I wrote concerning Jean Scott's MS 5-play results:
All things considered, I fully expect 5-play $.25 MS to be as
volatile as $1 5-play. Off the top of my head I can't conceive why
it wouldn't be, although it certainly bears greater thought. I
don't see any reason that 5-play MS would disperse risk to a greater
extent than 5-play $1. And as I previously noted, standard MS has
modestly greater variance than single play at the same total wager.
This statement was prompted by Jean's report that in her $.25 MS
5-play to date she'd encountered far smoother results than in her $1
standard play. This ran counter to my initial assessment of MS 5-play
and I considered whether this may simply reflect short-term results or
if I had overlooked some component of variance in my cursory review of
the game.
In short, I'm now asserting that Jean's results very likely reflect
the nature of 5-play MS and that it takes a considerably smaller
bankroll to play than a comparable wager on standard 5-play (as in
Jean's case - $.25 MS 5-play vs. $1 std 5-play).
If you're interested in an more extended, moderately technical
explanation of relative contributions to variance, you may care to
peruse what follows:
···
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My initial off-the-cuff observation in the post above reflected an
assessment that:
-- Dispersion of risk in 5-play vs. single, as a consequence of the
wager being spread over numerous hands, would be comparable between
standard vp and MS.
-- The variance calculation of each hand of 5-play MS would be
identical to that of single play MS and wouldn't contribute to any
variance reduction.
-- In absence of any factor that would reduce 5-play MS variance vs.
single play MS, the modestly increased variance of single line MS vs.
single line std. vp would carry over to 5-play of each.
In retrospect, I've overlooked a key difference related to covariance
-- the fact that in single line play each resulting hand of one deal
is related to another by the initial hold.
Multiplay covariance skews individual hand results on the draw so that
all multiplay hands tend toward a result as strong as the initial hold
rather than being independent of each other -- a factor that serves in
increase variance. (Get several strong holds in a row and you end up
with an extended string of strong final hands.)
I initially glossed over the covariance factor of MS 5-play vs.
standard 5-play and assumed that it was comparable -- a glaring
oversight as it turns out. It now occurs to me that as you play up
each level, you're playing fewer and fewer hands (as hands drop out
due to missed wins & free rides). This means that fewer of the hands
played on the upper levels are related to other played hands, thus
reducing the covariance factor.
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To illustrate the significance of this, imagine a standard 5-play game
which is constructed with no covariance at all. Now, we're talking
about a trivial game -- this would mean that the result of each line
of multiplay would be unrelated to another, something that could be
the case only if each line was dealt separately rather than originaing
with a single dealt hand for which a unique draw was performed to form
the final hands.
The point is that the element of covariance in standard multiplay is
the only contributor of increased variance in multiplay. In absence
of this, multiplay would have the same bankroll requirement of single
play at the same denomination, rather than being higher. (.25 3-play
has a higher bankroll requirement than $.25 single play strictly as a
consequence of covariance, not the fact that you wager more per play)
Thus, as the nature of MS reduces the role of covariance in
determining total variance, the expected swings in play are reduced.
And the covariance factor drops off very quickly. On average, you
only expect to play half the hands on the 2nd level that you've played
on the first, and that relationship continues up each level.
So, I can readily see where covariance for MS multiplay could be much
smaller than standard vp multiplay. And as noted above, as covariance
approaches 0, overall variance at a given denomination approaches
single line variance at the same denomination (I've strayed from
expressing variance in terms of total wager, but I think the point is
better made this way).
--> In other words, it may well be that you need only a moderately
greater bankroll to play 5-play $.25 than you need for single play $.25.
I don't have the statistical background to calculate MS multiplay
variance (it doesn't lend itself to the brute force method I've used
for single play MS). Perhaps Jazbo Burns will take it up at some
point. (He provides a comprehensive discussion of standard vp
multiplay variance and bankroll requirements at:
http://jazbo.com/videopoker/nplay.html
- Harry
