vpFREE2 Forums

Risk of Ruin before Royal simulation

#1

I previously mentioned in this forum that my trainer/analysis
program, Optimum Video Poker, now includes a feature to calculate the
Risk of Ruin Before Royal (RORBR), and that my method is based upon
the Poisson Distribution. This raised a long discussion on the topic.
At least two other methods of calculating the RORBR have been
proposed. Unfortunately, no two of the methods gave the same results,
so it was impossible to determine which (if any) of the methods was
"exact," or even which is best.

What seemed to be needed was a reliable simulation to give a basis of
comparison, so I undertook the task. It took a few hours to program
such a simulation in FutureBASIC, then it took only a few minutes to
run on my Macintosh. The program generated a table giving the RORBR
for starting bankrolls of 50 to 2000 betting units in 50-unit steps,
assuming perfect play on 9/6 Jacks or Better, running 1000 simulated
sessions for each starting bankroll.

I ran the program ten times, and none of the RoR values varied more
than 0.4%, so it appears that averaging 1000 sessions for each table
entry was sufficient. I could run more sessions for each entry, but
this accuracy of a little better than two significant digits seems
adequate for practical use by players and also for checking
algorithmic methods.

A commonly asked question is, "How big a bankroll do I need for at
least a 50% chance of hitting a royal flush?" The simulation shows
that the answer for 9/6 JoB is about 740 betting units. A bankroll
equal to the 800-unit payoff for a royal gives only a 52.4% chance of
hitting at least one royal.

The Session Bankroll feature of Optimum Video Poker shows 36.2% RORBR
for a 1000-unit bankroll and 13.1% for a 2000-unit bankroll. The
simulation gives 38.4% and 13.9%, respectively, confirming that OpVP
gives very good approximations. The problem is that the use of the
Poisson Distribution limits this feature's use to bankrolls
sufficient to play at least one royal cycle without hitting a royal.
(Note that the results are more accurate for larger bankrolls.)

How do the other methods measure up?

···

--
Dan Paymar
Author of best selling book, "Video Poker - Optimum Play"
Editor/Publisher of VP newsletter "Video Poker Times"
Developer of VP analysis/trainer software "Optimum Video Poker"
Visit my web site at www.OptimumPlay.com

"Chance favors the prepared mind." -- Louis Pasteur

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

#2

The chances of getting a royal on one hand of 9/6 JOB with maxER
strategy is .00247583% . To get a 50% chance of hitting a royal, you
have to play ln(100%-50%)/ln(100%-.00247583%)= 27,996 hands. Use
http://www.lotspiech.com/GamblersRuin.html with stake=740 x $1.25
=$925, retire with $1500, 27996 hands. Result is 33% busted. These
people that busted didn't get to play 27,996 hands, let's assume the
average they got to play was 20,000 hands, for a chance at a royal of
1 - (100%-.00247583%)^20000= 39%. Average chances: .33(.39) + .67(.5)
= 46% instead of 50%

···

--- In vpFREE@yahoogroups.com, Dan Paymar <Dan@O...> wrote:

A commonly asked question is, "How big a bankroll do I need for at
least a 50% chance of hitting a royal flush?" The simulation shows
that the answer for 9/6 JoB is about 740 betting units.

#3

> A commonly asked question is, "How big a bankroll do I need for at
> least a 50% chance of hitting a royal flush?" The simulation shows
> that the answer for 9/6 JoB is about 740 betting units.

The chances of getting a royal on one hand of 9/6 JOB with maxER
strategy is .00247583% . To get a 50% chance of hitting a royal, you
have to play ln(100%-50%)/ln(100%-.00247583%)= 27,996 hands.

Here we go again...

It is true that if you play 27,996 consecutive hands then you will
attain a 50% chance that no royal appeared during that length of
play. However, this doesn't factor in the risk of going broke in any
way. To guarantee with absolute certainty that you will play
27,996 hands one must start with a bankroll of 27,996 units.

Use
http://www.lotspiech.com/GamblersRuin.html with stake=740 x $1.25
=$925, retire with $1500, 27996 hands. Result is 33% busted. These
people that busted didn't get to play 27,996 hands, let's assume the
average they got to play was 20,000 hands, for a chance at a royal of
1 - (100%-.00247583%)^20000= 39%. Average chances: .33(.39) + .67(.5)
= 46% instead of 50%

A 731 unit bankroll has exactly 49.9788758% chance of going broke before
hitting a royal flush. There are now three independent simulations
(including Dan's) that support this result. Dan's figure a 740 from
simulation is much closer to my figure of 731 units than was his original
figure of 984 units.

Are you seriously suggesting that your approximation, which includes
"let's assume the average they got to play was 20,000 hands..." (pulled
from thin air, perhaps) should be taken as adequate "proof" that my
exact figure must be flawed, despite confirmation from three independent
simulations that were specifically designed for this purpose? Get real!

Every carefully designed simulation so far has confirmed my number.
I think at this point it is fair to say that those who wish to refute the
result with nothing more than a rough approximation and some hand
waving are in serious denial.

The math doesn't lie, and the simulations confirm it.

···

On Thursday 10 November 2005 05:56 pm, nightoftheiguana2000 wrote:

--- In vpFREE@yahoogroups.com, Dan Paymar <Dan@O...> wrote:

#4

I previously mentioned in this forum that my trainer/analysis
program, Optimum Video Poker, now includes a feature to calculate the
Risk of Ruin Before Royal (RORBR), and that my method is based upon
the Poisson Distribution. This raised a long discussion on the topic.
At least two other methods of calculating the RORBR have been
proposed. Unfortunately, no two of the methods gave the same results,
so it was impossible to determine which (if any) of the methods was
"exact," or even which is best.

What seemed to be needed was a reliable simulation to give a basis of
comparison, so I undertook the task. It took a few hours to program
such a simulation in FutureBASIC, then it took only a few minutes to
run on my Macintosh. The program generated a table giving the RORBR
for starting bankrolls of 50 to 2000 betting units in 50-unit steps,
assuming perfect play on 9/6 Jacks or Better, running 1000 simulated
sessions for each starting bankroll.

I ran the program ten times, and none of the RoR values varied more
than 0.4%, so it appears that averaging 1000 sessions for each table
entry was sufficient. I could run more sessions for each entry, but
this accuracy of a little better than two significant digits seems
adequate for practical use by players and also for checking
algorithmic methods.

A commonly asked question is, "How big a bankroll do I need for at
least a 50% chance of hitting a royal flush?" The simulation shows
that the answer for 9/6 JoB is about 740 betting units. A bankroll
equal to the 800-unit payoff for a royal gives only a 52.4% chance of
hitting at least one royal.

The Session Bankroll feature of Optimum Video Poker shows 36.2% RORBR
for a 1000-unit bankroll and 13.1% for a 2000-unit bankroll. The
simulation gives 38.4% and 13.9%, respectively, confirming that OpVP
gives very good approximations.

My exact formula gives 38.7207% and 14.9929%.

Note that the RORBR for 2000 units is exactly the square of the RORBR
for 1000 units. Risk works that way -- if you double the number of units
in the starting bankroll, you square the probability of failure.

The problem is that the use of the
Poisson Distribution limits this feature's use to bankrolls
sufficient to play at least one royal cycle without hitting a royal.
(Note that the results are more accurate for larger bankrolls.)

How do the other methods measure up?

Thanks Dan, I think you've confirmed my result, along with the other
simulations. Now I think you should ask yourself a serious question
about your program -- are you satisfied with using an approximation
that you now admit is only good for large bankrolls, or would you rather
use an exact computation that works for any size bankroll? Seems like
an easy choice to me.

···

On Thursday 10 November 2005 02:49 pm, Dan Paymar wrote:

#5

Something about your numbers seems odd to me. Your figure for a 50%
RORBR was 984 units, and above you give a RORBR of 36.2% for a 1000-unit
bankroll. It seems extremely improbable that adding a mere 16 units to a
984-unit bankroll would change the RORBR figure from 50% to 36.2%.

The following table compares the three figures you've provided with
the corresponding values from my calculations (which I still claim to
be exact):

bank Paymar Jacobs Paymar/Jacobs

···

On Thursday 10 November 2005 02:49 pm, Dan Paymar wrote:

The Session Bankroll feature of Optimum Video Poker shows 36.2% RORBR
for a 1000-unit bankroll and 13.1% for a 2000-unit bankroll. The
simulation gives 38.4% and 13.9%, respectively, confirming that OpVP
gives very good approximations. The problem is that the use of the
Poisson Distribution limits this feature's use to bankrolls
sufficient to play at least one royal cycle without hitting a royal.
(Note that the results are more accurate for larger bankrolls.)

-----------------------------------------------
2000 13.1 14.99 0.874
1000 36.2 38.72 0.935
984 50.0 39.31 1.272
-----------------------------------------------

If my numbers a truly correct as I believe they are, then your formula
is more accurate for the 1000 unit bankroll than it is for values above
or below 1000 units. This might also suggest that your formula isn't
really computing RORBR, but something quite different. Perhaps we
should examine a few more data points?

#6

Steve Jacobs wrote:

This might also suggest that your formula isn't
really computing RORBR, but something quite different. Perhaps we
should examine a few more data points?

Steve has shown that the results [his and Dan's] diverge as starting
bankrolls grow. I expected that behavior and also that the results
of the two methods will differ even more for positive EV games (I
can't wait for these results!)

Why? Because I believe there is a fundamental difference between what
Steve is computing and what other methods come up with-- nonetheless,
the methods should produce similar numerical results for certain
cases. Really. I plan to clearly show this (using some simple
examples) and also "validate" the "Sorokin" RoR formula (not RoRBR)
in a series of upcoming submissions to this group (I will start with
the plain old RoR).

To sum up, Perhaps Steve is the one who "isn't really computing RoRBR
but something quite different" ?
Perhaps also Dan is making a numerical error and there is, in fact,
no fundamental difference in what they are computing?
I don't think so.
But I guess I should hedge my bets—since I've been wrong before. We
shall see. BTW, I need to stress that I don't think Steve is making
ANY numerical errors. None. Moreover, it does not matter to my
argument whether he or Dan has made numerical errors, and frankly, I
don't care if they did or didn't. I am just concerned with the
understanding the underlying concept of Steve's approach. Simply
claiming that some result is "exact" doesn't prove OR disprove the
soundness of the concept. I am not trying to pick on Steve either--
but rather attempting to support his claims through a validation of
his concept (and so far, I have been unsuccessful).

I do want to say this:
Steve's method provides an elegant shortcut to producing an estimate
of RoRBR, that, in many cases, is very accurate—a remarkable
achievement. He should be congratulated by all for his deep insight.

This still, however, leaves open the other question: how to validate
the simulations. When I used to do simulations like this for a
living, it was not sufficient to say "the different simulations give
the same result, therefore they validate each other". After all, the
underlying algorithms or the implementations of said algorithms might
be flawed in each case. The only time one would result to such
methods (of "validation") would be when no better method was
feasible. And when that occurred, well, the term that was used was
not "validated" but rather "consistent with each other"-- either both
wrong or both right. Unless at least one of the simulations was
validate, labeling something "consistent" was a euphemism for "I
don't really know what I am doing so don't ask any tough questions."

So what are the better methods of validation? In general, simulations
are "validated" buy using them to compute quantities that are already
well known and by doing a parameter sensitivity analysis (second
step). In best cases, (at least) 3 types of quantities are
considered:

Type 1 is a quantity that is easily computable "by hand" using first
principles. Type 2 is a quantity that is not easily computable using
the same algorithm, but whose value is established via some other
method. Type 3 is a quantity that can be computed using the "same
algorithm" as used in the simulation, but "by hand" (this doesn't me
not using a computer)

An example of a Type 1 quantity for this case would be the RoRBR for
a very simplified version of VP. The simplified version of VP would
have to capture all the "physics" of the complicated VP to completely
validate the simulation. The goal is that by using the simplified
game, the RoRBR can be computed using first principles (accepted
analytic methods)

An example of a Type 2 quantity would be the "Sorokin" RoR, in so far
as we believe in the accuracy and meaning of the quantity. For
example, the simulation could be used to compute the RoR for a
positive EV game and that value can be compared to the RoR from the
formula. But first, the validity of the RoR formula should be
established (so I will try to do this first).

An example of a Type 3 quantity in this case depends on the
particular simulation being validated. The usual way to come up with
a trusted quantity of this type, is to produce a flow chart of the
algorithm (or better yet, a complete description of the algorithm
using one of those special languages developed for such things) used
in the simulation, and then have someone else (1) validate the "flow
chart" and then use the "flow chart" to compute the said quantity.
The next step is to us the simulation to compute the same quantity
and then compare results.

#7

A 731 unit bankroll has exactly 49.9788758% chance of going broke before
hitting a royal flush. There are now three independent simulations
(including Dan's) that support this result. Dan's figure a 740 from
simulation is much closer to my figure of 731 units than was his

original

figure of 984 units.

OK, I see what you are up to now. Using wizardofodds.com hand
probability numbers for 9/6 JOB I get: 0.99905165413274, 731 bankroll
is 0.99905165413274^731= 0.4997887585222, 50% is
ln(.5)/ln(0.99905165413274)= 730.5546225935

···

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