A silly, simple question that shows my ignorance

[Peter]

I wonder if some of you more math-inclined people might be able to stick with me for a
bit. Try as I may, I have still not been able to get my head around the concept of
"variance" as computed in WinPoker and defined in other places. Either I am really thick,
or, am not seeing the forest for the trees. Let me try to explain what I know about
variance and then maybe someone might be able to help me make the necessary "leap to
understanding", as it pertains to Video Poker.

Suppose that we have a lab experiment. In the classroom, we have a 2 x 4 (i.e., a board
made of oak, or some such) that is approximately 1 meter long. How is that for mixing
the British and metric systems? LOL. There are 1,000 students in the class and each is
given a tape measure and asked to "measure" the length of the board, recording his/her
measurement to 0.1 mm. Therefore, we have 1,000 independent measurements of the
length of the 2 x 4.

The first thing done, is to record each measurement into an Excel spreadsheet, for
convenience. We then add the 1,000 measurements and divide by the number of
measurements, i.e., 1,000 (of course, we can simply use the "average" function in Excel to
do the same thing). This gives us the "mean" or the "average" of the "distribution of
measurements". If the measurements are done correctly and carefully our distribution of
measurements might actually be a "normal distribution".

The next step is to subtract the "mean" (or the "average") from each of the individual
measurements and square each result (that is, multiply each result by itself). Summing
these 1,000 "squares", gives us something that is usually called the "sum of the squares of
the residuals". Now, if we divide that sum by 999 (i.e., n – 1, where n is the number of
measurements), we get what is formally known as the "variance" of the "distribution of
measurements". If we take the square root of this number we have what is called the
"standard deviation". Of course, we could have used Excel's "standard deviation" function
on the distribution and square it to get the variance, too. Let's not even talk about
skewness and kurtosis, i.e., the third and fourth "moments" of the distribution. LOL.

Now, what do we know about the "variance"? In some ways the variance gives us a kind of
description of the distribution of measurements. If the variance is small, we can maybe
assume that the students were very careful in their measurement, in that they all came up
with very similar results (i.e., most of the "residuals" were small, not far from the mean
result). If the variance is big, the students might be assumed to have been sloppy in
making their measurement and their individual measurements are all over the place.

Now, to ask what might be a really stupid question, what "distribution" does the "variance,
as defined in WinPoker" refer to, in light of our "board measuring" experiment? Am I even
asking the right question for me to get a better understanding of what variance is, as
pertains to Video Poker?

Thanks.

bl

[Jean-Baptiste_Queru]

If 1000 players each play 1 hand of VP, you'll have 1000 measurements.

Let's assume JoB for a second.

Quite a few players will hit nothing, i.e. they "measure" 0. Some will
hit a high pair, i.e. they "measure" 1. Some will hit two pairs, i.e.
they "measure" 2. And so on. Put all the numbers together, use your
usual formula, and you get the variance for that set of samples.

In VP however, we know precisely what the theoretical distribution is
supposed to be, so we can actually compute the theoretical variance of
the game - it's the variance that you get when each possible deal is
considered the same number of time, and where for each deal each
possible draw is considered the same number of times.

However, you need to be careful: when your students do their
measurements, you'll probably get a distribution that's fairly
"normal" (i.e. a bell curve). In VP however the distribution is
absolutely not a bell curve, i.e. the usual measurements done on a
bell curve don't apply to a single sample (e.g. you can't say that 16%
of players will be more than 1 standard deviation below the average).

A nice property of statistics is that the average of "enough" samples
of any distribution is close to a bell curve (that's why the normal
distribution has been studied so much). That means that if you get
your 1000 players to each play 10000 hands and consider their final
result, you'll have something that looks more like a bell curve. How
much is "enough" is a fairly open to interpretation. I personally
assume that 10 million samples are likely to be "enough" for VP,
though I have not studied that issue in detail. (I will, when I find
some free time during which I decide to not practice DB).

JBQ

···

On 8/22/05, bornloser1537 <bornloser1537@yahoo.com> wrote:

I wonder if some of you more math-inclined people might be able to stick with me for a
bit.

[Harry_Porter]

bornloser1537 wrote:

Try as I may, I have still not been able to get my head around the
concept of "variance" ...
Now, what do we know about the "variance"? In some ways the
variance gives us a kind of description of the distribution of
measurements ...
Now, to ask what might be a really stupid question, what
"distribution" does the "variance, as defined in WinPoker" refer to?
Am I even asking the right question for me to get a better
understanding of what variance is, as pertains to Video Poker?

I'll see if I can offer up some helpful comments.

From a practical vantage point, vp variance gives us a relative
measure of the degree to which the game return is dependent upon
infrequent hands with higher payoffs (e.g. RF or q-A in DB). The
higher the variance, the greater the risk that play deficient in these
hands will yield a significant loss and, conversely, the greater the
likelihood that a good run of such hands will advanced the credit
meter wildly.

Variance is a value that can be calculated for any data population or
sample. When it's known that a population adheres to a "normal"
distribution, variance is helpful in describing that distribution.
However, statistical methods are of greatest value only when a
population or sample meets a sufficient size.

The calculation is perfomed in a manner analogous to what you
described in your example. However, hand payoffs and their related
frequencies are used in the calculation in lieu of observed data points.

Video poker (and any other form of card playing) is a natural
phenomenon that adheres to the basics of a normal distribution.
However, when looking at a variable such as "total return", you're
looking at not just one variable but actually the incidence of all
possible hand results at the same time. This means that while very
frequently occurring hands such as "hi pair" will serve to normalize a
portion of total return very quickly, hands such as "royal flush" take
a very long time to occur with sufficient expected frequency to ensure
a smooth contribution to total return.

Consequently, in the context of most video poker discussion, it's
typically not prudent to refer to game variance as anything more than
a rough cut measure of how choppy the play experience will be. Even
that can prove to be misleading.

Novice Pick'Em players occasionally express surprise at how much more
quickly losses can rack up vs. a game like Jacks or Better, despite
the fact that Jacks variance is 30% higher. The problem is that the
overall variance calculation factors in the effet of hands that seldom
occur.

If you want to get a feel of relative volatility of two games over the
course of any single hour of play, subtract out the variance
contribution of any hand that occurs less often than once every
1000-2000 hands. When you do this for these two games, Pick'Em now
shows up as having a much higher "short-term" variance.

- Harry

[Jerrod_Ankenman]

I wonder if some of you more math-inclined people
might be able to stick with me for a
bit. Try as I may, I have still not been able to
get my head around the concept of
"variance" as computed in WinPoker and defined in
other places. Either I am really thick,
or, am not seeing the forest for the trees. Let me
try to explain what I know about
variance and then maybe someone might be able to
help me make the necessary "leap to
understanding", as it pertains to Video Poker.

<snip>

Now, to ask what might be a really stupid question,
what "distribution" does the "variance,
as defined in WinPoker" refer to, in light of our
"board measuring" experiment? Am I even
asking the right question for me to get a better
understanding of what variance is, as
pertains to Video Poker?

In video poker, the "population" is a distribution of
outcomes. For example, in Jacks or Better with perfect
play, the distribution is like so:

ROYAL FLUSH 3995 0.002%
STRAIGHT FLUSH 245 0.011%
4 OF A KIND 120 0.236%
FULL HOUSE 45 1.151%
FLUSH 25 1.101%
STRAIGHT 15 1.123%
3 OF A KIND 10 7.445%
TWO PAIR 5 12.928%
JACKS OR BETTER 0 21.459%
NOTHING -5 54.543%

**description of calculating mean and variance - skip
if you like**
If you weight these values and take the weighted
average (the "mean" of one 5-coin hand of video
poker), you get can obtain a mean of -.022805. This is
how much you rate to lose by playing 1 5-coin hand. So
if you are playing $1 JoB and play perfectly, each
hand costs you about 2.28 cents.

Now we normalize this loss to single credits and add
it to 1 to get ER = 99.5439%.

To calculate the variance, we take the expected
outcomes (-.022805) and for each of the outcomes, we
subtract it from the payout, square the result, and
multiply the squared result by the probability of that
event occurring.

So for example, a flush is worth 25 credits. So we
take 25 - (-.022805), square it, getting 630.71, and
multiply that by the chance of getting a flush,
.01101, and obtain 6.947008, which is the variance
term for flushes.

We add up all the variance terms for all of the
different payouts, and we get 487.875. This is the
variance of one hand of 5-coin video poker. For
whatever reason, it's common to normalize this to
single-coins, so we divide it by 25 (because we
squared all the coins previously), and we get 19.515,
which is the single-coin variance.

**end description of calculating mean and variance**

Now you have these two numbers: 99.54% for ER, and
19.515 for variance. You can use the first number
quite easily; multiply your total coin-in by the ER,
subtract your coin-in from the result, and that's your
expected win or loss for playing the game for that
much coin-in.

The second number really can't be used very well
directly. If you have a zillion hands, you can use it
to find how likely your observed outcome is to fall
within certain parameters.

The second and more important way in which you can use
it has to do with it being shorthand for how volatile
a game is. So virtually all video poker games pay out
within a fairly narrow band (96%-101% or so). But some
games have a pretty volatile payscale. For example,
15/10 Loose Deuces has a variance of 70.305. This is
because the "normal" payouts are reduced in favor of
"jackpot" payouts.

Imagine a really extreme volatile video poker game.
We'll call it Royal Flush Poker. So this game has a
payscale like this:

ROYAL FLUSH 115,000
NOTHING 0

This game pays out 99.65%, by the way. The variance
here is 22,918.24.

On the other side, imagine a game where Royals pay out
a pedestrian 500 credits, but one pair or aces or tw
pair payout 10 instead of 5, and flushes pay 25
instead of 30:

ROYAL FLUSH 500
STRAIGHT FLUSH 250
4 OF A KIND 125
FULL HOUSE 45
FLUSH 25
STRAIGHT 20
3 OF A KIND 15
TWO PAIR 10
ACES 10
NOTHING 0

This game pays out 99.9914%, and has a variance of
3.929.
(pretty amazing that aces being upgraded from 5 to 10
offsets the royal payout being reduced by 3500
credits!)

These two are extremes, but you should be able to see
how the volatility of the game is represented by the
variance.

Looking at some real games:

Double Double Bonus: 41.98
10/7 Double Bonus: 28.25
8/5 Bonus: 20.90
9/6 Jacks: 19.51
FPDW: 25.83
15/10 Loose Deuces: 70.30

You can see that the games where the payouts are more
concentrated in royals and other jackpot-type hands
(such as the 2500 payout in Loose Deuces) have higher
variance. This is because they are more volatile in
terms of the dispersion of observed outcomes from the
mean.

So you can use those variance numbers as a proxy for
volatility, at least as comparison on a relative
scale.

Jerrod Ankenman

···

--- bornloser1537 <bornloser1537@yahoo.com> wrote:

____________________________________________________
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[snowbird010339]

Thanks, Harry I now have amuch better grip.
Ed

···

I'll see if I can offer up some helpful comments.

From a practical vantage point, vp variance gives us a relative
measure of the degree to which the game return is dependent upon
infrequent hands with higher payoffs (e.g. RF or q-A in DB). The
higher the variance, the greater the risk that play deficient in these
hands will yield a significant loss and, conversely, the greater the
likelihood that a good run of such hands will advanced the credit
meter wildly.

- Harry