Match Doubling Problem - Maths Test
Michael Howard
strategy for the following position.
The bots disagree so this one is for humans only.
I know the answer (I think) but wat to see the arithmatic you would use at
the table. Please indicate which equity table or algorithm you use.
The Fish leads 12-2 in a 25pt match.
+12-11-10--9--8--7-------6--5--4--3--2--1-+ O: JellyFish pips 14
| O | | O |
| | | O | +-+
| | | O | |8|
| | | O | +-+
| | | O |
^| |BAR| |
| | | |
| | | |
| | | |
| | | X X X |
| | | X X X X X |
+13-14-15-16-17-18------19-20-21-22-23-24-+ X: Snowie2 pips 26
JellyFish - 12, Snowie - 2 in a 25 point match.
JellyFish to move. Cube Action?
Double?
Take/Drop?
Redouble?
--
Michael Howard
Donald Kahn
he really has only 5 on the ace point, this is an obvious double/pass
at almost any score. If he has six on the ace point, that's a
different matter. The double or redouble is very strong. There is
still lots of points to go, and O can use all of them. Actually, I
would pass as X, even though both bots call it a take. Reasoning: O
has a 4 roll position, at worst. X needs a big double to get off in
3, AND no misses. The bots say it is a take, so I guess I have to
believe them.
dk
Michael Howard
The Fish doubled expecting a pass but Snowie correctly took and redoubled
putting the match on the line and of course reducing O's equity.
I'm not sure I would have doubled as, at 90% sure to win this game puts the
score at 20-2, and therefore the match equity for Jellyfish is already in
the 95%+ range.
My calculations based on a 90% chance of winning this game are roughly:-
Equity at 20-2 = 99%. So doubling point = risk/(risk+gain) = 90/(90+10) =
90%. Therefore it is correct to double and obviously correct to take and
redouble which reduces O's equity by a couple percent.
However, I could not bring myself to give X an outside chance to win whilst
in such a commanding lead.
Conversely I don't thing I would be steady enough to take either! A big
psychological problem hereat.
Michael
Donald Kahn <don...@salzburg.co.at> wrote in message
news:38f97f0b...@enews.newsguy.com...
Donald Kahn
wrote:
>Something wrong here, isn't there? Only 14 pieces shown for O, and if
>he really has only 5 on the ace point, this is an obvious double/pass
>at almost any score. If he has six on the ace point, that's a
>different matter. The double or redouble is very strong. There is
>still lots of points to go, and O can use all of them. Actually, I
>would pass as X, even though both bots call it a take. Reasoning: O
>has a 4 roll position, at worst. X needs a big double to get off in
>3, AND no misses. The bots say it is a take, so I guess I have to
>believe them.
>
>dk
I can't plead drunkenness, nor (I think) insanity. Of course O does
not have a "four roll position at worst". With six checkers on the
ace point and one on the 10-point, O has a five roll position 2/3 of
the time. That being the case, X can scrape up a take, I guess, but
it is no great bargain.
O cannot redouble from 4 to 8 here, because that is as good as
deciding the match with this game (X must make it 16 on his turn,
unless O has rolled 6-6 or 5-5), and O stands to lose much more than
he has to gain.
But 2 to 4 is a good redouble. To hold the cube would qualify as
"trying to nurse a lead" which is usually a mistake.
Donald Kahn
<mi...@howard666.freeserve.co.uk> wrote:
>The board setup is absolutely correct.
>
>The Fish doubled expecting a pass but Snowie correctly took and redoubled
>putting the match on the line and of course reducing O's equity.
>I'm not sure I would have doubled as, at 90% sure to win this game puts the
>score at 20-2, and therefore the match equity for Jellyfish is already in
>the 95%+ range.
>
>My calculations based on a 90% chance of winning this game are roughly:-
>
>Equity at 20-2 = 99%. So doubling point = risk/(risk+gain) = 90/(90+10) =
>90%. Therefore it is correct to double and obviously correct to take and
>redouble which reduces O's equity by a couple percent.
>
>However, I could not bring myself to give X an outside chance to win whilst
>in such a commanding lead.
>Conversely I don't thing I would be steady enough to take either! A big
>psychological problem hereat.
>
>Michael
Sorry to have answered a different question: I didn't see that that 8
meant the cube.
I don't agree with your answer. My view of the match equities, 5a
23a, which O gets by keeping the 8 cube and winning, is worth 95%.
5a 15a, which he gets by losing at the 8 cube, is worth 90%.
Doubling to 16 makes it win/lose. He is risking 90% match-winning
chances to gain 5%. Therefore his advantage in the game should be
18-1 to make this an even proposition; i.e. 94.7% winning chances.
Besed on 90% chance to win the game, which is right or very near it,
he does not have enough to turn it to 16.
If O's outside checker is on the 7 point, he is a 19-1 favorite and
this would qualify, except that Snowie 2.3 call this "no double,
take".
Snowie must use a different match table. Mine is based on an
extension of Neil's numbers, assigning the value 3 when the trailer is
20 points away, and 2 when 26 points away, therefore 2.5 when 23
away..
Donald Kahn
Michael Howard
This is getting better.
However, your statement about the score when O loses seems adrift.
If he wins with 8 cube score is 5a-23a.
If he loses it is 13a-15a. NOT 5a-15a.
This may change your calculation. But probably in the positive direction for
a double.
I too, often use Neil's numbers....which fail at long leads like this (so do
most other formulae).....can you give details:-
I have 9 8 6 5 4 3 2 2 2 2 2 2 ...then what?
Rgds,
Michael
Donald Kahn <don...@salzburg.co.at> wrote in message
news:38fa9425...@enews.newsguy.com...
Donald Kahn
<mi...@howard666.freeserve.co.uk> wrote:
>Donald,
>
>This is getting better.
>However, your statement about the score when O loses seems adrift.
>If he wins with 8 cube score is 5a-23a.
>If he loses it is 13a-15a. NOT 5a-15a.
>
>This may change your calculation. But probably in the positive direction for
>a double.
>I too, often use Neil's numbers....which fail at long leads like this (so do
>most other formulae).....can you give details:-
>
>I have 9 8 6 5 4 3 2 2 2 2 2 2 ...then what?
>
>Rgds,
>
>Michael
>
It must be a bad week for me. Can't seem to get this right. Of
course if he loses at 8 he is 13a 15a, which gives him 58%. So he is
risking 58% to gain 5%. This says his redouble to 16 is borderline at
92% winning chances in the game. He has about 90% so the redouble
isn't right. But as you say, less wrong.
Another couple of corrections and maybe I will be OK.
I haven't seen any authoritative estimate for Neil-type numbers beyond
his own table. So I invented the rest. Neil's percentages run from
10 down, against the trailer's score away 3 4 5 6 8 11 15. Therefore
4% corresponds with 15 away. So I said, in that case maybe 3%
corresponds with 20 - I think you can see why - and 2% corresponds
with 26. I have no idea if it's right, but it can't be far off. Do
you know what Snowie's table is?
dk
Michael Howard
We are getting quite close to agreeing now.
I like your method although our figures are slightly different partially due
to the tables we use.
I have 62% equity at 12a-10a which is the risk. This is close.
I slightly disagree with your assessment of the potential gain though.
When doubling to 16 a win gives O the match and therefore 100%.
If we agree O's chance of winning the game/match from here is 90% -ish, then
the gain must be 10%.
Using your method and my figures I get : - 62/(62+10) = 86% = Double.
Whichever way you do it this is still a pretty close decision.
I don't know for sure about Snowie's table but it's a fair bet he uses Kit
Wolsley's table which is the most widely respected calculation.
Best rgds,
Michael
Donald Kahn <don...@salzburg.co.at> wrote in message
news:38fb1864...@enews.newsguy.com...
Michael Manolios
Donald, a 1296 Level 6 JF rollout with Standard Deviation only
0.001 gives 93.5% cubeless for O (and I absolutely trust it in bear -
off positions). Note that the cubeless winning percentage is what we
need, since after the redouble to 32, the cube will be dead.
This seems to make it a clear redouble. JF goes further than that
and suggests it is a drop, but I haven't check the tables yet. See if
this new winning percentage makes a difference in your calculations.
In article <38fb1864...@enews.newsguy.com>,
don...@salzburg.co.at (Donald Kahn) wrote:
> It must be a bad week for me. Can't seem to get this right. Of
> course if he loses at 8 he is 13a 15a, which gives him 58%. So he is
> risking 58% to gain 5%. This says his redouble to 16 is borderline at
> 92% winning chances in the game. He has about 90% so the redouble
> isn't right. But as you say, less wrong.
> Donald Kahn
>> >The Fish leads 12-2 in a 25pt match.
>> > +12-11-10--9--8--7-------6--5--4--3--2--1-+ O: JellyFish pips 14
>> > | O | | O |
>> > | | | O | +-+
>> > | | | O | |8|
>> > | | | O | +-+
>> > | | | O |
>> > | |BAR| |
>> > | | | |
>> > | | | |
>> > | | | |
>> > | | | X X X |
>> > | | | X X X X X |
>> > +13-14-15-16-17-18------19-20-21-22-23-24-+ X: Snowie2 pips 26
>> > JellyFish - 12, Snowie - 2 in a 25 point match.
>> > JellyFish to move. Cube Action?
>> >Double?
>> >Take/Drop?
>> >Redouble?
--
Michael Manolios (mann on FIBS, Glass on GG)
We play one and only money game session through our whole life...
Sent via Deja.com http://www.deja.com/
Before you buy.
Michael Howard
I wish I could find a difinitive calculation for match equity doubling ...
need to get Kit's book on the subject.
Also need a better table.
The Janowski formula gives 58% for 12-10 and 27% for 12-18
T 20-2 the formula fails for a lead of 20 points and runs out at 98%.
We have been using a conservative 95%.
Using normal logic though (huh) and Kit's advice....
If the chance of winning this game are around 90% (The Fish says 92%) then
we currently have 95 x 0.9 = 85% chance of winning the match.
By cubing to 16 we are 100 x 0.9 = 90% to win the match. Presumably any
decision that increases the match equity must be right??
I am trying to keep to simple round numbers that could be used across the
board.
Anybody got Kit's table for 25 point matches or more?
Cheers,
Michael
Michael Manolios <mm...@tee.gr> wrote in message
news:8dgud8$l92$1...@nnrp1.deja.com...
Donald Kahn
wrote:
>
>
> Donald, a 1296 Level 6 JF rollout with Standard Deviation only
>0.001 gives 93.5% cubeless for O (and I absolutely trust it in bear -
>off positions). Note that the cubeless winning percentage is what we
>need, since after the redouble to 32, the cube will be dead.
> This seems to make it a clear redouble. JF goes further than that
>and suggests it is a drop, but I haven't check the tables yet. See if
>this new winning percentage makes a difference in your calculations.
>
Absolutely. I knew I couldn't get it right. I have been doing it
with O's outfield checker one pip back, on the ten point. Must be a
jinx. I give it up.
dk
Walter Trice
>Thanks Michael for your input.
>
>I wish I could find a difinitive calculation for match equity doubling ...
>need to get Kit's book on the subject.
>Also need a better table.
The basic concept for any cube decision is this: figure out what the
decision gains when it works, and what it costs when it doesn't work. When
the odds in favor of it "succeeding" are better than the cost/gain ratio,
the decision is correct.
For example, here you have a decision to double (or not). The GAIN is your
match equity when you double and win minus your match equity when you don't
double and win. The COST is your match equity when you double and lose minus
your match equity when you lose after not doubling. The ODDS in favor is
(probablity of winning the position) divided by (probability of losing the
position.)
Working with odds in these decisions is equivalent to working with
probabilities, but it is conceptually simpler, and often results in simpler
arithmetic as well.
However, these simple ratios neglect future cube value, so they are in many
cases only a first approximation to the truth.
>
>The Janowski formula gives 58% for 12-10 and 27% for 12-18
>T 20-2 the formula fails for a lead of 20 points and runs out at 98%.
>We have been using a conservative 95%.
>
>Using normal logic though (huh) and Kit's advice....
>If the chance of winning this game are around 90% (The Fish says 92%) then
>we currently have 95 x 0.9 = 85% chance of winning the match.
Not quite. Assuming you're 95% by winning 8 points and 58% by losing 8
points, and that your cubeless chance of winning the game is 90%, the match
equity if you never double would be .9 X. 95 + .1 X .58 = 91.3%.
>By cubing to 16 we are 100 x 0.9 = 90% to win the match. Presumably any
>decision that increases the match equity must be right??
>
>I am trying to keep to simple round numbers that could be used across the
>board.
>
>Anybody got Kit's table for 25 point matches or more?
The book that Jake Jacobs and I co-wrote, called "Can A Fish Taste Twice As
Good?" contains, among other goodies, a match equity table for equal players
that goes out as far as 25 point matches. This table agrees reasonably well
with Kit Woolsey's 15 point match table, and needless to say I have a
certain amount of confidence in the Trice/Jacobs table, since I invented the
methodology behind it and did the programming.
The key number you need for this problem that you won't find in the Woolsey
table is match equity at 5-away/23-away. Our answer is 1.8%, which is
somewhat smaller than the 5% you were assuming. Our 13-away/15-away number
is 58.4%. So, for the doubling decision the RISK would be .584, and the GAIN
would be .018, which means that to have a proper double you'd have to be a
58.4-to-1.8 favorite in the position. That is, you'd need a probability of
winning of 58.4/(58.4+1.8) = 97%. And that's assuming no future cube value!
In reality your cubeless probabiity of winning would have to be even higher.
In practice what I usually do in this situation (big cube + big lead) is to
sit on my hands until I'm pretty damned sure that my opponent should pass,
then double and hope he blunders and takes out of desperation. I believe
that if this position had come up in a game of mine, then if I had thought
about it at all I would have decided not to double by realizing that my
opponent's equity at 23-away/5-away had to be a good deal less than 5% and
that his chances in the position are probably more than 5% (observing that
he has several turns to roll boxes and become the fave, plus I could roll
lousy twice and miss.)
-- Walter Trice
PS: Hope nobody objects to my plugging my own book here. It just happens to
be the most immediately available source of 25 point match equity numbers.
>
>Cheers,
>
>Michael
>
>Michael Manolios <mm...@tee.gr> wrote in message
>news:8dgud8$l92$1...@nnrp1.deja.com...
>>
>>
>> Donald, a 1296 Level 6 JF rollout with Standard Deviation only
>> 0.001 gives 93.5% cubeless for O (and I absolutely trust it in bear -
>> off positions). Note that the cubeless winning percentage is what we
>> need, since after the redouble to 32, the cube will be dead.
>> This seems to make it a clear redouble. JF goes further than that
>> and suggests it is a drop, but I haven't check the tables yet. See if
>> this new winning percentage makes a difference in your calculations.
>>
>> In article <38fb1864...@enews.newsguy.com>,
>> don...@salzburg.co.at (Donald Kahn) wrote:
>>
>> > It must be a bad week for me. Can't seem to get this right. Of
>> > course if he loses at 8 he is 13a 15a, which gives him 58%. So he is
>> > risking 58% to gain 5%. This says his redouble to 16 is borderline at
>> > 92% winning chances in the game. He has about 90% so the redouble
>> > isn't right. But as you say, less wrong.
>>
>> > Donald Kahn
>>
>> >> >The Fish leads 12-2 in a 25pt match.
>> >> > +12-11-10--9--8--7-------6--5--4--3--2--1-+ O: JellyFish pips 14
>> >> > | O | | O |
>> >> > | | | O | +-+
>> >> > | | | O | |8|
>> >> > | | | O | +-+
>> >> > | | | O |
>> >> > | |BAR| |
>> >> > | | | |
>> >> > | | | |
>> >> > | | | |
>> >> > | | | X X X |
>> >> > | | | X X X X X |
>> >> > +13-14-15-16-17-18------19-20-21-22-23-24-+ X: Snowie2 pips 26
>>
>>
>> >> > JellyFish - 12, Snowie - 2 in a 25 point match.
>> >> > JellyFish to move. Cube Action?
>> >> >Double?
>> >> >Take/Drop?
>> >> >Redouble?
>>
Michael Howard
Please tell us all where we can get your book. All the online stores do not
have it in their catalogues.
Amazon Bibliography mentions it but has a dead link and no facility for
ordering.
Sort this out and sell some books!!
We could send you cheques for the manuscript I guess? - pounds sterling OK?
Michael
Walter Trice <w...@world.std.com> wrote in message
news:Ft880...@world.std.com...
Walter Trice
Michael Howard wrote in message <8diq4c$lhq$1...@news8.svr.pol.co.uk>...
>Walter,
>
>Please tell us all where we can get your book. All the online stores do not
>have it in their catalogues.
Carol Joy Cole, prime mover of the Flint Area Backgammon Club in Michigan,
sells the 'Fish' book as well as virtually everything else related to
backgammon. E-mail is c...@tir.com. The book is $30, and you can also get the
(imho totally marvelous) computer program (alas, purely a DOS artifact) for
another $30. Don't know what international shipping would be.
Before posting this I went looking for an online version of Carol's 'store'
and found nothing. Asked her why not (via e-mail) and she said that for one
thing she had about as much business as she could handle anyway, so was in
no rush!
-- Walter Trice
Sam Pottle
>
> The book that Jake Jacobs and I co-wrote, called "Can A Fish Taste Twice As
> Good?" contains, among other goodies, a match equity table for equal players
> that goes out as far as 25 point matches. This table agrees reasonably well
> with Kit Woolsey's 15 point match table, and needless to say I have a
> certain amount of confidence in the Trice/Jacobs table, since I invented the
> methodology behind it and did the programming.
Would you be willing to share a thumbnail sketch of that methodology
with us?
I'm curious about the construction of match equity tables, and I've run
across
only two approaches in my reading. One is the Woolsey/Heinrich
approach, which
uses a database of human expert matches to estimate the distribution of
outcomes for games at any given score (leader wins 1 point, trailer wins
2,
etc.). This makes sense, but I can't go through it myself without
access to
the data.
The other approach is used by Tom Keith, and I've also found it used by
Kleinman. This model assumes (a) perfectly efficient cubes, and (b) a
fixed
gammon rate. The whole table can then be derived from these
assumptions.
The first assumption is awful, of course, but by underestimating the
gammon
rate (Keith uses 20%), you can get pretty good results, because the
errors
tend to cancel. Still, it's not the most intellectually satisfying
model.
If you are doing something different I would be most interested to hear
about
it. If what I'm asking for is what you're selling (in the book), then I
beg
your pardon. Perhaps I should just go buy it.
Sam
Walter Trice
Sam Pottle wrote in message <38FD557B...@lunabase.org>...
>Walter Trice wrote:
>>
>> The book that Jake Jacobs and I co-wrote, called "Can A Fish Taste Twice
As
>> Good?" contains, among other goodies, a match equity table for equal
players
>> that goes out as far as 25 point matches. This table agrees reasonably
well
>> with Kit Woolsey's 15 point match table, and needless to say I have a
>> certain amount of confidence in the Trice/Jacobs table, since I invented
the
>> methodology behind it and did the programming.
>
>with us?
Yes. There's nothing about it that I would consider a "trade secret," and
the mathematics, though complicated-looking when you put all the variables
in, is really elementary stuff.
In the special case where the players are equal in strength, all I am doing
amounts to a small modification of what Tom Keith describes. Your objection
was that he assumed perfect cube efficiency. Well, that can be fixed by the
simple expedient of adding "something" onto the cubeless probability of
winning that a player is assumed to need in order to acquire 1 incremental
match point worth of equity.
To make this concrete, let's briefly go way way back to Danny Kleinman's
analysis of 2-away/3-away. You can find this in his article "The Biased Cube
in the 3 Point Match", in "Vision Laughs at Counting With Advice to the
Dicelorn." Kleinman reasoned that with his assumed gammon rate of 20% the
leader needs a cpw of 1/3 to take, whereas the trailer needs 1/4. So the
leader would have to move from 1/2 to 3/4 to cash, but the trailer would
only have to move the smaller distance from 1/2 to 2/3. The ratio of these
"probability distances" is 2 to 3 -- that is (2/3 - 1/2) is 2/3 of (3/4 -
1/2), so (invoking the "continuous model") the trailer is 3/2 as likely as
the leader to gain the equity equivalent of 1 point. If the leader wins he'd
be at 75%, whereas if he loses he goes to 50%, so the leader's equity at the
start of the 2-away/3-away game is .4*.75 + .6*.5 = .6.
The modification amounts to assuming that a player always "loses his market"
by a certain amount of cpw. That is, the ratio (2/3 - 1/2) to (3/4 - 1/2) is
replaced by (2/3 + x - 1/2) to (3/4 + y - 1/2). In theory, the numbers x and
y should differ from each other, and should vary in different match and game
circumstances, but I wound up concluding that in practice it wouldn't hurt
too much to make them a constant. I refer to this adjustment as an
"overshoot factor" -- i.e., the amount of cpw by which a player is assumed
to "overshoot" his cash-point. The method of table construction described by
Norman Zadeh in his 1977 article in the journal Management Science uses the
same method to adjust for "discontinuity." [Note: Zadeh's method and mine
differ more significantly in the general case of unequal players, which we
are not considering here.]
Your other comment on Tom Keith's approach was that he assumed a fixed
gammon rate. Naturally when you define a model or write a program you don't
have to do this -- you just make it a variable, or a "user input." Since the
gammon rate has more influence on the match equity numbers than anything
else, making it a variable is a smart thing to do! The additional factors I
take into account amount to "bells and whistles" -- they include free drop
vig, backgammon rate, and an assumed reduction in the gammon rate whenever
the cube level goes up (for reasons like, for instance, when the cube goes
to 4 there's a good chance somebody just hit a shot in a contact bearoff and
CAN'T win a gammon.)
The "tricky" part of writing a match equity table program is the recursive
procedure Tom Keith describes, where you start by calculating take-points
for the highest possible live cube, and use these to keep adjusting things
for recube vig while working your way down to the center-cube situation.
This is all a bit sketchy. Feel free to ask specific questions if you think
the answers would make things clearer.
-- Walter Trice
Philippe Michel
>[...]
>have to do this -- you just make it a variable, or a "user input." Since the
>gammon rate has more influence on the match equity numbers than anything
>else, making it a variable is a smart thing to do! The additional factors I
>take into account amount to "bells and whistles" -- they include free drop
>vig, backgammon rate, and an assumed reduction in the gammon rate whenever
>the cube level goes up (for reasons like, for instance, when the cube goes
>to 4 there's a good chance somebody just hit a shot in a contact bearoff and
>CAN'T win a gammon.)
My understanding is that Snowie is able to do cubefull rollouts for
match play. Wouldn't it be possible to use it to do rollouts from the
starting position at -2:-1 and -3:-1, plug the results in its match equity
table, rollout at -3:-2, and so on...
This way, one wouldn't need any assumptions on gammon rates or winning
rates (which must vary depending on the score and not even be equal for
both players at lopsided scores) or other factors. In fact, one would be
able to estimate them.
Michael Howard
The rollout is from the current position to the end of the current game, not
the end of the match.
Thus you only get the winning chances for the game. It does not help at all
with calcualating the overall equity for the match which is completely
dependent on the distance from the winning post of each player at any given
time "factored" by their current winning chances in the game in hand.
Philippe Michel <Philipp...@syseca.thomson-csf.com> wrote in message
news:8e76k7$d5h$1...@news.syseca.fr...
> In article <FtB2E...@world.std.com>, Walter Trice <w...@world.std.com>
wrote:
> >[...]
> >have to do this -- you just make it a variable, or a "user input." Since
the
> >gammon rate has more influence on the match equity numbers than anything
> >else, making it a variable is a smart thing to do! The additional factors
I
> >take into account amount to "bells and whistles" -- they include free
drop
> >vig, backgammon rate, and an assumed reduction in the gammon rate
whenever
> >the cube level goes up (for reasons like, for instance, when the cube
goes
> >to 4 there's a good chance somebody just hit a shot in a contact bearoff
and
> >CAN'T win a gammon.)
Walter Trice
Philippe Michel wrote in message <8e76k7$d5h$1...@news.syseca.fr>...
the
>>gammon rate has more influence on the match equity numbers than anything
>>else, making it a variable is a smart thing to do! The additional factors
I
>>take into account amount to "bells and whistles" -- they include free drop
>>vig, backgammon rate, and an assumed reduction in the gammon rate whenever
>>the cube level goes up (for reasons like, for instance, when the cube goes
>>to 4 there's a good chance somebody just hit a shot in a contact bearoff
and
>>CAN'T win a gammon.)
>
>My understanding is that Snowie is able to do cubefull rollouts for
>match play. Wouldn't it be possible to use it to do rollouts from the
>starting position at -2:-1 and -3:-1, plug the results in its match equity
>table, rollout at -3:-2, and so on...
>
>This way, one wouldn't need any assumptions on gammon rates or winning
>rates (which must vary depending on the score and not even be equal for
>both players at lopsided scores) or other factors. In fact, one would be
>able to estimate them.
>
Absolutely! And I hope someone will do this and document the results.
However, there will still be some degree of statistical error in the
resulting table. (Until we have super-super-super-duper computers than can
do rollouts so fast that the errors can be pushed down to a decimal place
where it really doesn't matter.) I would like to use a table of that type
for testing a mathematical model. The advantage to being able to calculate a
table from a consistent, logical model, is that you can be sure that when
you see 'patterns' in the results you are not just looking at randomness and
round-off.
Also a Snowie-vs.-Snowie table wouldn't solve the problem of tables for
unequal players. And it is also possible that there will always be
differences between human play and perfect play that will imply different
match equities for the best human players, even equals, than for perfect
play. But still there is no question that a table generated as you describe
would be VERY interesting.
-- Walter Trice
Michael Howard
are you saying that Snowie 3 Pro CAN take a match score such as
25away-24away and rollout the entire match to a conclusion?
If so that is totally brilliant.
Except it would take a long time.
However, surely the newsgroup could each cooperate to build a table??
All those with the full Snowie could select match scores to work on and post
the results.
In a relatively short while we could have our very own rbg match equity
table.
Of course it would need some erudite person such as yourself to specify
parameters and ensure conformity etc.
How about it guys??
Michael
Walter Trice <walte...@worldnet.att.net> wrote in message
news:XGHN4.24770$PV.17...@bgtnsc06-news.ops.worldnet.att.net...
>
> Philippe Michel wrote in message <8e76k7$d5h$1...@news.syseca.fr>...
> >In article <FtB2E...@world.std.com>, Walter Trice <w...@world.std.com>
> wrote:
> >>[...]
> >>have to do this -- you just make it a variable, or a "user input." Since
> the
> >>gammon rate has more influence on the match equity numbers than anything
> >>else, making it a variable is a smart thing to do! The additional
factors
> I
> >>take into account amount to "bells and whistles" -- they include free
drop
> >>vig, backgammon rate, and an assumed reduction in the gammon rate
whenever
> >>the cube level goes up (for reasons like, for instance, when the cube
goes
> >>to 4 there's a good chance somebody just hit a shot in a contact bearoff
> and
> >>CAN'T win a gammon.)
Walter Trice
Michael Howard wrote in message <8e92jq$s9e$1...@news6.svr.pol.co.uk>...
>Walter,
>
>are you saying that Snowie 3 Pro CAN take a match score such as
>25away-24away and rollout the entire match to a conclusion?
Well, not exactly. First of all, I don't KNOW what any version of Snowie can
do because I don't own any of them. I am ASSUMING, based on what other
people have said, that you can set it up to roll out the starting position,
using the cube, from any match score, including Crawford and post-Crawford
scores. (A pretty big assumption, btw. I do know that with Jellyfish you
can't even set it up to roll out the starting position properly, i.e.
leaving out the doubles.)
Second, as Philippe Michel suggested in his original post, the efficient way
to go about it would be to roll out match situations rather than whole
multi-game matches. For example, to roll out the situation where player A
leads player B in the Crawford game at 1-away/2-away, you would tell the
computer to roll out the game at that score. The result would 3
probabilities for the possible outcomes: x=(prob. that A wins), y=(prob.
that B wins 1 point), z=(prob. that B wins 2 points). Then you would
calculate the match winning chance for A at 1-away/2-away Crawford as x +
y/2.
Likewise for the 'longer' scores: the procedure would always be to roll out
the game at the score, get the probability distribution for number of points
won or lost, then use these to compute the match equity in terms of other
match equities.
The big problem is having the computer time and power to get results that
are accurate enough to be interesting. If you'd like to kick off the
project, feel free to roll out 1-away/2-away until Snowie tells you the
standard deviation is less than .0001 ;-)
-- Walter Trice