Yes, you're on the right track. The 134,459 figure is the number of
5-card hands that are distinguishable assuming all ranks are distinct
while all suits have equal priority. The payoff schedule isn't taken
into account when breaking down the hands into these 134,459
equivalence classes.
Jeff Lotspiech mistakenly claims that all 134,459 patterns must be
tested in order to guarantee that an optimal strategy is found, but
that isn't true. When finding an optimal strategy, we really only
care about the best possible draw for each hand -- the 31 other
ways to play the hand are irrelevant. You only have to look at
all 134,459 patterns if you need to attach some kind of relevance
to the 31 sub-optimal draws.
If you ignore suits completely, there are only 6175 hands that
can be distinguished based on rank pattern. But for 9/6 JoB,
where we only need to distinguish high ranks (AKQJ) from
low ranks (T98765432) there are only 28 hands that include
a pair or bigger group, and 16 patterns with 5 distinct ranks.
So, for draws to "grouped" hands, there are only 44 distinct
starting hands.
The real trick is to partition the hands into equivalence classes
that are derived from the payoff schedule. The optimal strategy
for 9/6 JoB ends up including only 1292 draws that can be
distinguished based on the resulting probability distribution for
payoffs. So, if you had an omniscient oracle that could partition
the starting hands into 1292 piles and tell you "these are the
differences that matter" then you'd only have to test one hand
from each pile in order to find the optimal strategy. Of course,
other games have different levels of complexity. For 10/7
Double Bonus, the max-EV strategy has only 785 classes of
starting hand, but the "best shot at royal" strategy has 827
classes and the min-risk strategy has 781 equivalence classes.
Clearly, the optimal partitioning of hands depends not only
on the payoff schedule but on the player's objective, but that
would be hard to take into account without help from an
omniscient oracle.
···
On Tuesday 16 November 2004 09:52 pm, ckbrune wrote:
--- In vpFREE@yahoogroups.com, Steve Jacobs <jacobs@x> wrote:
> The 134,459 starting hands can be reduced to a much
> smaller set of starting hands by considering all of the
> different symmetries in ranks and suits. My program
> reduces this number to something like 8,000 cases,
> but I believe it is possible to reduce the number of
> cases even more.
Steve-
I agree that VP analysis is fun to program.....I've spent countless
hours optimizing my programs to run as fast as possible.
Which brings me to my question: how are you able to reduce the
analysis down to less than 134,459 unique starting hands? Using
only 8,000 starting hands would reduce the calculations by a factor
of over 16X when compared with the 134,459 figure, so that's a major
improvement.
I know that the 134,459 figure takes into account all possible suit
symmetries, so you must also do another reduction based on ranks.
If my understanding of VP analysis is correct, then this additional
reduction will vary depending on the pay categories of the game
being analyzed; ie, a DOUBLE DOUBLE BONUS POKER paytable would not
have the same rank reductions as a JACKS OR BETTER game because the
FOUR OF A KIND category is broken into five separate categories in
DDBP. Likewise, the reductions for TENS OR BETTER would be
different from JACKS OR BETTER because two 10s is a paying pair in
one game, but not the other.
Am I on the right track? Could you give an example of how these
reductions work?