Steve,
I really don't understand why you are arguing with me, and likewise,
how you really anyone to believe your claims SIMPLY because you say
they are correct.
The fact that I'm responding to your post does not imply that I disagree
with what you are saying. I do agree with much of what you say.
I don't expect anyone to take my word for it. In fact, I encourage anyone
who is so inclined to develop the math themselves and compare their
results with mine. I value independent verification (or refutation) of what
I'm doing. Nobody should accept what I say (or what anyone else says)
as indisputable fact.
BTW, I already know that EV isn't evervything, I
(LOL) I can go on the record of saying that, well, more than 20 yrs (
but, my isn't bigger. oh well, can't win them all).
Oh my, that comment sure makes me want to continue this discussion...
Back to the point:
Can you explain some things for me:
You claim that "Any other strategy will not perform as well, when
measured in terms of" the goal of the min-cost-Royal strategey and
you make a similar statement fot min-RoRBR strategy.
I think I know what you are saying mathematically (and for these 2
cases I beleive you are correct, in a sense),
I believe I'm correct in a mathematical sense.
but the statement that
says "an optimal strategy [for meeting a certain goal in VP] is
better than all other stratagies" is false in general,
What definition of "optimal" leads you to make such a statement?
and I think
the folks who do try to read and understand your posts should
understand that [you didn't actually write anything that contradicts
this statement about optimal strategies, but what you did write did
suggest to me, at least, such a contradiction. What about the more
casual reader?]. It's easy to find examples. There are plenty. One
you have discussed is the min-ROR strategey,
'min-RoR is "highest probability of playing forever without going
broke"'
Minimizing probability of ruin is mathematically equivalent to maximizing
probability of playing forever without going broke. Discussions of RoR
are generally geared toward games which favor the player, so that
RoR is less than 100.0000000%
However, the concept can be generalized to negative games, and
the value of a "risk parameter" will be made as small as possible
by using a min-risk strategy.
For a negative game, like JoB, RoR is always zero, regardless of
strategy. So, there is a huge set of min-RoR stratgies (classified
by involving playing until going broke and not stoping in betweem)
have no effect on the RoR, and they are all equally optimal, or
perhaps equally non-optimal. That is, in terms of the goal of the min-
RoR strategy, they all equally suck.
That isn't true. Solving the risk polynomial for a negative game will
give a value of R that is greater than one. The implication of this
is that it becomes meaningless to talk about playing forever, because
the player is guaranteed to eventually go broke. However, the
concept of risk does not have to be applied only to endless games.
Risk can be measured by looking at the probability of turning a
starting bankroll of N units into a goal bankroll of G units. This
is equivalent to pitting the player against a casino with a tiny
bankroll, and the risk parameter gives a measure of the probability
that the player will prevail. A risk of R=1.000001 is very close to
a fair game, and is much better than a risk of R=1.5. The playing
strategy with the lowest risk parameter will give the player the best
chance of reaching the goal bankroll, even if the game is negative.
For risk parameter R, the probability of turning N units into G units
is given by:
prob(success) = (1 - R^N) / (1 - R^G)
Caveat: the formula above is only exactly correct for games that
are like a simple coin flip. For games with payoffs larger than 1:1,
the min-risk strategy is only approximately optimal for turning N
units into G units. For video poker, the "best shot at G units"
strategy is very complex, because the strategy changes as one
gets closer to reaching the goal. If the goal bankroll is 10,000
units and you get up to 9,999 units, the strategy becomes radically
different than either max-EV or min-RoR. I believe (but haven't
proved, even to myself) that "best shot at G units" strategy will
be identical to min-risk strategy whenever the bankroll is "far
enough" from the target bankroll (so that hitting a royal would
still fall well short of reaching G units).
But, had the goal been, play as
long as possible without going broke, then, well, there very well be
a single optimal strategy, but then again, maybe not.
Yes, it is theoretically possible to have more than one strategy (and
I've seen real world examples of this for max-EV strategy), but this
is a fine point that doesn't happen all that often with real games.
It also
depends on how one compares strategies-- and if how one compares
stratgeies "properly", one will find that there is usually a class of
equivelent stratgies that are optimal, where the differences between
strategies have not effect on the goal.
Usually? Did you mean to say "hardly ever"? For the specific alternate
strategies that I've described, I don't know of a single example of this. If
you have one, I'd be interested in seeing it. Generally, optimal strategies
for VP are unique -- deviating any single playing decision at all will result
in a reduction in performance that can be exactly quantified.
[For VP it is relatively
simple, but for other games, like heads-up no-limit hold'em, the
optimal stratgey involves random play, and the distribution of that
random play matters. Really.]
I have not made any claims whatsoever about applying these methods
to live poker. That is a whole different realm.
Conclusion: specification of a goal is
not always sufficent to produce AN optimal strategy, and sometimes it
depends on the game (with min RoR, Job vs FPDW, for example).
That is a huge nit-pick at best. In real world VP games, the kinds of
strategies I've discussed virtually _always_ produce a unique playing
strategy. Of course, that strategy tends to change whenever you
change the payoff schedule -- JoB vs FPDW are completely different
games, so one should expect different strategies. But changing from
JoB to FPDW doesn't involve the kind of paradigm shift that would go
along with live poker, so I'm not sure what you're point is when you
say "it depends on the game".
I do like you analogy to driving (one of my favorite activities). And
the point you draw from it is, IMHO, excellent. I do have quite a bit
to say about of this analogy can be used mathematically in
application to gambling (Feynman path integral approach) . But until
you try to apply some rigour to this analogy, I'll keep my fingers
idle.
I wasn't really planning to take the driving analogy any further in
a mathematical sense.
Are you, by any chance, a physicist? I have some half-baked ideas
about how these alternate strategies might relate to the Heisenberg
Uncertainty Principle, but I don't know any genuine physicists who
are also interested in gambling. I'd really like to talk to someone
who fits that description.
···
On Thursday 08 December 2005 10:00 am, cdfsrule wrote:
