Better measure for variance – how to calculate bankroll

[blaw57]

Two related questions here – first, what's the formula to calculate
risk of ruin (and thus bankroll) for a given game and desired risk
level? I presume it's based on a game's ER, variance and acceptable
risk level?

Second question is more involved and more philosophical – is there a
better way to measure a game's riskiness than Variance?

Here are the problems I have with variance:

1. Highly degree of dependence on a single high term. For example,
take the relatively low-variance 9/6 jacks-or-better game (19.50 for
the basic game) and add a high (say 200% of reset) progressive
jackpot. The result is a variance number of 67.00 for the game, which
is higher than most of the high variance basic games (Triple Bonus,
Super Aces, etc). Yet it still plays like normal JOB as long as you
don't hit the royal, very small swings, little threat of massive loss.

2. Game return irrelevant. The variance calculation doesn't factor in
game return. Thus, 9/6 JOB with 0.5% cashback has a variance of 19.5
and a return of 100%, while 6/5 JOB at the same casino has a variance
of 19.0 and return of 95%. Which is more of a risk to your bankroll?
And while the answer to that question is obvious on its face, it
might not be so obvious comparing Super Double Bonus (var 38, er
100.10) to Super Aces (var 63, er 100.35).

By the way, I'm not doubting the correctness of the Variance
calculation or the truth of what it is saying. Your final return
playing 200% progressive JOB truly will be as various, as defined by
that ratio, as SAB. If you were to run a trial of 10,000 sessions of
10,000 hands of each game, the final results would calculate out to
similar standard deviations. However, I believe the clustering
patterns would be different – SAB would show many more big losers and
more big winners, while 200% JOB would show a lot of small losers and
a handful of extremely big winners. Just because they average out to
the same standard deviation does not mean they are equally risky.
Furthermore, the JOB group will total a much larger net win due to
the ER advantage.

So, what's a formula that shows a satisfactory result in this
intuitive case, that can be applied to other cases that are not so
intuitive?

[Harry_Porter]

blaw57 wrote:

What's the formula to calculate risk of ruin (and thus bankroll) for
a given game and desired risk level? I presume it's based on a
game's ER, variance and acceptable risk level?

I'll address conceptual issues here. I'll let someone else provide
the actual formula.

Risk of Ruin is a function of bankroll, game ER, and variance.
Bankroll is a function of desired ROR, game ER, and variance.

Is there a better way to measure a game's riskiness than Variance?

Here are the problems I have with variance:

1. Highly degree of dependence on a single high term. For example,
take the relatively low-variance 9/6 jacks-or-better game (19.50 for
the basic game) and add a high (say 200% of reset) progressive
jackpot. The result is a variance number of 67.00 for the game,
which is higher than most of the high variance basic games (Triple
Bonus, Super Aces, etc). Yet it still plays like normal JOB as long
as you don't hit the royal, very small swings, little threat of
massive loss.

There's a better, more accurate way to think of variance: It's the
risk that your actual play results will vary from the ER of a game.

It makes sense that 9/6 JB with a high progressive meter should have a
high variance. With a large amount of return dependent upon hitting a
RF, failure to achieve the statistically expected number of RF's in
your play will cause return to fall short of ER by a margin far
greater than in a 4000 cr. RF game. Likewise, hit more RF's and your
return will greatly exceed game ER in the case of a progressive. A
high variance is strictly representative of this simple fact.

2. Game return irrelevant. The variance calculation doesn't factor
in game return.

That's right. Variance isn't directly dependent upon game return
(although it does measure the risk of variance from game return). But
game risk, as noted, isn't solely a function of variance. In
determining ROR, game return isn't irrelevant.

If you were to run a trial of 10,000 sessions of 10,000 hands of
each game, the final results would calculate out to similar standard
deviations. However, I believe the clustering patterns would be
different – SAB would show many more big losers and more big
winners, while 200% JOB would show a lot of small losers and
a handful of extremely big winners. Just because they average out to
the same standard deviation does not mean they are equally risky.

Granted, the clustering of results between these two games would be
markedly different over the course of 10,000 hands. Variance,
however, is a long-term measurement -- well in excess of 1 million
hands.

The behavior of any game varies from another over the shorter term.
That's why, even when you have two games of similar variances, the
amount you need to walk into the casino to ensure a desired amount of
play can greatly differ.

A prime example is 9/6 JB and pick'em. Pick'em has a far smaller
variance, however, for any given casino trip you should allow roughly
double the cash for pick'em than JB to have a decent shot of lasting
the same number of hands for each game.

So, what's a formula that shows a satisfactory result in this
intuitive case, that can be applied to other cases that are not so
intuitive?

I hope I've provided additional food for thought that will reshape
your intuition in this case. Variance is a strong predictor of game risk.

However, because it's a long-term measurement, it needs to be treated
largely as a benchmark when your play of a given game is likely to
fall considerably shorter -- and, as discussed, it should be
remembered that in the very short run it's a rough predictor at best.

- Harry

[Steve_Jacobs]

Two related questions here � first, what's the formula to calculate
risk of ruin (and thus bankroll) for a given game and desired risk
level? I presume it's based on a game's ER, variance and acceptable
risk level?

Any kind of "measure" of a game, whether ER or variance or RoR or
the probability of having a bankroll survive until hitting a royal flush,
is computed from the probability distribution. Technically, these measures
aren't properties of the game itself, but depend both on the payoff schedule
and the playing strategy. Different playing strategies are required to
maximize (or minimize) different properties.

So, RoR isn't based on ER or variance, but is an independent measure.

Second question is more involved and more philosophical � is there a
better way to measure a game's riskiness than Variance?

Yes. In fact, variance is a crude measure of risk. But, riskiness itself
is kind of a nebulous concept. Risk of ruin is really a measure of the
probability that you _won't_ play indefinitely without losing your entire
bankroll. RoR is related to an entire family of risk measures that all
involve a monetary goal. If you want to build a bankroll up to 10,000
units or go bust trying, then RoR is the thing you want to measure and
the thing you want to tune your playing strategy to optimize. But, if
you really want to just have a bankroll last as long as possible to give
you the best shot at surviving until you hit a royal, then you need a
slightly different measure and a slightly different strategy.

Here are the problems I have with variance:

1. Highly degree of dependence on a single high term. For example,
take the relatively low-variance 9/6 jacks-or-better game (19.50 for
the basic game) and add a high (say 200% of reset) progressive
jackpot. The result is a variance number of 67.00 for the game, which
is higher than most of the high variance basic games (Triple Bonus,
Super Aces, etc). Yet it still plays like normal JOB as long as you
don't hit the royal, very small swings, little threat of massive loss.

Agreed. Variance doesn't really describe risk so much as it describes
volatility.

2. Game return irrelevant. The variance calculation doesn't factor in
game return. Thus, 9/6 JOB with 0.5% cashback has a variance of 19.5
and a return of 100%, while 6/5 JOB at the same casino has a variance
of 19.0 and return of 95%. Which is more of a risk to your bankroll?
And while the answer to that question is obvious on its face, it
might not be so obvious comparing Super Double Bonus (var 38, er
100.10) to Super Aces (var 63, er 100.35).

Good observation. I personally have found that I no longer have much
use for variance. There are a lot of formulae out there that have been
developed over the years to relate RoR to some combination of ER and
variance, but they are all approximations. The exact computation of
RoR for a given strategy doesn't use (or need) variance.

···

On Thursday 24 June 2004 05:24 am, blaw57 wrote:

[nightoftheiguana2000]

Two related questions here – first, what's the formula to

calculate

risk of ruin (and thus bankroll) for a given game and desired risk
level? I presume it's based on a game's ER, variance and acceptable
risk level?

Bankroll (11%ror)=VAR/(ER-1) bets
double it for 1% ror (ror=risk of ruin)

Second question is more involved and more philosophical – is

there
a

better way to measure a game's riskiness than Variance?

variance is variance, it's a mathematical term, it's not the same
as "riskiness" whatever that may be

···

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

[fordscks_jason_c_vp]

"Here are the problems I have with variance:
1. Highly degree of dependence on a single high term. For example,
take the relatively low-variance 9/6 jacks-or-better game (19.50 for
the basic game) and add a high (say 200% of reset) progressive
jackpot. The result is a variance number of 67.00 for the game, which
is higher than most of the high variance basic games (Triple Bonus,
Super Aces, etc). Yet it still plays like normal JOB as long as you
don't hit the royal, very small swings, little threat of massive
loss."

Steve Jacobs wrote:
"Agreed. Variance doesn't really describe risk so much as it
describes volatility."

blaw57 wrote:
"2. Game return irrelevant. The variance calculation doesn't factor
in game return. Thus, 9/6 JOB with 0.5% cashback has a variance of
19.5 and a return of 100%, while 6/5 JOB at the same casino has a
variance of 19.0 and return of 95%. Which is more of a risk to your
bankroll? And while the answer to that question is obvious on its
face, it might not be so obvious comparing Super Double Bonus (var
38, er 100.10) to Super Aces (var 63, er 100.35)."

Steve Jacobs wrote:
"Good observation. I personally have found that I no longer have
much use for variance. There are a lot of formulae out there that
have been developed over the years to relate RoR to some combination
of ER and variance, but they are all approximations. The exact
computation of RoR for a given strategy doesn't use (or need)
variance."

I want to add my two cents, TomSki developed his TSI Index
calculator to address the difficulties of comparing (1) different
games at (2) different denominations with (3) different bankrolls in
(4) single vs multiple line games with (5) different cash back or
other ev enhancements (such as comps or promotions, etc) and (6)
different game speed (i.e. hph) before the advent of TITO. This
Index is TomSki's independently derived version. [Disclaimer: TomSki
and I equally share rights to the first generation of the index.]
TomSki, against my wishes, and in his typical goodhearted nature,
allowed this program to be download freely [the TSI index, as I
understood it, was originally to be bundled with his VPSM program to
form a vp killer app]. The software and can be found on www.vid-
poker.com and possibly on other sites. If anyone has time, you
should read what Skip wrote about the TSI Index, and the warning
about high variance games (i.e. from progressives) on small bankroll.

I don't think TomSki was satisfied with his work, but it clearly
shows his attempt to create an index on vp game selection based on
ev, variance, bankroll, etc. If TomSki were to work on TSI Index
today, he probably would include the concept of N(0), as well as
some of his favorite topics such as confidence intervals and trip
simulations. What was missing in TSI, IMHO, was the the effect of
strategy alterations on the index, i.e. altering strategy to change
the ev and variance such that the delta of the ev in relation to the
delta of the variance hit the player's desired threshold
[this "mapping" can be done with fortan akin to solving an equation
with two unknowns]. Keep this in mind, that we were talking about
1999 before the advent of Frugal VP. As good as TomSki was in vp
theory, he (nor I) is no substitute for a guy like Steve Jacobs in
terms of math prowess.

To this day, I still marvel at TomSki's TSI Index and VPSM; these
are truly ground breaking pieces of work in the field of VP.

···

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

On Thursday 24 June 2004 05:24 am, blaw57 wrote:

[fordscks_jason_c_vp]

"2. Game return irrelevant. The variance calculation doesn't factor
in game return. Thus, 9/6 JOB with 0.5% cashback has a variance of
19.5 and a return of 100%, while 6/5 JOB at the same casino has a
variance of 19.0 and return of 95%. Which is more of a risk to your
bankroll? And while the answer to that question is obvious on its
face, it might not be so obvious comparing Super Double Bonus (var
38, er 100.10) to Super Aces (var 63, er 100.35)."

Steve Jacobs wrote:
"Good observation. I personally have found that I no longer have
much use for variance. There are a lot of formulae out there that
have been developed over the years to relate RoR to some combination
of ER and variance, but they are all approximations. The exact
computation of RoR for a given strategy doesn't use (or need)
variance."

In a separate reply I talked about the TomSki's TSI Index. I forgot
to mention how versatile the TSI index is besides game selection. I
recall TomSki telling me Liam Daily was playing with an earlier
version of TSI index commented it gave the wrong answer when
comparing a particular hand. I was thinking what?? And then it
dawned on me what Liam was trying to do.

Here's the question and how TomSki's TSI Index answer the question.
Q: Supposed you are playing 3-play 9/6 JOB on $1 and are dealt a pat
flush with four cards to a royal flush. What is the correct play?

Under 9/6 JOB basic strategy or under a Max EV strategy, the correct
play was to break the pat flush and go for the royal flush, i.e. the
classical example of throwing away a guaranteed winner for a bigger
jackpot.

I told TomSki that can't be the correct answer because the correct
answer is dependent on her goal(s) and constraints, with her
bankroll being a constraint. TomSki understood this and ran the
particular hand in TSI and found that unless her bankroll was $XXX,
she should kept the pat flush rather than go for the royal flush.
That means, if the game was 10-play or 20-play instead of 3-play, it
may result in different answers. This is important because it
points out as your bankroll changes, the decision rule for a
particular hand *may* change.

I will continue to argue that how players break-up pat hands is
strategy dependent, and therefore should be dependent on one's goal
(s) and constraint(s). The easiest case to see is PE, keeping a high
pair over any 3-card RF; in the case of 10/7 DB, it's keeping 3 aces
in a full house versus going for quad aces, (I forgot the deuces
wild example), etc.

I don't believe the power of TomSki's TSI index has been fully
tested.

···

--- In vpFREE@yahoogroups.com, Steve Jacobs <jacobs@x> wrote:
On Thursday 24 June 2004 05:24 am, blaw57 wrote:

[blaw57]

Thanks for the comments. I'll track down that TomSki download and
check it out. If anybody could mention the names or sources or the
other measures you referred to, I'd appreciate it, and I'll try to
track those down too.

I did run my simulation, 10,000 hands of each, and I plotted the
final outcomes. I wish I could post the graph here, it really
demonstrates the point I was trying to make about the different
character of the outcomes. IMHO, the JOB outcomes definitely present
a more favorable picture overall – more potential return for less
risk.

Several people made the point, and I agree, that riskiness is a
nebulous concept. I'll give some more thought to exactly how I'm
defining "risk" – right now, the best I can do is to say that if I'm
playing wuarters, I don't want to lose anywhere close to $1000 in a
long 10,000 hand session, but I want to have a shot at a big win once
every 5 trips or so, and to have the wins overall total more than the
losses.

Anyhow, here are the results of the simulation. To even things out, I
made believe that the casino I was visiting had a suicidal SAB promo
going on, where they paid you 1.6% cashback, versus zero on JOB.
Also, on the JOB game with the 200% royal I stuck to normal 9/6
strategy, so both games have an ER of 101.54%. After running through
100 million hands of each, it turned out that they hadn't fully
normalized yet – the JOB ran 0.06% over ER, and SAB was 0.10% under.
So just to equalize things even more, I went back and retroactively
gave an additional 0.16% cashback to the SAB hands, so both games had
a realized return of 101.60% over 10,000 sessions of 10,000 hands
each.

Here are the results.

Max win – SAB $4185 JOB $5790
Max loss – SAB $2405 JOB $1167

% Losses over $1000 – SAB 9.2% JOB 0.1%
% Losses over $750 – SAB 16.1% JOB 2.0%

% Wins over $1250 – SAB 13.8% JOB 21.3%
% Wins over $2500 – SAB 1.8% JOB 3.3%

Max royals – SAB 3 JOB 3
Max quad aces/total 4K – SAB 9/45 JOB na/47

Now, everything so far makes JOB look better. But with identical
returns and similar variances, obviously things have to even out
somewhere. That somewhere, for this comparison, was in the middle –
SAB players won more, and won small amounts more.

% who made money on the session – SAB 55.0% JOB 29.1%
% who made $0-1250 – SAB 41.2% JOB 7.9%

(Note, this is considerably skewed by my fake 1.76% cashback on SAB
versus zero on JOB. Without this, only 46% of SAB players made money.
In other words, 9% of SAB sessions would lose from $1 to $175, but in
this example they were covered by my cashback assumption to even the
overall yields. Still, 45% versus 29% is a big difference, which just
goes to reinforce the idea that JOB without a royal is a slow bleed.)

In other words, to put it words, most (54%) JOB progressive players
lost $0-500. Very few lost more than $750, and very few made less
than $750. A minority of players won any money, but of those who did,
a significant part won very large amounts.

SAB players were more evenly spread across the graph, with 57%
falling between winning $750 and losing $750. Those on the wings
tended to lose more than JOB progressive players but win less.

[vpFREE4]

The TomSki Index is included in the "Bankroll Links" Directory
of vpFREE Links:

http://members.cox.net/vpfree/Bank.htm

<a href="http://members.cox.net/vpfree/Bank.htm">
http://members.cox.net/vpfree/Bank.htm</a>

···

On 25 Jun 2004 at 1:39, blaw57 wrote:

Thanks for the comments. I'll track down that TomSki
download and check it out.

[nightoftheiguana2000]

if you're trying to compare games to see which is better i recommend
this formula:
(ER-1)^2 x 10^7/VAR
it's the bankroll growth index
some numbers:
FPDW+.25%cb 39
PE+.75%cb 32
FPDW 22
9/6DDB(rf=1700) 22
AA 18
8/5Bonus(rf=1500) 18
9/6JOB(rf=1300) 17
NSUD(rf=1300) 17
10/7DB(rf=1100) 17
KOBJoker 15
15/10LooseDeuces 12
10/7/80DB 9
8/5SAB+.5%cb 3
10/7DB 1
10/6DDB .1

the scale is linear so this means, for example, that playing fpdw+.25%
cb, your bankroll would grow 390 times faster than it would playing
10/6DDB for the same risk of ruin

positive gambling is still gambling and always involves some risk of
ruin, which is why negative gambling is a fool's folly

[fordscks_jason_c_vp]

--- In vpFREE@yahoogroups.com, "nightoftheiguana2000"
<nightoftheiguana2000@y...> wrote:

if you're trying to compare games to see which is better i

recommend

this formula:
(ER-1)^2 x 10^7/VAR
it's the bankroll growth index
some numbers:
FPDW+.25%cb 39
PE+.75%cb 32
FPDW 22
9/6DDB(rf=1700) 22
AA 18
8/5Bonus(rf=1500) 18
9/6JOB(rf=1300) 17
NSUD(rf=1300) 17
10/7DB(rf=1100) 17
KOBJoker 15
15/10LooseDeuces 12
10/7/80DB 9
8/5SAB+.5%cb 3
10/7DB 1
10/6DDB .1

"the scale is linear so this means, for example, that playing
fpdw+.25% cb, your bankroll would grow 390 times faster than it
would playing 10/6DDB for the same risk of ruin"

First, can you explain how that formula is derived and who derived
it. Second, I have problems with the formula in that I am trying to
determine if it's better to keep a pat hand or go for bigger hit, as
in being dealt a pat flush with a 4-card RF draw in triple play in
9/6 JOB. Well, according to your formula, you shouldn't break the
pat flush because the variance is so low for a pat winning hand.
Maybe you can do the math for this hand, and show us the results.

There's conceptual problems with your formula since there is such a
strong bias toward low variance games (as in 10^7/variance). I am
almost willing to bet given the identical game, with the only
difference being the strategy used (along with resulting different
return and variance), AND assuming both games have over 100% return,
your index will throw-off strange answers. Thanks.

[nightoftheiguana2000]

First, can you explain how that formula is derived and who derived
it.

I did it, using math.

Second, I have problems with the formula in that I am trying to
determine if it's better to keep a pat hand or go for bigger hit,

as

in being dealt a pat flush with a 4-card RF draw in triple play in
9/6 JOB. Well, according to your formula, you shouldn't break the
pat flush because the variance is so low for a pat winning hand.
Maybe you can do the math for this hand, and show us the results.

Strategy is a different issue. The strategy used determines the er
and variance, once you have those you can plug them into the
formulas. By default I assume max er strategy, but there are others.
FrugalVP will give you the er and variance for your own defined
strategy, which is a very useful feature, otherwise you can do your
own math but it can become tedious. There is a sorokin strategy which
has the lowest bankroll requirement because of a reduced variance
which comes with the cost of a reduced er. There is an optimum
bankroll growth strategy which is roughly halfway between the sorokin
and max er strategy.

There's conceptual problems with your formula since there is such a
strong bias toward low variance games (as in 10^7/variance). I am
almost willing to bet given the identical game, with the only
difference being the strategy used (along with resulting different
return and variance), AND assuming both games have over 100%

return,

your index will throw-off strange answers. Thanks.

Nope. The math is correct.

···

--- In vpFREE@yahoogroups.com, "fordscks" <jason_c_vp@y...> wrote:

[blaw57]

--- In vpFREE@yahoogroups.com, "nightoftheiguana2000"
<nightoftheiguana2000@y...> wrote:
> if you're trying to compare games to see which is better i
recommend
> this formula:
> (ER-1)^2 x 10^7/VAR
> it's the bankroll growth index
> some numbers:
> FPDW+.25%cb 39
> PE+.75%cb 32
> FPDW 22
> 9/6DDB(rf=1700) 22
> AA 18
> 8/5Bonus(rf=1500) 18
> 9/6JOB(rf=1300) 17
> NSUD(rf=1300) 17
> 10/7DB(rf=1100) 17
> KOBJoker 15
> 15/10LooseDeuces 12
> 10/7/80DB 9
> 8/5SAB+.5%cb 3
> 10/7DB 1
> 10/6DDB .1

"the scale is linear so this means, for example, that playing
fpdw+.25% cb, your bankroll would grow 390 times faster than it
would playing 10/6DDB for the same risk of ruin"

First, can you explain how that formula is derived and who derived
it. Second, I have problems with the formula in that I am trying to
determine if it's better to keep a pat hand or go for bigger hit,

as

in being dealt a pat flush with a 4-card RF draw in triple play in
9/6 JOB. Well, according to your formula, you shouldn't break the
pat flush because the variance is so low for a pat winning hand.
Maybe you can do the math for this hand, and show us the results.

There's conceptual problems with your formula since there is such a
strong bias toward low variance games (as in 10^7/variance). I am
almost willing to bet given the identical game, with the only
difference being the strategy used (along with resulting different
return and variance), AND assuming both games have over 100%

return,

your index will throw-off strange answers. Thanks.

I dunno, I think the formula has some merit, although it doesn't
overcome the problems I see in using the variance number. It appears
to basically be a ratio of profit to variance, with the profit number
emphasized by squaring. You can ignore the 10^7 factor conceptually,
that merely gives you a short number like 39 instead of a long
decimal like .00000039 to look at. Thus, the basic formula squares
the expected profit (which has the effect of increasing relative
discrepancies - while the ratio of 3% expected profit to 1% is 3:1,
the ratio of the squares is 9:1) and then divides by the variance. In
other words, it's a ratio of emphasized profit to variance. That
sounds perfectly valid to me, although I'd like to study the results
of some samples compared to other measures and see if it all appears
to make sense.

Hey Iguana/movie-star (hey everybody, next time you see an elderly
lady playing VP, and there are lots of them in Tunica, be nice to
her, it might be Elizabeth Taylor or some other former-sex-idol-movie-
star past her sexy days), could you fill in the ERs and Variances on
your table? I started doing it but it's time consuming running the
games through the analyzer, and you must already have the numbers
since you had to have them to calculate the ratios. I'd like to see
which cases there are that rank a lower ER game above a higher one,
to see if that makes sense intuitively.

I've been thinking more about the comments people made, and
researching the various ratios mentioned, and I still come back to
the idea that variance isn't the best base to use to measure risk.
Yeah, people say it's not a risk measurement, and risk is a nebulous
concept, but all the formulas I've looked at use it as a risk proxy.
The TomSki Index, for example, has as its only significant inputs
game return, variance and bankroll. He is clearly using variance as a
risk proxy.

And, the more I think about it, risk as used here is NOT a nebulous
concept. Risk is, quite simply, the chance that I will lose more
money than I was prepared to lose. No more, no less. That means I
want to know my chances of losing various amounts of money and,
since "prepared to lose" involves some hope of gain to justify any
loss, I also want to know my chances of making various amounts of
money.

So, I'm coming to think that no single ratio can be anywhere near as
helpful as a risk measure as would be a short summary of a single-
session-length game simulation. You'd need to have 6 or 8 standard
reference points that you use for all the sims (results of 99th
percentile player on both ends, % win and lose, and % who make or
lose more than certain threshold levels like 500/1000 betting units).
Then run sims on 20-30 different games (and repeat several times to
be sure the answers are reasonable) and report the results in a
consistent and comparable manner. I think that would be some useful
information to help players choose which game to play. Anybody
interesting in helping to compile something like that?

BTW, for anybody who owns Excel 2000 or later, I've got a VBA-macro-
based simulation that fills out a worksheet with game results that
I'd be happy to share. It's very simple and flexible. This is a good
way to do sims because you have the detail in a worksheet where you
can make adjustments after the fact, like the cashback I added to my
SAB-JOB comparison to make the realized returns equal, or set your
groupings in different ways or run SDs or just study individual
results. To use this model you have to input the paytable and hand
probability table, but that's easy to get from
http://www.gamblingtools.net/vp/vpanalyzer.html or similar places.
(My model does, however, use the much-maligned microsoft not-so-
random number generator, but I'd argue that's not a problem in this
case, as long as you do certain things to avoid known problems. It's
also just a 7-digit random number, which skews results slightly for
low probablilty hands like royals, but it's relatively easy to
measure the skew and compensate for it.) Probably everybody here
who's interested in that kind of thing has written their own, but if
not I'd be glad to email you a copy of this one.

···

--- In vpFREE@yahoogroups.com, "fordscks" <jason_c_vp@y...> wrote:

[nightoftheiguana2000]

Hey Iguana/movie-star (hey everybody, next time you see an elderly
lady playing VP, and there are lots of them in Tunica, be nice to
her, it might be Elizabeth Taylor or some other former-sex-idol-

movie-

star past her sexy days), could you fill in the ERs and Variances

on

your table? I started doing it but it's time consuming running the
games through the analyzer, and you must already have the numbers
since you had to have them to calculate the ratios. I'd like to see
which cases there are that rank a lower ER game above a higher one,
to see if that makes sense intuitively.

1.01%var26 FPDW+.25%cb 39
0.70%var15 PE+.75%cb 32
0.76%var26 FPDW 22
1.60%var116 9/6DDB(rf=1700) 22
0.72%var27 AA 18
1.10%var73 8/5Bonus(rf=1500) 18
0.95%var55 9/6JOB(rf=1300) 17
0.95%var55 NSUD(rf=1300) 17
0.92%var48 10/7DB(rf=1100) 17
0.65%var26 KOBJoker 15
0.97%var70 15/10LooseDeuces 12
0.52%var29 10/7/80DB 9
0.44%var63 8/5SAB+.5%cb 3
0.17%var28 10/7DB 1
0.07%var42 10/6DDB .1

···

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

[fordscks_jason_c_vp]

--- In vpFREE@yahoogroups.com, "nightoftheiguana2000"
<nightoftheiguana2000@y...> wrote:
"I did it, using math."

I'm re-reading Michael Canjar's work on "Gambler's Ruin Revisited:
The Effect of Skew and Large Jackpot," and I'm having a difficult
time understanding why you came up with factor: 10^(7/variance);
mainly the exponent part of your formula. I'm not disputing your
math, just trying to understand why the constant is "7/variance."

"Strategy is a different issue. The strategy used determines the er
and variance, once you have those you can plug them into the
formulas. By default I assume max er strategy, but there are others.
FrugalVP will give you the er and variance for your own defined
strategy, which is a very useful feature, otherwise you can do your
own math but it can become tedious. There is a sorokin strategy
which has the lowest bankroll requirement because of a reduced
variance which comes with the cost of a reduced er. There is an
optimum bankroll growth strategy which is roughly halfway between
the sorokin and max er strategy."

So, as I understand it, the max ev strategy will dominate the other
strategies when using your formula. While I never seen the sorokin
strategy, which is sometime I have pursued separately and in a
primative way -- can you point to literature on the sorokin
strategy. The only person that has written about this was Steve
Jacobs.

[nightoftheiguana2000]

10^7 is just a scaling factor to make the result easier to use:

(((ER-1)^2) / VAR) x 10^7

bankroll growth = (ER-1)/bankroll
but bankroll = VAR/(ER-1)
therefore bankroll growth = (ER-1)/VAR/(ER-1) = (ER-1)^2 /VAR

···

--- In vpFREE@yahoogroups.com, "fordscks" <jason_c_vp@y...> wrote:

I'm re-reading Michael Canjar's work on "Gambler's Ruin Revisited:
The Effect of Skew and Large Jackpot," and I'm having a difficult
time understanding why you came up with factor: 10^(7/variance);
mainly the exponent part of your formula. I'm not disputing your
math, just trying to understand why the constant is "7/variance."

[Steve_Jacobs]

--- In vpFREE@yahoogroups.com, "nightoftheiguana2000"
<nightoftheiguana2000@y...> wrote:
"I did it, using math."

I'm re-reading Michael Canjar's work on "Gambler's Ruin Revisited:
The Effect of Skew and Large Jackpot," and I'm having a difficult
time understanding why you came up with factor: 10^(7/variance);
mainly the exponent part of your formula. I'm not disputing your
math, just trying to understand why the constant is "7/variance."

The factor isn't 10^(7/variance), it is (10^7)/variance.

"Strategy is a different issue. The strategy used determines the er
and variance, once you have those you can plug them into the
formulas. By default I assume max er strategy, but there are others.
FrugalVP will give you the er and variance for your own defined
strategy, which is a very useful feature, otherwise you can do your
own math but it can become tedious. There is a sorokin strategy
which has the lowest bankroll requirement because of a reduced
variance which comes with the cost of a reduced er. There is an
optimum bankroll growth strategy which is roughly halfway between
the sorokin and max er strategy."

So, as I understand it, the max ev strategy will dominate the other
strategies when using your formula.

Nope. These formulas are approximations used for Kelly betting.
They are designed to maximize geometric growth rate (or,
equivalently, expected value of log(bankroll)).

Just as it is possible to compute RoR without using variance, the
log-optimal playing strategy and resulting growth rate can be
computed exactly without resorting to formulae which use
variance (or ER for that matter). As nightoftheiguana2000 says,
the Kelly strategy is roughly halfway between the max-er
strategy and the min-risk strategy.

The log-optimal strategy is most applicable to games where the
player has the luxury of choosing a bet size that is a precise
fraction of bankroll. For games like VP, where the allowed
betting denominations are spaced far apart, perfect log-optimal
play would require changing strategy as bankroll changes.
One extreme case is when you have a huge bankroll but
the largest denomination of machine allows you to bet only
a tiny fraction of the Kelly optimal wager. In this case, the
log-optimal strategy become identical with max-er strategy.
At the other extreme, your bankroll is so small that the smallest
denomination of machine still forces you to bet more than
the Kelly optimal bet fraction. When the bankroll shrinks to
the point where each play causes you to bet about 2X the
optimal bet, the log optimal playing strategy becomes
identical with min-risk strategy.

So, when bet size can't be adjusted to arbitrary sizes, the
perfect log-optimal player would use a strategy that ranges
between max-er (when the player is "rich") and min-risk
(when the player is "poor").

I suspect that some people use bankroll requirements to
compare different games and/or strategies, but this is
a misguided concept. If what you really want is to maximize
growth rate, then different games/strategies must be
compared on the basis of growth rate. It is not equivalent
to compare games on the basis of bankroll requirement.
In fact, if one blindly forges ahead and uses "minimum
bankroll requirement" as the basis for choosing the best
games and the optimal strategy, then you'll end up playing
min-risk strategy on the game that has the lowest risk of
ruin, rather than playing max-growth strategy on the game
with maximum growth potential.

While I never seen the sorokin
strategy, which is sometime I have pursued separately and in a
primative way -- can you point to literature on the sorokin
strategy. The only person that has written about this was Steve
Jacobs.

A Google search with key-phrases "Sorokin" and "risk of ruin" will
get a few online pages. I don't really care much for attaching
Sorokin's name to this concept, because the formula dates back
hundreds of years to Laplace, De Moivre, Lagrange, Bernoulli
(etc?) as pointed out by the Canjar paper.

If you compute RoR using that formula, then you get the probability
of eventually going broke if you start with one unit and play the
game repeatedly. The overall probability of ruin when starting
with a bankroll of B units and playing indefinitely is given by:

session_ruin = (RoR)^B

So B = ln(session_ruin) / ln(RoR)

This is the exact bankroll requirement for a specified risk of
going broke. This is different than the Kelly bankroll requirement,
which gives the bankroll needed so that betting a single unit
will allow the log-optimal strategy to yield maximum growth.

Bottom line: just as there is no single "best" way to play, there
is also no single formula for bankroll requirement. The optimal
playing strategy depends on your objective, and for each
different objective (and optimal strategy) we can get a new and
different kind of bankroll requirement that is framed in terms of
the new objective.

···

On Saturday 26 June 2004 10:54 pm, fordscks wrote:

[blaw57]

Good info here, thanks for the post, Steve.

I've got a basic problem with bankroll requirements that is probably
shared by a lot of people - my psychological bankroll is considerably
smaller than my physical bankroll, so it is the one that controls at
present, but it is not really defineable at any point in time and
changes constantly. In other words, I could afford to lose
considerably more than I am willing to lose at gambling. I basically
like casino gambling so long as I can reasonalby support positive or
at least neutral expectations, but I would quit and go back to
private games if I had negative expectations. In this light, bankroll
calculations are very interesting in the sense that they indicate how
far I might be down at any given moment without disproving my
positive expectations, but I really can't pick out a bankroll number
to plug into a formula that decides what games to play or what
strategy to use.

For the present I've resolved the issue by sticking to quarter and 20-
cent machines and assuming unlimited bankroll, thus accepting any
degree of volatility for games I consider to be "fun." This approach
is logically flawed in several ways, but I haven't come up with a
better one yet.

> --- In vpFREE@yahoogroups.com, "nightoftheiguana2000"
> <nightoftheiguana2000@y...> wrote:
> "I did it, using math."
>
> I'm re-reading Michael Canjar's work on "Gambler's Ruin Revisited:
> The Effect of Skew and Large Jackpot," and I'm having a difficult
> time understanding why you came up with factor: 10^(7/variance);
> mainly the exponent part of your formula. I'm not disputing your
> math, just trying to understand why the constant is "7/variance."

The factor isn't 10^(7/variance), it is (10^7)/variance.

> "Strategy is a different issue. The strategy used determines the

er

> and variance, once you have those you can plug them into the
> formulas. By default I assume max er strategy, but there are

others.

> FrugalVP will give you the er and variance for your own defined
> strategy, which is a very useful feature, otherwise you can do

your

> own math but it can become tedious. There is a sorokin strategy
> which has the lowest bankroll requirement because of a reduced
> variance which comes with the cost of a reduced er. There is an
> optimum bankroll growth strategy which is roughly halfway between
> the sorokin and max er strategy."
>
> So, as I understand it, the max ev strategy will dominate the

other

···

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

On Saturday 26 June 2004 10:54 pm, fordscks wrote:
> strategies when using your formula.

Nope. These formulas are approximations used for Kelly betting.
They are designed to maximize geometric growth rate (or,
equivalently, expected value of log(bankroll)).

Just as it is possible to compute RoR without using variance, the
log-optimal playing strategy and resulting growth rate can be
computed exactly without resorting to formulae which use
variance (or ER for that matter). As nightoftheiguana2000 says,
the Kelly strategy is roughly halfway between the max-er
strategy and the min-risk strategy.

The log-optimal strategy is most applicable to games where the
player has the luxury of choosing a bet size that is a precise
fraction of bankroll. For games like VP, where the allowed
betting denominations are spaced far apart, perfect log-optimal
play would require changing strategy as bankroll changes.
One extreme case is when you have a huge bankroll but
the largest denomination of machine allows you to bet only
a tiny fraction of the Kelly optimal wager. In this case, the
log-optimal strategy become identical with max-er strategy.
At the other extreme, your bankroll is so small that the smallest
denomination of machine still forces you to bet more than
the Kelly optimal bet fraction. When the bankroll shrinks to
the point where each play causes you to bet about 2X the
optimal bet, the log optimal playing strategy becomes
identical with min-risk strategy.

So, when bet size can't be adjusted to arbitrary sizes, the
perfect log-optimal player would use a strategy that ranges
between max-er (when the player is "rich") and min-risk
(when the player is "poor").

I suspect that some people use bankroll requirements to
compare different games and/or strategies, but this is
a misguided concept. If what you really want is to maximize
growth rate, then different games/strategies must be
compared on the basis of growth rate. It is not equivalent
to compare games on the basis of bankroll requirement.
In fact, if one blindly forges ahead and uses "minimum
bankroll requirement" as the basis for choosing the best
games and the optimal strategy, then you'll end up playing
min-risk strategy on the game that has the lowest risk of
ruin, rather than playing max-growth strategy on the game
with maximum growth potential.

> While I never seen the sorokin
> strategy, which is sometime I have pursued separately and in a
> primative way -- can you point to literature on the sorokin
> strategy. The only person that has written about this was Steve
> Jacobs.

A Google search with key-phrases "Sorokin" and "risk of ruin" will
get a few online pages. I don't really care much for attaching
Sorokin's name to this concept, because the formula dates back
hundreds of years to Laplace, De Moivre, Lagrange, Bernoulli
(etc?) as pointed out by the Canjar paper.

If you compute RoR using that formula, then you get the probability
of eventually going broke if you start with one unit and play the
game repeatedly. The overall probability of ruin when starting
with a bankroll of B units and playing indefinitely is given by:

session_ruin = (RoR)^B

So B = ln(session_ruin) / ln(RoR)

This is the exact bankroll requirement for a specified risk of
going broke. This is different than the Kelly bankroll requirement,
which gives the bankroll needed so that betting a single unit
will allow the log-optimal strategy to yield maximum growth.

Bottom line: just as there is no single "best" way to play, there
is also no single formula for bankroll requirement. The optimal
playing strategy depends on your objective, and for each
different objective (and optimal strategy) we can get a new and
different kind of bankroll requirement that is framed in terms of
the new objective.

[Peter]

Thanks for putting this into words as you have. This seems to be my case as
well and I seem to want to do the same thing as you do.

For what it is worth, I, in some ways, consider that I have an infinite bankroll,
bring a reasonable amount of money with me when I gamble, enjoy what I am
doing when I am doing it, but then quit and do something else when it
(temporarily) ceases to be fun. If I should run out of "cash" and still want to
gamble I head for the bank, the cashier's cage, or the money machine and
withdraw appropriately.

All of the machinations in attempting to "compute" how much I should bring
with me seems to be worthless.

In essence, I gamble only as long as it is fun. But, it is fun and, though I might
take a few hours off after a bad run, I am always back in to start up again. By
being careful of what I do and where I do it, my quest is to "break even". Over
the years of gambliing, my copious records (my wife thinks that I am nuts for
the detail I keep...LOL), I have, thankfully, succeeded.

.....bl

···

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

my psychological bankroll is considerably
smaller than my physical bankroll

I could afford to lose
considerably more than I am willing to lose at gambling.