If a single player plays 100 million hands, then you'd
expect the frequency of payoffs to come pretty close to what the
VP programs compute. Similarly, if you have 100 million players
each play a single hand, the frequency of payoffs averaged over
these players would be the same. Yet, each player is playing
a very "short run" of one single game.
I suppose that's true. And an infinite number of monkeys at typewriters
will write the Bible if given infinite time. That's my point: as true as
a statement, or mathematical model may be, it's got to have some practical
value. Variance is unpredictable, unforecastable. It is just there. It
is what it is. It is, if anything, a deviation from the norm...but the
norm is made up entirely of deviations. The models will show the amount of
expected variance, but don't tell you that the complete probability
distribution is the sum of that variance.
It makes no difference whether we're talking about EV or ER or
the complete probability distribution which describes the game
(as played by some chosen strategy).
I think that it does make a difference, and I purposely avoided mentioning
EV because of that difference. Assuming perfect play ER remains constant
while EV is dependent on other factors. The ER is a direct reflection of
the complete probability distribution. In my posts I complained that some
players use models of ER as if they were products with some kind of implied
guarantee. I did not mention EV because it doesn't apply to my argument.
Yes, there are fluctuations. That comes with uncertainty and ANY
model that isn't deterministic. However, this has nothing to do
with any sort of approximation, and IMO short-term vs. long-term is
usually a meaningless distinction. As more games are played, we
expect the relative frequency of outcomes to grow closer to the exact
percentages predicted, but in an absolute sense the numbers become
less certain. If you play a single game, you know the number of payoffs
for "high pair" will either be zero or one. If you play 100 million
games,
we might observe 21.3% of the outcome as "high pair" events, but the
absolute uncertainty in this value will be proportional to sqrt(100
million),
so the std. dev. in absolute terms has grown to something like 4600 units
for the "high pair" payoffs. So, as the number of trials grows, absolute
uncertainty increases while relative (percentage) uncertainty decreases.
Going from short term to long term just trades one form of uncertainty for
another, but in either case we are limited to how accurately we can pin
down the results.
So in either case there is a relatively large amount of absolute/relative
uncertainty, which would seem to indicate that the models are very nice
indicators of what will happen over time but not very good indicators of
what will happen at any given point in the timeline of those 100 million
hands. Within the context of my premise that players who rely too heavily
on these models as a basis for their expectations are misinterpreting
their meaning, that seems to support it, and moreso in the short term.
I understand your aversion to using that
word in connection with gambling, but we can predict with
complete certainty that an infinite number of payoffs will never
occur. Knowing which payoffs are possible, and knowing how
often those payoffs can be expected, is far removed from
"virtually no" information.
Nice twist. But no, in the short term there are no reliable expectations.
Frequency distribution is a long term measure. True, there is information,
but it's tantamount to saying that the average temperature in Los Angeles
is 68, without mentioning that some days are 105 and others are 40. Let me
put it this way: When it comes to playing vp, always bring a sweater.
lb
"variance giveth, variance taketh away"