John wrote:
Steve (and anyone else who wants to chime in, of course ;), I have a
question about how EV is computed for use in developing a strategy.
I'll accept that secondary invitation to chime in ... hopefully
intelligently, though likely not at authoritatively as Steve
might/will. EV, and variance, are hot buttons for me and prime
motivators in selecting plays.
As I understand it, EV for a given set of dealt and held cards is
the *average* of all possible outcomes,
That's correct. You can look at the hand analysis in WinPoker and
Frugal VP to see this in action. For a given deal, a count is made of
all the possible winning hands on the draw, and their value
accumulated, for each of the 32 possible holds. The hold with the
highest EV wins as best.
Developing a strategy is simply a matter of sorting all possible
holds by descending EV.
It's much more difficult than that. The goal of a formal strategy is
to represent the possible holds for all conceivable dealt hands (2.6
mil for a non-wild card game) by a grouping of hold types (1 pair, 3
flush, etc.) and rank their value accordingly.
However, the value of each hold is dependent upon the discards in the
hand. Holding 3RF has weaker value if you're discarding a similarly
suited flush card than if not.
There are occasions where hold "A" is stronger than hold "B", when
they're contained in separately considered hands, but hold "B" is
stronger than "A" if they're contained in the same hand. These types
of considerations can pose complications when preparing a ranked
strategy table.
It's the case, for example, that some of the strategies in Frugal
Video Poker are more accurate than those in the prior released VP
Strategy Master, because these considerations (and others) are more
accurately reflected. (We're not talking about light-years of
improvement, but definitely not insignificant.)
Aren't averages considered "brute force" values, statistically
speaking? Shouldn't something like the "mean" and standard
deviation (variance?) be used instead? That way, someone could
calculate a proper strategy for them based on an acceptable variance
level or value.
Evaluating EV's is, at heart, a counting exercise and nothing more.
To the extent that values are expressed as averages, rather than
absolutes (both are valid), they reflect the same "brute force"
method. "Mean", as interpreted for vp analysis, has no practical
application beyond the weighted average value of possible holds -- a
straight arithmetic average.
Consideration of standard deviation/variance puts the analysis into an
entirely different framework -- one that's multivariate and requires
that the player make judgemental decisions.
Steve Jacobs has discussed various alternate strategies, each of which
has a different goal than the "Max-EV" that's reflected in almost all
published strategies. He has written a discussion on this topic that
can be found at:
http://members.cox.net/vpfree/FAQ_S.htm
A game I think is a good candidate for which to discuss an
"alternate" (non-max EV) strategy is Multi-Strike (any other
high-variance game would qualify as well). Lowering the variance
without affecting the EV too much would be desirable for most
people.
I agree. As you say, any higher variance game is a potential
candidate for alternate strategies which decrease variance without
undue ER impact.
One factor is that for all but the more advanced players, the analysis
and playing considerations involved are almost esoteric when compared
to the challenge of just getting the basics down. Plus, there are no
ready made software tools to assist with this (although a good deal of
the legwork can be accomplished with modified vp software use,
augmented by a good spreadsheet program).
But, I'll offer up my personal perspective on this, which largely
isn't shared by Steve Jacobs:
If you set a tight tolerance on ER reduction, the extent to which you
can reduce the variance of most games is relatively nominal. I won't
suggest insignificant, since it's possible to shave as much as 10% of
the variance at the extreme in some cases.
However, more often than not, a game that presents an uncomfortable
variance play at a given bankroll isn't going to be made appreciably
more comfortable under an altered strategy to reduce variance. So,
I've largely lost interest in this area of game exploration and almost
entirely stick to Max-EV strategies.
As an example, I once sought to make a dent in 10/7 DB variance by
holding pat FH's over 3 Aces. The EV difference is very small, but
obviously this, in a single instance, makes a big variance reduction.
But, I ultimately decided that the relative ravages of this game over
gentler games such as JB and BP were such that the game wasn't any
more comfortable. In fact, the downside potential made me crave the
variance adding shot for a 800 cr. win instead of just taking a no
risk 50 cr. win.
But this is a personal take. The variance/ER tradeoff is one of
individual preference and to be decided by each player.
- Harry
lowering the variance on some games is worth considering, as long as it has an acceptably small effect on EV. Others want to protect bankroll, etc.