A litle bit more than my $0.02:
There's been a lot of talk about comparing games, variance, RoR,
kelly bankroll, cost of DiaD, and the like. Since we all have our
own risk-tolerances and each of us values money differently, there
isn't a one-size-fits-all answer to these "subjective" questions.
That said, there are indeed mathematically correct (rigorous,
accurate) methods and mathematically incorrect methods to go about
addressing the problems. Likewise, using approximations can be a
valid and helpful technique, so long as one understands the
assumptions made in using the approximations.
The answers to many of the questions I've read recently can be found
using the probability density function or PDF. So long as one
actually knows the PDF that is! Luckily, it is a relatively straight
forward process to compute the PDF's for single line VP (though it
can be a slow process, and to speed things up a tiny approximation or
two is made) and we need not speculate. [There are also some methods
to quickly compute an approximate PDF for multiline VP if the
covariances are known already]
My rant aside, there are plenty of times when we might might want to
compare two variances (or other metrics) and that is fine. Go ahead
and do so. The challenge with statistics is not computing them per
se (especially given today's computers), but rather understanding
what they (the statistics) mean. But comming up with meaningful
statistics or metrics is no mean feat (pun intended).
So, I'll leave to the group as a whole to come up with the important
metrics, and I'll try to help compute them when I get a chance.
Other comments below:
A Mean does not have to belong to the set of values, but Median and
Mode do,
right?
Useing "sets" needlessly complicates the issue. You are 2/3 right
(the mean doesn't have to belong to the set either!) but the problem
isn't really with the statistics, but with the PDF, which doesn't
lend itself to rigourous definition or "meaning" if we are dealing
with actual data (so called student populations). If we are dealing
with the so called parent population, things are easier, but still no
need to worry much about sets. Just use the "set" of all real
numbers if needed. If one must deal with student data, the CDF is a
much better choice, but most people don't learn about them in school.
So if you play a small finite number of trials (hands) the left
side of the curve will be fairly smooth and even alot of the right
side too. But there will be spikes on the far right side, where the
royals hang out. The graph of results is lumpy.
(1) I already submitted a plot of PDFs for Pickem. I will do the same
for JoB. Critically speaking, the PDFs are never smooth, since some
values (like 0.5 coin) are impossible. But that technicality aside,
you are correct. For 1 hand the PDF has non-zero values at
precisely 1 place for each unique entry in the pay table. Hence it
must be "spikey" in nature (or stick-like)
(2) The PDF's are always "lumpy" regardless of the number of hands
that are played. Really. To understand this, consider the first
hand. There is a tiny lump for the first royal. Now consider the
next hand. There is an even tinier lump for the possibility of
getting two RF's in a row. It could happen. Really. Hence, for
every hand, the PDF gets another lump, located at (for Job) 4000*(the
hand number). So no matter what, the PDF always has a very, very
long tail to the right (positive side). The part to the left is
always shorter; it stops at (-coins bet)*(number of hands). In other
words, VP PDFs have very long positvie tails. But don't take my word
for it, compute some for yourself. (BTW, I'd perfer another word
other than lumpy. Any ideas?)
Playing 800 independant hands (bet units) on $5 Jacks 9/6 for $20k,
gets a real lumpy curve.Playing 4,000 independant hands (bet units)
on $1 Jacks 9/6 for $20k coin-in has been a possible play, and starts
the smoothing.
You are right: smoothing is happening all the time, with every hand,
BUT at the same rate for every hand. Really. That said, smoothing
appears to be occuring more in the area we care about at a faster and
faster rate. This is our eyes playing tricks on us when we look at a
graph... the rate isn't changing.
But, 80,000 independant hands (bet units) on nickel Jacks 9/6 for
$20k coin-in...really smoothing the curve, but starts running into
other problems.
What problems, exactly?
The set of possible outcomes increases with more hands played,
which smoothes out the curve, the three "M"s start to converge.
Yes. Yes. Yes. But they never really do converge for VP. This is
due to the long tail issue. Hence the mean will always be to the
right (more positve) than the mode or median. Always. (But admitedly
less and less so as we approach infinity)
Some people think this behavior is an example of the central limit
theorem. Well, I'm not sure what the CLT is actually, but what is
happening isn't the CLT. Trust me. Nonetheless, I don't object to
calling it the CLT, since the CLT conjors up a mental image that many
people can understand. [The at the pickem PDF's. Sure they are
getting smoother and nicer looking, but there really isn't a central
limit developing]
But Harrahs tosses in another constraint, 24 hrs. More importantly
is the player's contraint of time allotted/fatique.
BTW, In message #480017 I gave some statistics for JoB for up to 4000
hands. I assumed that the player has enough backroll to play that
number of hands without EVER going bust.
This is why a multi-play game helps. The results of each hand are
now dependant upon the initial deal, but you get alot more different
possibles results...a trade-off. Not as good as independant ones,
but helps get the three "M"s closer together.
yes. But put another may, multiplay reduces the variance, though by
never as much the equivelent number of single-line hands would (for
the same total bet per play). There is a proven mathematical theory
behind this, don't worry. Now, just to confuse people, the variance
itself of our sessions (for single line play) would produce a normal-
like distribution given enough sessions. This is an example of the
CLT. [If my memory serves me correclty, the variance of a
uncorrelated random process follows a chi-square distribution of
order n, where n is the number of hands here. As n gets large, the
chi-squared distribution becomes "normal". Someone might want to
check me on this. Perhaps it is only true for normal processes?]
This has been a qualitative study, our math majors will now (or
have) make it quantitative.
Sorry, not a math major exactly, so I will stick to rigorous theory
and poor spelling at the moment and leave numbers for someone else.
