Steve Jacobs wrote:
> > Has anyone ever published a min risk, or a min hands between royal
> > strategy or anyothers besides max ER?
>
> Thanks for asking. Here's the min-risk strategy of 9/6 JoB:
>
> Jacks or Better 8/5
I presume that the above header is incorrect where it indicates "8/5",
and that the data in the table and strategy chart which followed was
actually for "9/6", ... correct? The other table for min_cost_royal
strategy does indicate "9/6".
Indeed, the header is incorrect. The strategy is for 9/6 JoB.
snipped data and a very good illustration and information. Thanks, Steve.
You're welcome.
I'll start with a couple of questions:
1. What do you mean by "cost", as in ... "cost: 0.9999999999989" ... at
the bottom of the min-risk distribution table?
That number is meaningless when virtual payoffs are being used.
I define cost as the number of units "paid" by losses, divided by the
number of units "bought" as winnings. For 9/6, the max-EV strategy
has a cost of 1.008432571589, meaning that on average each units you
receive back from the machine comes in exchange for 1.008432571589
units that you fed into the machine.
For 9/6 JoB, the game is so close to breakeven that the min-cost strategy
is identical to the max-EV strategy. But, for 8/5 JoB, max-EV has a cost
of 1.041733 while min-cost reduces this value to 1.0417056. These
strategies are only a tiny bit different. The min-risk 8/6 JoB strategy has
a cost of 1.042609.
2. What is the actual ER for each of the two strategies you gave? I
know how to derive them from the data in your tables, but if you already
made the calculations, which I presume you have, then you can save me
the time and trouble, and possible error, in calculating them myself.
The min-cost strategy has an ER of 99.54390436%, same as max-EV.
The min-risk strategy has an ER of 99.53515116%, only 0.009%
different than the max-EV strategy.
One thing worth noting here is that the differences between the strategies
become more pronounced as game EV strays further away from a
breakeven game. If there were a real game that just happened to have
an ER of exactly 100.00000% then the max-EV, min-cost and min-risk
strategies would all be identical. As a result, if one strategy is favorable
to the player then the others will be as well, they are simply favorable
in different ways.
For 8/5 JoB the differences are more noticeable, since it is not as close
to breakeven.
I thought that the min-risk strategy might be a variable one that
changes as the goal and initial bankroll vary. Perhaps that was just an
assumption on my part due to the percentages of 'Risk of Ruin' itself
changing as changes are made to the stake and goal (the 'retire at'
value in Jeff Lotspiech's calculator). Obviously, it would be better if
the strategy does not vary, since that would make it a whole lot simpler
to learn. Since you did not indicate a particular stake or goal, I will
assume that the strategy is actually independent of either of them and
therefore remains the same, with only the resulting RoR percentiles
varying; if I'm wrong, please advise what stake and goal were used for
the example you posted.
The precise answer is a little complicated. The min-risk strategy is
almost independent of stake and goal, and whenever the goal is
"far enough" from the original stake, then the difference is probably
small enough to ignore. Technically, when the distance to the goal
is less than a royal flush away, you could reduce the risk slightly
by treating the royal payoff as if it was only barely large enough to
reach the goal, and alter the strategy accordingly.
Now, were your adjustments for your virtual paytable obtained through
trial and error, or by a precise mathematical method used to derive them
-- similar to my approach to calculate exact adjustments for MultiStrike
( http://www.velek.com/ms )?
That depends on what you mean by "trial and error". For risk in
particular, there is no "closed form" solution that allows you to simply
compute the virtual payoffs directly, and then plug them into a VP
program to compute the strategy. Instead, you have to iterate through
the process a few times to converge on the exact solution. I described
this some in a post this morning, so perhaps you've already seen that.
If your min-risk strategy is indeed
independent of how many hands will be played, then I think this is
essentially along the same lines as a concept that I had been
experimenting with last year, but using trial and error because I
couldn't figure out how to come up with any sort of an equation that
would include so many variables (each value in a paytable being a
variable). I just played around in WinPoker, reducing the values of
Royals first, and later, combinations of both Royals and STFL's --
figuring that those hands are the least frequent. I used Excel
spreadsheets (lost all that data when my hard drive crashed in the
fall), but I recall that I was able to increase my return for the more
common hands at only a slightly higher cost from the Royal/STFLs. I
thought this would provide a solution to helping my bankroll last a bit
longer, but I'd sure like to know how to get a precise answer without
having to spend hours and hours of trial and error for only a very rough
answer.
It sounds like you were trying to do something similar to min_cost_royal,
but lumping royals and STFL's both together. The question then is how
to "weight" the relative importance of royals vs STFL's.
I suspect that for any reasonable weighting, you'd end up with a
strategy that is very close to min_cost_royal, which is already very
close to min-risk.
Anyway, I stopped working on it even before my computer crash,
once I became convinced of full-coin play which changed my point of view
quite a bit; now I've been playing full-coin almost exclusively (no need
to discuss my exceptions). But since I am a rec player and may never
accumulate the amount of play to ever really see long-term results, and
since I'm very interested in stretching my stake (especially now that
I'm playing full-coin), I think that maybe min-RoR is actually the way I
ought to _personally_ play. It might not be for everyone -- especially
pros -- but I'm interested.
Aha, a convert. Welcome to the Church of VP Blasphemy 
I appreciate all the time you have spent on this, Steve. I do have what
I hope will be a final question or two in this thread, but which will
appear in my reply to another of your posts in this thread. I hope
you'll take the time to answer.
I'm trying to answer all the posts that I can. I've probably missed a couple
after running out of time, so if _anyone_ asked a question that I haven't
covered then please ask again and I'll try to answer.
This has been a great thread, and I appreciate all the feedback from
everyone who has participated.
···
On Friday 20 February 2004 09:56 am, Bill Velek wrote:
> On Thursday 19 February 2004 08:14 pm, deuceswild1000 wrote:
