousy dollar away from your goal. The only game available

is a

dishonest coin-flip that loses 55% of the time and wins 45% of the

time.

If you "flat bet" one dollar at a time, hoping to win quickly and

go home,

then your probability of reaching $256 is about 82%. If you use the
OPTIMAL martingale instead, you only lose if you lose 8 bets in a

row,

which has a probability of (0.55)^8 = 0.84%, so that your

probability of

···

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

On Saturday 26 July 2003 11:49 am, Michael Boutot wrote:
winning increases to 99.16%.

Of course, in those 18% of cases where the flat-bettor does lose,
they get to play an average of 2550 trials, while the progression
bettor faces at most 8 wagers. But, when you are stranded in
Podunk Nebraska, why would you want to take a LONG TIME
while REDUCING the probability of getting on the plane?

Mathematical fact: progressions can be useful, even beneficial,
depending on the task at hand. While often a recipe for disaster,
an honest disclosure of mathematical truths calls for this fact to
be acknowledged. Those who consistently/vehemonently paint
progressions as evil/misguided/wrong in all situations, are either
lacking in facts or intentionally misleading.

[ckbrune]

I understand that minimizing ROR and maximizing EV may not result in
identical playing strategies. We're all familiar with maximizing EV
(ie, plug the numbers into WinPoker and see what comes out), but how
do you approach the problem of minimizing ROR? I don't think it's
the same thing as the Min Cost strategy, is it?

One way to decrease the ROR without sacrificing EV is the special
case when there are 2 or more ways to play a hand, all of which have
identical EV's. Take the hand AH KS QS JH 9S in FPDW as an example.
In these special cases, you would play to minimize the variance of
that hand (hold AKQJ or KQJ9 in this case and forget the KQ9 SF).

It seems to me that the "characteristic equation" can only be derived
and the ROR calculated after the entire game has been analyzed. I
suppose that you could make a strategy deviation from the max EV
strategy, get a new "characteristic equation" for the modified
strategy, and then calc the new ROR, which may turn out to be lower
than the ROR that you get by using the Max EV strategy if you're
lucky. But how will you ever know that ROR has been minimized?

Just curious.....thanks.
-Chris

> The most common statistic displayed is the variance of a game.

I've

> maintained that standard deviation, the square root of variance,

is a

> more accurate measure of a game's risk. However, at least one

source

> who I greatly respect has maintained that variance is the more

pertinent

> measure.

I fail to see how either can be "more pertinent" than the other.

They each

encompass the same information. Taking sqrt(V) to get S.D.

just "stretches"

the axes a bit, but in terms of accuracy they must inherently each

be just

as accurate as the other.

> In the wee hours of a sleepless night I've had a chance to

examine this

> question more closely. A key aid in my review has been a Risk of

Ruin

> Spreadsheet that was developed by Dunbar (first name a mystery).

The

> source of his work is a short-term risk of ruin formula detailed

in Don

> Schlesinger's BJ Attack (1997), p. 162. The spreadsheet allows

you to

> determine the ROR for any specified bankroll given the standard
> deviation and effective return of a game. While the formula was

devised

> for the specific purpose of evaluating blackjack, it is readily

put to

> use in looking at video poker.

Don's formula is also an approximation based on EV and variance.

This

is essentially equivalent to using a normal approximation in place

of the

games true probability distribution. I seriously question the

accuracy

of such approximations in the short term. They are fine for

looking at

long term results as the number of plays approach infinity.

The exact ROR for VP games can be computed in a more direct fashion,
using the "characteristic equation" which represents the game. I

don't

have time right now to go into detail (and I'll be at BARGE next

week,

and thus offline) but I feel this is a vastly superior approach for

answering

questions that involve risk of ruin. In addition, this approach

allows any

game to be modelled with a "risk-equivalent coin".

> My review revealed that both variance and standard deviation are
> relevant measures. If you're interested in the bankroll to

support play

> of a game, then variance is the direct measure that is

proportional to

> bankroll requirements.

I disagree. I'd say that variance gives a "good approximation" but

is

not a "direct measure". The true, exact ROR value (I'll call it

XROR to

distinguish it from the other forms of ROR such as Don's approximate
formula,) computed from the characteristic equation, is a direct

measure.

Anything else is not direct except perhaps for special cases.

> However, if you're starting from a fixed
> bankroll, standard deviation (the square root of variance) is the

value

> that will allow you to directly determine how the ROR of one game
> compares to another as a function of return. These comparisons

are most

> readily made when either std. deviation or return is a common

value.

> Variation of both variables presents a more complex exercise.

XROR and EV are somewhat independent measures. Higher EV doesn't
imply lower XROR, nor vice-versa. You can compares games on one
basis or the other, but it is important to understand that these

are like

apples and oranges. An analogy is height and weight. They measure
different things, and greater height doesn't necessarily imply

greater

weight, even though they often track closely.

> The explanation is fairly straightforward. As in finance, if you
> evaluate the risk associated with two alternate games the same

principal

> applies and standard deviation will give you a relative measure

of risk

> of ruin as a function of return.

The problem with this is that ROR isn't a function of return. It

is easy to

construct biased-coin games where game A has 10X EV of game B, but
game B has much lower risk of ruin.

> However, consider how the return of a
> game translates to potential swings in the results you might
> experience. Any variation in actual return of play from the

theoretical

> ER will produce a compounding effect with each hand played.

Therefore,

> the potential downside that you face grows proportionally with the
> number of hands played. This suggests that standard deviation, a
> measure that pertains to return, won't reflect that compounding.

On the

> other hand, variance, the square of standard deviation, will

measure the

> potential growth of the absolute shortfall in cash return over

time.

> Thus bankroll, the cushion against that shortfall, is more

directly

> reflected by the variance statistic.

To throw another curve in here, consider the fact that for most VP

games,

the playing strategy which is optimal from an XROR perspective is
significantly (read "measureably") different than the playing

strategy which

maximizes EV. The goals "minimize ROR" and "maximize EV" are simply
different, and require different tactics.

> However, if what you're looking for is how alternate games will

strain

> your existing bankroll (relatively speaking), then standard

deviation

> will give you that indication (although alternate game returns do
> complicate that calculation). The decision of what measure to

apply in

> looking at a game will depend on your intent.

Since variance/std_dev are approximate, I prefer working directly
with XROR. A big advantage of this approach is that it applies

directly

to problems of the form "given bankroll B and target T, what is the
probability of reaching target T before losing the entire

bankroll". The

XROR approach allows these problems to be framed in terms of a

repeated

coin flip using a risk-equivalent coin that "matches" the games

exact

probability distribution. This can give some interesting insights

in what

XROR "means" and the qualitative differences between XROR and EV.
The difference might be summarized by asking "do you want to reach
the target most quickly (max-EV) or do you want the highest

probability

of reaching the target before going broke (min-ROR)". The

difference

in these goals is subtle, and some may ask "what's the difference?"

> I hope this discussion clarifies more than it obscures. I'm

interested

> in any feedback, either in support of my assertions or in

argument, from

> any knowledgeable sources.

This is an area that I've been spending a lot of time thinking about
in recent months. I'd be very interested in discussing/exploring

this

further. I think this area is poorly understood, even by experts,

and

there are some surprising conclusions that pop out of the math. One
example of this: gambling experts will tell you that progressions
cannot alter EV. They go on to conclude "progressions are

worthless,"

but this is simply false. Progressions can have an impact on ROR.
In fact, when playing an unfavorable coin-flip game with the goal of
maximizing the probability of reaching a target bankroll, the

optimal

betting strategy takes the form of a Martingale progression. You

bet

just enough that winning reaches the goal. If you lose, you're now
twice as far from the goal and thus bet twice as much. Repeat until
you either reach the goal or lose your bankroll. This is the

optimal

strategy from a ROR perspective, and any betting strategy which
deviates from this will result in an increase in ROR.

So, the fact that progressions cannot alter EV does NOT imply that
progressions are worthless. I have yet to see a single "expert"

point

this out. In fact, experts are often smug about this to the point

that

they won't respond to any article that asks about progressions, and
they belittle those who ask. I think this ultimately stems from

the fact

···

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

On Saturday 26 July 2003 06:04 am, Harry D. Porter wrote:
that gambling experts are so focused on EV that the entire topic of
ROR has received much less study, and is poorly understood.

[Harry_Porter]

lawrenceboxer wrote:

As related functions, don't both variance and SD pertain to longterm
ER, while ROR limits itself to a function of a fixed amount? My
argument a few months ago was that ROR assumes an increase in time
also, as a function of increased bankroll, but that's off topic.

My main question is, just how valuable are either ER, variance, or
to some extent ROR, to a short term player, defined as one who plays
perhaps several one-hour sessions a day for a few days, three or
four times a year? That discription would probably include the
majority of this group.

We're talking about, say, 50 hours of play a year? You know, for such
a player I think even the concept of ER is of limited value. Luck in
any given year, or any number of successive years has just way too
strong of an effect for ER to be predictive of results with any
desirable confidence.

(Watch the group membership count drop by the minute on that one

So let me limit my discussion to a more active player, say an active
weekend player who logs about 400 hours a year or more.

Now, for such a player likelihood of play conforming to ER is
considerably higher. We know from experience (in my case, a little
too well) that play is still subject to considerable volatility. A
large factor in that volatility will be experience in hitting royals.
After all, we're talking about 2% of the return. However, even after
you back out the effect of royal experience on return, you're still
looking at considerable fluctuation in return.

Now, while admittedly SD and var. aren't going to be highly reliable
is explaining that volatility, I think most players recognize that
even for a single session the degree of variance of a game highly
correlates to just how rough a ride to expect. So I have no problem
with a statement that SD and var. have a strong role in the
description of the short term behavior of game play.

My posted discussion of my review of this subject was an attempt to
make an initial examination of how the variables of standard deviation
and variance relate to game behavior in even the short to medium term
from a practical sense (expressed in terms of ROR and bankroll
requirements). On it's own, it's not complete guidance, just an
exploration of the general nature of those relationships.

- Harry

[Steve_Jacobs]

I understand that minimizing ROR and maximizing EV may not result in
identical playing strategies. We're all familiar with maximizing EV
(ie, plug the numbers into WinPoker and see what comes out), but how
do you approach the problem of minimizing ROR? I don't think it's
the same thing as the Min Cost strategy, is it?

I believe that minimizing ROR is equivalent to one form of min-cost strategy,
but I haven't been able to prove this in games as general as VP. For a
coin-flip game, ROR and cost are the same number, so they are clearly
equivalent and have the same optimal strategy. For a game with a simple
N:1 payoff, such as betting a single number or group of numbers on a
roulette wheel, the equation for risk or ruin can be put into a form where
a monotonic function of ROR is on one side of the equation while the ratio
p(lose)/p(win) is on the other side of the equation. This implies that
reducing the p(lose)/p(win) will reduce ROR. Cost for this game is
(1/N)*p(lose)/p(win), so cost is proportional to the lose/win ratio which
"controls" risk of ruin. Minimizing the lose/win ratio is thus equivalent
to minimizing both cost and ROR.

I'm working to prove that min-cost and min-ROR are mathematically
equivalent in the general case, but I haven't got there yet.

One way to decrease the ROR without sacrificing EV is the special
case when there are 2 or more ways to play a hand, all of which have
identical EV's. Take the hand AH KS QS JH 9S in FPDW as an example.
In these special cases, you would play to minimize the variance of
that hand (hold AKQJ or KQJ9 in this case and forget the KQ9 SF).

The interplay between ROR and variance is interesting. In a +EV game,
reducing variance tends to reduce ROR. In a -EV game, reducing variance
tends to increase ROR. I used to describe this by saying "in -EV games,
variance is your friend," but now I believe that ROR is the real issue and
variance is a side effect.

It seems to me that the "characteristic equation" can only be derived
and the ROR calculated after the entire game has been analyzed.

That is true. To compute exact ROR, you need the complete probability
distribution for the game, and to compute that for VP you need to know
the strategy. The computation of exact ROR is itself an iterative process,
and I suspect this characteristic carries over to finding the min-ROR
strategy - you have to iterate in order to find the optimal strategy.

I
suppose that you could make a strategy deviation from the max EV
strategy, get a new "characteristic equation" for the modified
strategy, and then calc the new ROR, which may turn out to be lower
than the ROR that you get by using the Max EV strategy if you're
lucky. But how will you ever know that ROR has been minimized?

Well, if I can prove that min-ROR is equivalent to min-cost, then the
problem is solved because I know how to compute a min-cost strategy.
The method is roughly like this: pretend that all payoffs are scaled in
such a way that the max-EV strategy with scaled payoffs is exactly
breakeven. In other words, pick a scaling factor and compute a max-EV
strategy. If the optimal strategy gives a positive EV, the scaling factor
is too high, while a negative EV means the scaling factor is too low.
If you iterate until you find the scaling factor which makes the game
exactly breakeven, then that strategy miminizes cost.

···

On Saturday 26 July 2003 07:34 pm, ckbrune wrote:

[Lawrence_Boxer]

We're talking about, say, 50 hours of play a year? You know, for

such

a player I think even the concept of ER is of limited value. Luck

in

any given year, or any number of successive years has just way too
strong of an effect for ER to be predictive of results with any
desirable confidence.

I suppose I could have edged that play time up a bit, but even so I
think you are correct that there is little reason for most
recreational players to be overly concerned with ER or variance (not
to put works in your mouth. You've already got tonight's bedroom
banter to worry about <g>.

But as I said before, that's not an argument against optimal play.
Thanks again for the interesting analysis.

lb

[Harry_Porter]

lb -

I suppose I was a little loose in my statement. My intent was to
suggest that the concept of ER was of limited value to the more
limited player in predicting what actual results they'd realize in
their play. Luck plays an overwhelming role in the short term.

However, that by no means should suggest that the concept of ER is
irrelevent to such a player.

Understand that there are variables, such as variance, over which the
player has only limited control. There are other variables such as ER
which are firmly within the players control.

At question is the extent to which large uncontrollable variables such
as variance (to a considerable extent a "luck" equivalent) compete
with controllable variables (such as ER) in having an effect on play
outcome.

In the relatively short-term, few argue that luck is the predominant
factor.

Nonetheless, that's not to suggest that the role of ER is
insignificant. In fact, it's always a significant aspect of play,
it's just in the short term it's overwhelmed the the aspects of luck.

But the point is that were you to assume that two players experience
the same degree of good or poor luck, then with strong confidence the
player of the higher ER game will come out on top.

For this reason, it is a very important component of the play strategy
of even the limited player. They simply must realize that they have
nominal confidence that they'll realize results that conform to ER,
and the more volatile the game they play, the even lower that confidence.

It remains that for the very active player, ER is of stronger use as a
predictor of their results.

- Harry

···

--- In vpFREE@yahoogroups.com, "lawrenceboxer" <ljboxer@e...> wrote:

>
> We're talking about, say, 50 hours of play a year? You know, for
such
> a player I think even the concept of ER is of limited value. Luck
in
> any given year, or any number of successive years has just way too
> strong of an effect for ER to be predictive of results with any
> desirable confidence.
>

I suppose I could have edged that play time up a bit, but even so I
think you are correct that there is little reason for most
recreational players to be overly concerned with ER or variance (not
to put works in your mouth. You've already got tonight's bedroom
banter to worry about <g>.

But as I said before, that's not an argument against optimal play.
Thanks again for the interesting analysis.

lb

[Lawrence_Boxer]

Again, I think that we're pretty much in agreement. I think that
optimal play takes ER into consideration, since ER is based on long
term optimal play. If we assume that the goal is to extend play as
long as possible to take advantage of volatility's friendlier gifts,
then yes, ER is a consideration. To be more precise I should have
said that choosing a game based on it's theoretical ER is not of
prime importance to an infrequent player, due to variance playing
such a big role in outcome.

lb

···

I suppose I was a little loose in my statement. My intent was to
suggest that the concept of ER was of limited value to the more
limited player in predicting what actual results they'd realize in
their play. Luck plays an overwhelming role in the short term.

However, that by no means should suggest that the concept of ER is
irrelevent to such a player......

[Pat_O_Connell]

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

VP wrote:
> Podunk:
> an actual village in MA (and CT, MI, NY, VT) that has become
> figurative for some small unimportant backwater hick town.

Other podunk towns are even more figuratively named, particularly

some

quaint towns in the Amish country outside of Philly: Bird-in-Hand,
Blue Ball, and Intercourse.

Of course, to finish that tour you have to jump over to my former

turf

and visit Climax, MI (not too far outside of Kalamazoo, of course).

(Hey VP Pappy, there must be another choice 'ville or two in Mich.
that I've forgotten about in the 20+ yrs. I've been away, no? Those
UP'ers have some fairly whimsical notions as I recall.)

Don't know the UP very well, but I do know Indiana:
Bean Blossom, Gnaw Bone, Stone Head, Carp, and Pumpkin Center (two of
them).

[AJ1]

Are you saying that scaling ALL the payoffs by the same scaling
factor will change the max-EV strategy?

Thanks,

AJ

Stevew Jacobs said:
...

Well, if I can prove that min-ROR is equivalent to min-cost, then

the

problem is solved because I know how to compute a min-cost strategy.
The method is roughly like this: pretend that all payoffs are

scaled in

such a way that the max-EV strategy with scaled payoffs is exactly
breakeven. In other words, pick a scaling factor and compute a max-

EV

strategy. If the optimal strategy gives a positive EV, the scaling

factor

is too high, while a negative EV means the scaling factor is too

low.

If you iterate until you find the scaling factor which makes the

game

···

exactly breakeven, then that strategy miminizes cost.

[Steve_Jacobs]

Yes, it will, provided that you do NOT scale losses equally.

This implies that cashback can cause the optimal strategy to change.

···

On Saturday 26 July 2003 11:04 pm, AJ wrote:

Are you saying that scaling ALL the payoffs by the same scaling
factor will change the max-EV strategy?