Trailing when leader is 2 away is a mandatory double if.........
norman_newbie
I would like to try and prove here that when you are trailing in a
match and the leader is 2 away it is mandatory to double ONCE YOU ARE
IN THE DOUBLING WINDOW AND HAVE *ANY* MARKET LOSERS OR A SEQUENCE THAT
WILL CAUSE THE *LEADER* TO CASH THE GAME.
If this has already been covered excuse me but a post I sent to RGB
recently seemed to suggest that this idea is not yet clear.
Lets start with a well known situation.
Money game: You have a checker each on the 2 point and 5 point your
opponent has 2 checkers on his ace point. Equity is .056 and as this
is the last roll of the game you double because the cube is dead and
you simply double your stakes.
Now lets look at a theoretical situation. A race where your winning
chances are 60 pct so your equity is .2.If I were to say to you go
ahead and double because I'm taking the cube away after this roll.
You'd be a fool not to, as you will simply win .4 points per game.
Once I've taken the cube away it doesn't matter how you arrive at 100
pct whether in the mean time you go to 95 pct back to 15pct and then
all the way up to 100 pct. The only thing that matters is that you get
there.
Now lets look at the real world of backgammon. In a race where you
have 60 pct winning chances you wouldn't double because the equity you
give away by giving access to the cube solely to your opponent will
cost you more than leaving it in the middle where you both can access
it.
Now lets look at a specific match score. For example you are trailing
6 away 2 away.
When should you double? I say whenever you are WITHIN YOUR DOUBLING
WINDOW and have ANY market losers or sequences that would cause THE
LEADER to cash the game . At this match score the doubling window
according to Kit Woolsey's table opens at 58.8 pct and closes at 72.8
pct. (for the leader the doubling window opens at 71.4pct and closes
at 90 pct). So with 60 pct winning chances you should double because
risk against reward you are better off at the match score . But, I
here you all say no way, because you should wait until you are nearer
you opponents drop point for a more efficient double and wait until
you have some market losers.
However look at it this way. When you (or JellyFish) evaluates or
roles out a position and gives you an equity estimate this is
comprised of all the different outcomes of the games which are then
weighted and averaged to give an equity evaluation.
If your given equity in a position is .2 you can't improve your
equity merely by waiting .Sure it may improve but it could also get
worse and given the same position an infinite number of times to play
the equity will eventually go to .2. By doubling this position, it now
becomes a cubeless game in which you are favourite. The cube is dead
and can't be used against so in theory you have doubled your
favourable equity at no cost to you and this goes back to my example
above where I said I would take the cube away after the next roll.
What can happen in the game if you don't double and leave the cube in
the centre ?
The same sequences of rolls apply but the cube now becomes live.
As explained above your equity diverges from .2 by going down the
numerous branches but on an infinite number of games eventually comes
back to .2. so…..
If the next sequences leaves the equity within the doubling window
for both sides then it doesn't matter whether you doubled or not.
If you lose your market then you wish you would have doubled as you
have lost equity by not doing so.
If the LEADER gets to a cashing position, again you wish you would
have doubled as again you will have lost equity.
This is true because if the sequence gives you an equity of -.9 your
opponent will cash and claim 1 point so you've lost .1. If you had of
doubled, your loss would be only -.9 because he can't use the cube
against you to cash at the 2 level.
What happens when the position is not a race and gammons come into the
equation? Well gammons are hugely favourable to the trailer and this
causes his doubling window to open earlier. They are much less
favourable to the leader and by doubling the game you eliminate any
benefit the leader can gain from them, as gammons are only useful to
the leader if the game hasn't been doubled already. So by doubling you
again increase your equity.
One other subtle advantage of this theory is once you have eliminated
the cube the trailer can play his checkers as if it were double match
point. The benefit of course is that your can select plays which both
increase your gammon chances as well as your winning chances and as
gammons don't count against you it doesn't matter. Whereas the leader
cannot do the same.
When checking deja-news and some of the web sites I couldn't find any
material on this subject except some stuff on 2 away 4 away . One
interesting article by Kit Woolsey who mentioned that when playing
Mloner he noticed it was very aggressive doubling whilst trailing 4
away 2 away I wonder if the same applied when Mloner trailed by more
than 4 away ? Plus the benefit of it then switching into loner 1 point
mode at which it plays an awesome game..
The theory stated above is merely and extension of the 2 away 2 away
mandatory double which has been proved that if both players know what
they're doing is mandatory. It then follows that theory of the leader
being 2 away and you trailing merely states that once you are in the
doubling window i.e your risk against reward ratio at the match score
is in your favour you MUST double to lock in your favourable equity
and eliminate all unfavourable sequences which may follow. .
Kit Woolsey
normannewbie wrote:
: I would like to try and prove here that when you are trailing in a
: match and the leader is 2 away it is mandatory to double ONCE YOU ARE
: IN THE DOUBLING WINDOW AND HAVE *ANY* MARKET LOSERS OR A SEQUENCE THAT
: WILL CAUSE THE *LEADER* TO CASH THE GAME.
: If this has already been covered excuse me but a post I sent to RGB
: recently seemed to suggest that this idea is not yet clear.
<snip>
Rather than try to poke holes into Norman's seemingly persuasive
arguments, it is easier to show a simple counterexample.
Consider the following position:
13 14 15 16 17 18 19 20 21 22 23 24
+------------------------------------------+
| X X X | | O O O |
| | | O O O |
| | | O O O |
| | | O |
| | | |
| | | |
| | | |
| | | |
| | | X X X X X X |
| O | | X X X X X X |
+------------------------------------------+
12 11 10 9 8 7 6 5 4 3 2 1
Cube is in center, X is trailing 2 away, 4 away, X is on roll. Note that
O has 3 checkers off.
If X hits the shot, he wins about 86% of the time (at least that's what
my jellyfish rollout says, and I'm inclined to believe that is about right).
If X misses the shot, he always loses (yes, there are some rare sequences
where X could win, but these are so unlikely that it is reasonably
accurate to just assume that X always loses if he misses).
X has 27 hitting numbers. Thus, his winning chances are (27/36) * .85 =
.645 -- thus, X is well inside the doubling window.
Since O's take point is 20% at this match score, any hit is a market
loser, so X clearly has his share of market losers.
If X doubles and wins his equity is 50%. If he doubles and loses, his
equity is 0%. Therefore, since he has .645 winning chances, his overall
match equity if he doubles is 32.25%.
Suppose X doesn't double. 27/36 of the time he hits, and then he cashes,
to be behind 3 away, 2 away for 40% equity. 9/36 of the time he misses
and loses to be behind 1 away, 4 away, for 17% equity. Thus X's overall
equity if he doesn't double is:
(.75 * .40) + (.25 * .17) = .30 + .0425 = .3425
Since this is higher than X's equity if he double, it is correct not to
double.
What is happening here is that even though X has clear market losers (75%
of his rolls, to be exact), so if he hits the shot he wishes he had
doubled, he doesn't lose his market by a huge amount -- he still has to
win the game. On the other hand, when he misses the shot the double is
VERY costly, since he has no more chance in the match.
Kit
Kit Woolsey
Kit Woolsey (kwoo...@netcom.com) wrote:
: Cube is in center, X is trailing 2 away, 4 away, X is on roll. Note that
: O has 3 checkers off.
: If X hits the shot, he wins about 86% of the time (at least that's what
: my jellyfish rollout says, and I'm inclined to believe that is about right).
: If X misses the shot, he always loses (yes, there are some rare sequences
: where X could win, but these are so unlikely that it is reasonably
: accurate to just assume that X always loses if he misses).
: X has 27 hitting numbers. Thus, his winning chances are (27/36) * .85 =
: .645 -- thus, X is well inside the doubling window.
Oops -- should be .6375 (don't know what I was thinking). Of course,
this just makes my example even better (provided I didn't screw up
elsewhere).
Kit
: Since O's take point is 20% at this match score, any hit is a market
Kevin Bastian
Kit,
I've wondered this so many times, I decided to ask you!
Why do you use the term "match equity" to indicate one's winning
percentages in a match? It would seem to me that's a misleading
description. Game equity isn't winning percentages, it's what the game is
worth at any given moment, taking into account winning percentages,
gammons, backgammons. So I don't understand why winning percentages in a
match gets the term "equity". Why use the term "match equity" to describe
my 1-away 2-away chances as 70%? Isn't my match equity really .40?
(.70 wins - .30 losses = .40)
If I were playing someone a match for $100 and I was ahead 1-away 2-away
(Crawford), if my opponent offered to buy me out of the match, I should
want at least $40 (unless it was you, in which case I'd probably happily
call it even and consider myself lucky ;-)
So...there are really 2 questions:
1. Why use the term match equity to describe something that really isn't
equity, it's winning chances? (i.e., why not call it winning chances or
something like that?)
2. Why not use the equity concept in match calculations the same way it's
used in game calculations? (i.e., .40 instead of 70%)
Thanks,
KevinB
Kit Woolsey
Kevin Bastian (kbas...@DON.T.SPAM.ME.sctcorp.NO.SPAM.com) wrote:
: Kit,
: Thanks,
: KevinB
Couldn't really say. The term match equity has been used as long as I
can remember -- I don't believe I was responsible for coining it. And,
since it is what we have become used to using, that's why we continue to
use it.
Kit
Øystein Johansen
Kit Woolsey wrote:
> Consider the following position:
>
> 13 14 15 16 17 18 19 20 21 22 23 24
> +------------------------------------------+
> | X X X | | O O O |
> | | | O O O |
> | | | O O O |
> | | | O |
> | | | |
> | | | |
> | | | |
> | | | |
> | | | X X X X X X |
> | O | | X X X X X X |
> +------------------------------------------+
> 12 11 10 9 8 7 6 5 4 3 2 1
>
> Cube is in center, X is trailing 2 away, 4 away, X is on roll. Note that
> O has 3 checkers off.
>
> If X hits the shot, he wins about 86% of the time (at least that's what
> my jellyfish rollout says, and I'm inclined to believe that is about right).
>
If O has 3 checkers off, where is the last checker? Also off? (With 4
checkers off I will estimate ca. 79% wins for X after a hit)
-Øystein (oysteijo)
Chuck Bower
In article <01bd0533$10a8a280$6178...@REMOTE.sctcorp.com>,
Kevin Bastian <kbas...@DON.T.SPAM.ME.sctcorp.NO.SPAM.com> wrote:
(snip)
>1. Why use the term match equity to describe something that really isn't
>equity, it's winning chances? (i.e., why not call it winning chances or
>something like that?)
>
>2. Why not use the equity concept in match calculations the same way it's
>used in game calculations? (i.e., .40 instead of 70%)
If Yogi Berra read this newsgroup, he would respond: "It's
DejaNews all over again!". Larry Strommen (The Diceman) asked this
question about a year ago and it was answered by Walter Trice (with
a similar reason as Kit's, if my memory serves me).
This is a classic example of imprecise jargon becoming accepted.
It happens quite often in many fields (for example, I've heard
PhD physicists say "...weighs...grams". Of course they'd
likely mark their students down for this!) Better to spend your
time analyzing problems than to let these linguistic imperfections
rob you of valuable sleep time.
Chuck
bo...@bigbang.astro.indiana.edu
c_ray on FIBS
David desJardins
Kevin Bastian (invalid address) writes:
> 2. Why not use the equity concept in match calculations the same way it's
> used in game calculations? (i.e., .40 instead of 70%)
Of course, you could. It doesn't matter whether you use one or the
other. In game calculations, it does matter, because you sometimes
multiply the equity by the value of the cube, and that doesn't work on
percentages. But when you are doing match calculations it never matters
whether you use one or the other (since the value of the match never
changes), and the two statistics are related by a simple linear
transformation (weight = 2 * percentage - 1). So it's an arbitrary
choice, and percentages are what happens to have caught on.
David desJardins
Hank Youngerman
Probably one very good reason for using the term "Match Equity" as it
is commonly employed is that when you calculate cube decisions, these
are based on winning chances. So in this regard, the technically
correct form of match equity isn't meaningful.
It's also true that equity for a match and a game are inherently
different, in that a match has only two outcomes (+1, -1) while a game
can have 6 outcomes - or 36, if you factor the cube in.
Also, it is not uncommon for individual game positions to be sold,
settled, or played as a proposition, and in this case equity is
meaningful. But, while my experience in money play is quite limited,
I have never seen a match handled in this form. I suppose that it
occassionally happens that two players play a match for money, and
misestimate the amount of time they have to play, and rather than
continue later they agree to settle. But these are rather
far-fetched.
I must confess though that I had never considered the fact that the
two terms were not semantically identical.
James H. Cochrane
On 10 Dec 97 06:14:41 GMT, "Kevin Bastian"
<kbas...@DON.T.SPAM.ME.sctcorp.NO.SPAM.com> wrote:
>Kit,
>
>I've wondered this so many times, I decided to ask you!
>
>Why do you use the term "match equity" to indicate one's winning
>percentages in a match? It would seem to me that's a misleading
>description. Game equity isn't winning percentages, it's what the game is
>worth at any given moment, taking into account winning percentages,
>gammons, backgammons. So I don't understand why winning percentages in a
>match gets the term "equity". Why use the term "match equity" to describe
>my 1-away 2-away chances as 70%? Isn't my match equity really .40?
>
> (.70 wins - .30 losses = .40)
>
>If I were playing someone a match for $100 and I was ahead 1-away 2-away
>(Crawford), if my opponent offered to buy me out of the match, I should
>want at least $40 (unless it was you, in which case I'd probably happily
>call it even and consider myself lucky ;-)
But if this were a finals match for $100 in a tournament with no
second prize, you would want $70 for your position if you were to sell
to a bystander. That is, in this situation the winner collects BUT THE
LOSER DOESN'T PAY. In such a context the match equity is 70%, and the
term is used properly.
norman_newbie
On Wed, 10 Dec 1997 04:12:09 GMT, kwoo...@netcom.com (Kit Woolsey)
wrote:
>normannewbie wrote:
>: I would like to try and prove here that when you are trailing in a
>: match and the leader is 2 away it is mandatory to double ONCE YOU ARE
>: IN THE DOUBLING WINDOW AND HAVE *ANY* MARKET LOSERS OR A SEQUENCE THAT
>: WILL CAUSE THE *LEADER* TO CASH THE GAME.
>: If this has already been covered excuse me but a post I sent to RGB
>: recently seemed to suggest that this idea is not yet clear.
>
><snip>
>
>Rather than try to poke holes into Norman's seemingly persuasive
>arguments, it is easier to show a simple counterexample.
>
>Consider the following position:
>
>
> 13 14 15 16 17 18 19 20 21 22 23 24
> +------------------------------------------+
> | X X X | | O O O |
> | | | O O O |
> | | | O O O |
> | | | O |
> | | | |
> | | | |
> | | | |
> | | | |
> | | | X X X X X X |
> | O | | X X X X X X |
> +------------------------------------------+
> 12 11 10 9 8 7 6 5 4 3 2 1
>
>
>Cube is in center, X is trailing 2 away, 4 away, X is on roll. Note that
>O has 3 checkers off.
>
>If X hits the shot, he wins about 86% of the time (at least that's what
>my jellyfish rollout says, and I'm inclined to believe that is about right).
>
>If X misses the shot, he always loses (yes, there are some rare sequences
>where X could win, but these are so unlikely that it is reasonably
>accurate to just assume that X always loses if he misses).
>
>X has 27 hitting numbers. Thus, his winning chances are (27/36) * .85 =
>.645 -- thus, X is well inside the doubling window.
>
>Since O's take point is 20% at this match score, any hit is a market
>loser, so X clearly has his share of market losers.
>
>If X doubles and wins his equity is 50%. If he doubles and loses, his
>equity is 0%. Therefore, since he has .645 winning chances, his overall
>match equity if he doubles is 32.25%.
>
>Suppose X doesn't double. 27/36 of the time he hits, and then he cashes,
>to be behind 3 away, 2 away for 40% equity. 9/36 of the time he misses
>and loses to be behind 1 away, 4 away, for 17% equity. Thus X's overall
>equity if he doesn't double is:
>
>(.75 * .40) + (.25 * .17) = .30 + .0425 = .3425
>
>Since this is higher than X's equity if he double, it is correct not to
>double.
>
>What is happening here is that even though X has clear market losers (75%
>of his rolls, to be exact), so if he hits the shot he wishes he had
>doubled, he doesn't lose his market by a huge amount -- he still has to
>win the game. On the other hand, when he misses the shot the double is
>VERY costly, since he has no more chance in the match.
>
>Kit
Im back with my tail a little bit between my legs however as the
result of your example was a close call I investigated it a bit
further.First I'd like to check some maths with you.The doubling
window opens at 62.9 for X as I figured he risks 17 to gain 10.
Secondly I did some rollouts (14400) of the position results were as
follows cubless X wins 62.1 pct O wins 37.9 pct. If this is right then
the position doesnt fall into the doubling window and so doesnt
qualify. I'm not sure whats happening here but I think X does win some
positions even when he misses (maybe when O rolls 21 or 11 and X hits
again ? Or just by rolling big numbers.
Just to send me away could you prove BEYOND DOUBT theory is not
correct ?
Kit Woolsey
normannewbie wrote:
: Im back with my tail a little bit between my legs however as the
: result of your example was a close call I investigated it a bit
: further.First I'd like to check some maths with you.The doubling
: window opens at 62.9 for X as I figured he risks 17 to gain 10.
: Secondly I did some rollouts (14400) of the position results were as
: follows cubless X wins 62.1 pct O wins 37.9 pct. If this is right then
: the position doesnt fall into the doubling window and so doesnt
: qualify. I'm not sure whats happening here but I think X does win some
: positions even when he misses (maybe when O rolls 21 or 11 and X hits
: again ? Or just by rolling big numbers.
: Just to send me away could you prove BEYOND DOUBT theory is not
: correct ?
I had made an error when I copied the position -- my description which
said O has 3 men off was more accurate. Try the rollouts again with X
O having one more checker on his two point, and I think they will agree
pretty closely with my estimates.
Kit
David Montgomery
In article <kwoolseyE...@netcom.com> kwoo...@netcom.com (Kit Woolsey) writes:
[ Kit gave an example where player at -4 shouldn't double player at -2
despite being in doubling window and having a lot of market losers. ]
> What is happening here is that even though X has [...] market losers
> (75% of his rolls, to be exact), so if he hits the shot he wishes he
> had doubled, he doesn't lose his market by a huge amount
Here's a different example. Each side has 15 checkers on the acepoint,
side needing 4 points is on roll. Here again, the trailer shouldn't
double, despite being well inside the window (67%) and having market
losers (doublet/non-doublet).
This example is different because now you do have huge market losers
(doublet/non-doublet puts you at 91.4%), but they don't occur that
often (less than 14% of the time).
If you don't have both big market losers and a solid chance to lose
your market, it's rarely right to double.
(My calculations indicate that with 14 each on the ace, the trailer
still shouldn't double, but with 12 each on the ace, the trailer
should cube. Both of these results were so close that it probably
depends on what match equity table you use.)
David Montgomery
monty on FIBS
mo...@cs.umd.edu
David Montgomery
In article <348dc8ab...@news.demon.co.uk> norman newbie writes:
>I would like to try and prove here that when you are trailing in a
>match and the leader is 2 away it is mandatory to double ONCE YOU ARE
>IN THE DOUBLING WINDOW AND HAVE *ANY* MARKET LOSERS OR A SEQUENCE THAT
>WILL CAUSE THE *LEADER* TO CASH THE GAME.
>lets look at a theoretical situation. A race where your winning
>chances are 60 pct so your equity is .2.If I were to say to you go
>ahead and double because I'm taking the cube away after this roll.
>You'd be a fool not to, as you will simply win .4 points per game.
Yes. And if you had this kind of choice in *any* match or money
situation, that is, you are in the doubling window and after this roll
the cube will be frozen -- *no matter what you do* -- then you should
double.
The problem is in real life this isn't the choice you have. The choice
is to double now (which freezes the cube if your opponent needs 2) or
not to double, and maybe you or your opponent will double later.
This changes everything.
>If the LEADER [the player at -2] gets to a cashing position, again you
>wish you would have doubled
No. If things go against you, so that your opponent can double, you
should be glad you did not double. This is true whether you can take
the cube or not.
Here's an example. You need 4, your opponent needs 2. You think about
doubling but don't. The game goes against you.
If your opponent doubles and you take, you are *better* off, because now
you can redouble and win 4 points. If you had doubled, you could only
win 2. You're way better off not having doubled.
If your opponent doubles and you pass, you are *better* off, because
you have less than a 17% chance of winning the game, which means if you
had doubled you would have less than an 8.5% (17% * 50%) chance
of winning the match. Since you didn't double, you can pass and have
17%. You're way better off not having doubled.
Think about these two scenarios and you'll see that you really are
better off not having doubled when your game goes bad. The point is,
there really is a downside to doubling, not just an upside.
Another way of looking at it: If you would really wish that you had
doubled when things go against you, then you could just double! If
you start losing, you should just grant your own wish and double --
but you're not even in the doubling window, so this can't be right!
>One other subtle advantage of this theory is once you have eliminated
>the cube the trailer can play his checkers as if it were double match
>point. [...] Whereas the leader cannot do the same.
No, you select plays that maximize (your chance of winning) + (your gammon
price)*(your chance of winning a gammon). The leader tries to minimize
the same quantity.
>The theory stated above is merely and extension of the 2 away 2 away
>mandatory double which has been proved that if both players know what
>they're doing is mandatory. It then follows that theory of the leader
>being 2 away and you trailing merely states that once you are in the
>doubling window you MUST double to lock in your favourable equity
>and eliminate all unfavourable sequences which may follow. .
Doubling doesn't eliminate unfavorable sequences. It doubles the
losses that come from them.
---
The doubling window calculation assumes that this is your last
chance to double. In real life, it usually isn't.
Since you can still double later if you wait, waiting does better
than the doubling window calculation says. Since waiting does
better, it's usually not right to double near the bottom of
the doubling window. This is true whether you are playing a
money game, trailing an opponent that needs 2, or leading needing 2.
Alexander Nitschke
Here are some calculations to David's example:
14 each on the ace:
Cubeless game.
Game Winning Probability: 68.23305%
Cube in center.
Match Equity (no double): 34.52719%
Match Equity (double accepted): 34.11653%
Match Equity (double declined): 40.50000%
Player 1 should not double to 2.
Last roll doubling window: 65.7% to 81.0%.
This is a rather clear no double.
12 each on the ace:
Cubeless game.
Game Winning Probability: 69.75290%
Cube in center.
Match Equity (no double): 35.03619%
Match Equity (double accepted): 34.87645%
Match Equity (double declined): 40.50000%
Player 1 should not double to 2.
Last roll doubling window: 65.7% to 81.0%.
Now it is getting more close, but it still is no double.
10 each on the ace:
Cubeless game.
Game Winning Probability: 71.74503%
Cube in center.
Match Equity (no double): 35.86739%
Match Equity (double accepted): 35.87252%
Match Equity (double declined): 40.50000%
Player 1 has an initial double, which player 2 should take.
Last roll doubling window: 65.7% to 81.0%.
This is really on the hair line, since the match equities are almost
equal.
8 each on the ace:
Cubeless game.
Game Winning Probability: 74.52632%
Cube in center.
Match Equity (no double): 37.06242%
Match Equity (double accepted): 37.26316%
Match Equity (double declined): 40.50000%
Player 1 has an initial double, which player 2 should take.
Last roll doubling window: 65.7% to 81.0%.
Now it is again close, but it is a double.
6 each on the ace:
Cubeless game.
Game Winning Probability: 78.78087%
Cube in center.
Match Equity (no double): 37.91898%
Match Equity (double accepted): 39.39043%
Match Equity (double declined): 40.50000%
Player 1 has an initial double, which player 2 should take.
Last roll doubling window: 65.7% to 81.0%.
The leader has a surprisingly clear take here. This is a money game
pass. The last roll take point (which is also the real take point here,
because the leader can't redouble) is down to 19% because the second
point wins the match.
4 each on the ace:
Cubeless game.
Game Winning Probability: 86.11111%
Cube in center.
Match Equity (no double): 37.40278%
Match Equity (double accepted): 43.05556%
Match Equity (double declined): 40.50000%
Player 1 has an initial double, which player 2 should drop.
Last roll doubling window: 65.7% to 81.0%.
This is of course a clear pass.
On additional comment: In money game 15 checkers each on the ace point
is an initial double. So the trailer must be more reluctant to double
than in money game.
--
Alexander
Chuck Bower
In article <349503CD...@ww.tu-berlin.de>,
Alexander Nitschke <alexander...@ww.tu-berlin.de> wrote:
(big snip)
>On additional comment: In money game 15 checkers each on the ace point
>is an initial double. So the trailer must be more reluctant to double
>than in money game.
I believe you, but this surprises me. I thought the "rule" was to
wait until it's five roll vs. five roll for the initial double, and
four roll vs. four roll for the redouble. Could someone please elaborate.
Are we to throw out EVERYTHING we learned in the Dark Ages of BG???
Stephen Turner
Chuck Bower wrote:
>
> I thought the "rule" was to
> wait until it's five roll vs. five roll for the initial double, and
> four roll vs. four roll for the redouble. Could someone please elaborate.
> Are we to throw out EVERYTHING we learned in the Dark Ages of BG???
>
You're right about the redouble but not about the initial double. The 15 vs 15
position has an equity of 0.4503 if you don't double, 0.4519 if you do. I'd
be surprised if you could find a book which claimed the opposite. Robertie,
AFAIR, says that the 10 vs 10 position is a double, but doesn't talk about
higher positions.
--
Stephen Turner sr...@cam.ac.uk http://www.statslab.cam.ac.uk/~sret1/ Statistical Laboratory, 16 Mill Lane, Cambridge, CB2 1SB, England
"The Bishop of Huntingdon and Postman Pat each opened new school extensions"
(Cambridge Weekly News, 28-May-97)
David Montgomery
In article <349503CD...@ww.tu-berlin.de> alexander...@ww.tu-berlin.de writes:
[
At -4:-2, Alexander's program gets
No Double, 6+ roll position
Borderline, 5 roll position
Double-Take, 3-4 roll position
Double-Drop, 2 roll position.
(Pure n-roll positions, all on ace point.)
]
Thanks much for the results Alexander. Could you tell us
what match equity table you used? I would be interested
in seeing the results for
- Woolsey's "Inside BG"/_How to Play Tournament BG_ table
- Jacobs/Trice's _Fish_ table for equal players
- Ortega/Kleinman's _Cubes and Gammons_ table
I'm curious how much change you get from using the different
tables. Seeing this would give us a sense of the kind of
precision we're really talking about -- somewhat less than
the 5 printed decimal places I'm sure :-).
David Montgomery
> 8 each on the ace:
>Cubeless game.
> Game Winning Probability: 74.52632%
>Cube in center.
> Match Equity (no double): 37.06242%
> Match Equity (double accepted): 37.26316%
> Match Equity (double declined): 40.50000%
>
>Alexander
I couldn't get my numbers to come out like Alexander's, so I'm
wondering where the error is. Below are my calculations for a
4 roll position when the match trailer doesn't double. Maybe
someone can point out an error.
All match equities are presented from the match leader's point
of view. The relevant equities are:
-1:-4 Crawford, match equity=t. t is around .83
-2:-2 match equity=.50
-2:-3 match equity=f. f is around .60
t and f are intended to mnemonics for when the player needing [t]wo
wins and for when the player needing [f]our wins.
We have to work back from three later positions. First up is
Position A
----------
-2 on roll, 2 rolls against 1
1/6 roll doubles, obtain match equity t.
5/6 non-doublet, obtain match equity f.
=> t/6 + 5f/6
Position B
----------
-4 on roll, 3 rolls against 2
1/6 roll doubles, get position A, match equity is t/6 + 5f/6.
5/6 non-doublet, pass cube, match equity is t.
=> 5t/6 + (t + 5f)/36
Position C
----------
-2 on roll, 4 rolls against 3
1/6 roll doubles, get position B, match equity is 5t/6 + (t+ 5f)/36
5/6 non-doublet, take redouble, match equity is 1 - 0.5*(.78781) = .6061
.78781 comes from your cubeless chance of winning a 3 roll position.
.5 comes from the -2:-2 score that results when -4 wins
subtract from 1 to put in terms of the match leader's equity
=> 5t/36 + (t + 5f)/216 + .5051
Position D
----------
-4 on roll, 4 rolls against 4, player doesn't double.
1/6 roll doubles, cash position next time regardless, match equity is f
5/6 non-doublet, get position C, match equity is 5t/36 + (t+5f)/216 + .5051
=> f/6 + 25t/216 + (5t + 25f)/1296 + .4209
Here are some values for position D with different tables:
Woolsey, f=.60, t=.83
Match leader's equity is .6317, trailer's is .3683
Jacobs/Trice, f=.603, t=.822
Match leader's equity is .6313, trailer's is .3687
Ortega/Kleinman, f=.598, t=.831
Match leader's equity is .6315, trailer's is .3685
Alexander Nitschke
Match leader's equity is .6294, trailer's is .3706
Notice how close the values are for all the published tables.
Althought Alexander's numbers are only about .2% different
from mine, and all agree that this is a cube, the .2%
difference changes the cube action on some of the other
positions.
Alexander Nitschke
David Montgomery wrote:
>
> In article <349503CD...@ww.tu-berlin.de> alexander...@ww.tu-berlin.de writes:
> At -4:-2, Alexander's program gets
> No Double, 6+ roll position
> Borderline, 5 roll position
> Double-Take, 3-4 roll position
> Double-Drop, 2 roll position.
> (Pure n-roll positions, all on ace point.)
> ]
>
> Thanks much for the results Alexander. Could you tell us
> what match equity table you used? I would be interested
> in seeing the results for
>
> - Woolsey's "Inside BG"/_How to Play Tournament BG_ table
> - Jacobs/Trice's _Fish_ table for equal players
> - Ortega/Kleinman's _Cubes and Gammons_ table
>
> I'm curious how much change you get from using the different
> tables. Seeing this would give us a sense of the kind of
> precision we're really talking about -- somewhat less than
> the 5 printed decimal places I'm sure :-).
>
I used the match equity table from Tomas Szabo which is published in the
Internet on http://sg3.organ.su.se/~tsz/equity.html . I used the table
with a gammon rate of 26%, a gammon rate which is consistent with JF
evaluations and rollouts of the starting position.
I would like to implement other match equity tables, but from your three
tables mentioned I have only the first one from Kit Woolsey, and I don't
like this much although it is a quasi standard. The reasons for my
dislike are: The gammon rate is 20% - too low and giving significantly
different results especially at 4-away scores. Secondly the table has
only a precision of full percentages, not tenths of a percentage.
Obviously this is not precise enough for a program like mine.
I'm pleased if you send me the other two tables, and I try to find time
to bring them into my program. Then I will give the results with these
tables too.
--
Alexander
Stig Eide
>I used the table with a gammon rate of 26%, a gammon rate which is
>consistent with JF evaluations and rollouts of the starting position.
The old version of JF had 27% gammons. But I checked JF3.0, and it
estimates around 24% gammons. But then again, the human gammon rate
is different. I would guess higher. (Btw, you don't need an equity-
table with decimals, just that you don't know the exact gammon rate
should convince you about that).
It would be interesting to hear what the other programs estimated
gammon rate is.
Stig Eide
Alexander Nitschke
I made a lot rollouts of opening replies to all opening moves. The
summary of all these rollouts gives a gammon rate for the starting
position of 26.4%.
Of course you are right with your remark about the precision of the
equity table. But I have written a program for bear off positions which
gives equities to arbitrary precision. In this light I hope you
understand that I want a match equity table with decimal precision.
--
Alexander
Alexander Nitschke
David Montgomery wrote:
>
> In article <349503CD...@ww.tu-berlin.de> alexander...@ww.tu-berlin.de writes:
> At -4:-2, Alexander's program gets
> No Double, 6+ roll position
> Borderline, 5 roll position
> Double-Take, 3-4 roll position
> Double-Drop, 2 roll position.
> (Pure n-roll positions, all on ace point.)
> ]
>
> Thanks much for the results Alexander. Could you tell us
> what match equity table you used? I would be interested
> in seeing the results for
>
> - Woolsey's "Inside BG"/_How to Play Tournament BG_ table
> - Jacobs/Trice's _Fish_ table for equal players
> - Ortega/Kleinman's _Cubes and Gammons_ table
>
> I'm curious how much change you get from using the different
> tables. Seeing this would give us a sense of the kind of
> precision we're really talking about -- somewhat less than
> the 5 printed decimal places I'm sure :-).
>
> David Montgomery
> monty on FIBS
> mo...@cs.umd.edu
I used the match equity table from Tomas Szabo which is published in the
Internet on http://sg3.organ.su.se/~tsz/equity.html . I used the table
with a gammon rate of 26%, a gammon rate which is consistent with JF
evaluations and rollouts of the starting position.
I would like to implement other match equity tables, but from your three
Samuel Pottle
In article <673qit$b7t$1...@dismay.ucs.indiana.edu>,
Chuck Bower <bo...@bigbang.astro.indiana.edu> wrote:
>
> I believe you, but this surprises me. I thought the "rule" was to
>wait until it's five roll vs. five roll for the initial double, and
>four roll vs. four roll for the redouble. Could someone please elaborate.
>Are we to throw out EVERYTHING we learned in the Dark Ages of BG???
Actually, Kleinman discusses this in _Vision Laughs At Counting_, so it's
not exactly new theory. Until I read Kleinman I thought as you did, probably
because Robertie's treatment in _Advanced Backgammon_ only went up to the
5-roll v. 5-roll position. Robertie never actually says, though, that the
5 roll position is the earliest initial double.
How should we approach the double/no-double decision in a pure 8-roll v.
8-roll position?
Well, first of all, we're a solid favorite (about 67% cubeless), so an initial
double is at least worth thinking about. We lose our market whenever we roll
doubles and our opponent rolls non-doubles (about 14% of the time). These
are big market losers, too -- our winning chances in the resulting 6-roll v.
7-roll position are over 90%.
Average sequences (both players roll doubles, or both roll non-doubles) wash;
the resulting 6-roll v 6-roll or 7-roll v. 7-roll positions are still
double/take.
What about our poor sequences? We'll be sorry we doubled whenever we roll
non-doubles, opponent rolls doubles, and we roll non-doubles (about 11.5%).
Opponent will then be on roll in a 6-roll v. 6-roll position, leaving her
with a strong advantage, but still a roll or two away from a recube. In
fact, if we've doubled, opponent will hold the 2-cube, while if we haven't,
opponent will give us an initial double (which we'll take), so that we will
be holding a 2-cube. So in this variation, having doubled costs us
precisely the value of cube ownership of a 2-cube in a 6-roll v. 6-roll
position.
How much is that cube ownership worth? One clue that it's pretty valuable
is the fact that opponent shouldn't redouble before reaching a 4-roll v.
4-roll position, while she has a correct initial double already. Also, that
recube is pretty efficient when it comes, and it frequently does come.
So, which costs more, the big market loser, or the big loss of cube
ownership? Let's assume that they're about the same. Then the fact that
our bad sequences are only 5/6 as numerous as our good ones tips the balance
in favor of doubling.
Let's put some numbers to these notions. Our equity after a good sequence is
about .80 (1.60 on a 2-cube), so losing our market costs us about .60 when
it happens. Our equity with opponent on roll in a 6-roll v. 6-roll position
is about -.30 (-.60 on a 2-cube) if we own the cube, and about -.65 (-1.30)
if opponent owns the cube. So having doubled costs us about .70 in these
variations. Since .60 is bigger than 5/6 of .70, doubling turns out to be
correct by a small margin.
The margin is indeed quite small, as David Montgomery pointed out: .452 if
you double, .450 if you don't. But the double is technically correct.
Bonus questions:
(Assume we're allowed more than 15 checkers per side.)
-- What is the smallest number n for which an n-roll v. n-roll
position is not a proper initial double?
-- What is the smallest number n for which an n-roll v. (n+1)-roll
position is a take?
Sam Pottle (starbird)
"The difference between a human being and, say, a dinner plate,
is that the plate lacks the illusion of free will."
Philippe Michel
In article <6741rh$l...@twix.cs.umd.edu>,
David Montgomery <mo...@cs.umd.edu> wrote:
>In article <349503CD...@ww.tu-berlin.de>
>alexander...@ww.tu-berlin.de writes:
>[
> At -4:-2, Alexander's program gets
> No Double, 6+ roll position
> Borderline, 5 roll position
> Double-Take, 3-4 roll position
> Double-Drop, 2 roll position.
> (Pure n-roll positions, all on ace point.)
>]
>
>Thanks much for the results Alexander. Could you tell us
>what match equity table you used? I would be interested
>in seeing the results for
>
>- Woolsey's "Inside BG"/_How to Play Tournament BG_ table
>- Jacobs/Trice's _Fish_ table for equal players
>- Ortega/Kleinman's _Cubes and Gammons_ table
>
>I'm curious how much change you get from using the different
>tables. Seeing this would give us a sense of the kind of
>precision we're really talking about -- somewhat less than
>the 5 printed decimal places I'm sure :-).
I've a program similar to Alexander's that uses Woolsey's table.
It confirms yous calculations and it is clear that the 4-, 5- and 6-rolls
positions are close enough that the equity table used can make a difference.
Eval <s=-4 1|14 ... 14|1 s=-2>...
Undoubled: 34.23 Double(2) accepted: 34.12 declined 40.00
Value = 34.23
Eval <s=-4 3|12 ... 12|3 s=-2>...
Undoubled: 34.84 Double(2) accepted: 34.88 declined 40.00
Value = 34.88
Eval <s=-4 5|10 ... 10|5 s=-2>...
Undoubled: 35.65 Double(2) accepted: 35.87 declined 40.00
Value = 35.87
Eval <s=-4 7|8 ... 8|7 s=-2>...
Undoubled: 36.83 Double(2) accepted: 37.26 declined 40.00
Value = 37.26
Eval <s=-4 9|6 ... 6|9 s=-2>...
Undoubled: 37.34 Double(2) accepted: 39.39 declined 40.00
Value = 39.39
Eval <s=-4 11|4 ... 4|11 s=-2>...
Undoubled: 36.81 Double(2) accepted: 43.06 declined 40.00
Value = 40.00
Philippe Michel
In article <6767rb$rmu$1...@balder.adm.ku.dk>,
Stig Eide <ok...@stud.ibt.ku.dk> wrote:
> [...] (Btw, you don't need an equity-
>table with decimals, just that you don't know the exact gammon rate
>should convince you about that).
Moreover, the gammon rate (and the single wins rate as well) ought to
vary depending on the match score.
For what it's worth, I've had Jellyfish play out the -2:-1 Crawford
score about 200 times and the results were:
Leader wins: 55%
Trailer wins single game: 34%
Trailer wins gammon: 11%
giving the trailer 28% match winning chances.
This is an extreme case, and the sample is too small for these numbers
to be considered accurate, but it is clear to me that if Woolsey's table
(based on a sample of roughly similar size as far as I know) may not be
accurate even to 1%, computing a table from scratch starting with simply
a "precise" gammon rate isn't likely to be better.
On the other hand, massive rollouts with software more versatile than
stock Jellyfish may give interesting insights (it should be easy to do
this, at least for the -1:-n and possibly -2:-n scores).
Stig Eide
In article <6787ld$ess$1...@syseca.syseca.fr>,
mic...@syseca.fr (Philippe Michel) wrote:
>
>Eval <s=-4 3|12 ... 12|3 s=-2>...
>Undoubled: 34.84 Double(2) accepted: 34.88 declined 40.00
>Value = 34.88
Is the undoubled figure cubeless?
That is, does this program (only) check for double now or never?
Stig Eide
Philippe Michel
In article <678pu3$4ie$1...@balder.adm.ku.dk>,
No, this is the match winning chance if the player on roll doesn't
double now, but with a live cube. Cubeless match winning chances don't
have much sense if the cube isn't already dead.
Chuck Bower
In article <6789g4$fa5$1...@syseca.syseca.fr>,
Philippe Michel <mic...@syseca.fr> wrote:
(snip)
>For what it's worth, I've had Jellyfish play out the -2:-1 Crawford
>score about 200 times and the results were:
>
>Leader wins: 55%
>Trailer wins single game: 34%
>Trailer wins gammon: 11%
>
>giving the trailer 28% match winning chances.
>
>This is an extreme case, and the sample is too small for these numbers
>to be considered accurate, but it is clear to me that if Woolsey's table
>(based on a sample of roughly similar size as far as I know) may not be
>accurate even to 1%, computing a table from scratch starting with simply
>a "precise" gammon rate isn't likely to be better.
(snip)
Would you please give more detail. In particular, what does "play
out" mean. Did you perform an interactive rollout? If "yes" did you
always make the "best" play for your side ("best" determined using JF's
level-7 evaluation)?
Based on these numbers (which seem to be statistically lacking, as
you point out) trailer has upped his/her gammon fraction from near
1 out of 4 games to 1 out of 3, but traded some wins in the process.
One interesting observation that can be taken from your numbers is
that the player trying to avoid gammons (leader) picks up ground relatively
speaking compared to the player going for extra gammons. To see this,
note that when JFv3.0 level-6 plays against itself using money strategy
and at this match score, then the trailer wins the match about 32% of
the time.
The Woolsey-Heinrich table says 30% for this match score. Theoretical
tables based on 26% gammon fraction say 32%. Your number goes in the OTHER
direction! Could just be statisitics, though... Please give more details.
Philippe Michel
In article <678vn9$iao$1...@dismay.ucs.indiana.edu>,
Chuck Bower <bo...@bigbang.astro.indiana.edu> wrote:
>In article <6789g4$fa5$1...@syseca.syseca.fr>,
>Philippe Michel <mic...@syseca.fr> wrote:
>
> (snip)
>>For what it's worth, I've had Jellyfish play out the -2:-1 Crawford
>>score about 200 times and the results were:
>>
>>Leader wins: 55%
>>Trailer wins single game: 34%
>>Trailer wins gammon: 11%
>>
>>giving the trailer 28% match winning chances.
>>
>>This is an extreme case, and the sample is too small for these numbers
>>to be considered accurate, but it is clear to me that if Woolsey's table
>>(based on a sample of roughly similar size as far as I know) may not be
>>accurate even to 1%, computing a table from scratch starting with simply
>>a "precise" gammon rate isn't likely to be better.
> (snip)
>
> Would you please give more detail. In particular, what does "play
>out" mean. Did you perform an interactive rollout? If "yes" did you
>always make the "best" play for your side ("best" determined using JF's
>level-7 evaluation)?
I started a 2pt match with initial score -2:-1 Crawford, saved the initial
position, had Jellyfish complete it (with its "running man" button), reloaded
the saved position, etc... So it played at level 5 only, but used the right
gammon prices (or so I think - the result seems to suggest it anyway).
> Based on these numbers (which seem to be statistically lacking, as
>you point out) trailer has upped his/her gammon fraction from near
>1 out of 4 games to 1 out of 3, but traded some wins in the process.
Huh ? 11 out of 45 is still near 1 out of 4. What surprised me was that
the leader won so many games. 55% wins instead of 50 is a huge plus.
> One interesting observation that can be taken from your numbers is
>that the player trying to avoid gammons (leader) picks up ground relatively
>speaking compared to the player going for extra gammons. To see this,
>note that when JFv3.0 level-6 plays against itself using money strategy
>and at this match score, then the trailer wins the match about 32% of
>the time.
The trouble is that using money strategy is inadequate at this score.
In his original article about his equity table, Kit Woolsey points
that in his sample, the leader won a little more than 50% of the games
(but not significantly more, his sample was small too) in the
Crawford-even-number-to-go games. Summing all these results tend to
lessen the influence of the importance of gammons compared to the -2:-1
score only.
Moreover, I'm inclined to think that Jellyfish is at least as good at
finding a good balance between winning gammons and avoiding losses or
vice-versa at these scores with lopsided gammon prices than even the best
players in the 80s.
Regarding the fact that the leader picks up ground compared to the trailer,
it seems to me that in this situation, the onus is on the trailer. The fact
that the leader has to avoid gammons more than usual is a handicap, but
it is more than compensated by the fact that he can disregard his own gammons
and concentrate on the wins.
Maximizing his winning chances is an excellent way to avoid gammons that
he can apply all the time. On the other hand, the situations where the
leader has to sacrify winning chances to save the gammon are more sporadic
IMHO.
<hint>
As I wrote in my firt posting, it should be easy for a backgammon software
developper who has already implemented a rollout module to tweak it to
give statistically accurate equities and gammon rates for these match scores
</hint>
Stephen Turner
Samuel Pottle wrote:
>
> Bonus questions:
>
> -- What is the smallest number n for which an n-roll v. n-roll
> position is not a proper initial double?
>
It's n = 9 (the smallest n which can't occur in real backgammon).
33 v. 33 is a beaver.
> -- What is the smallest number n for which an n-roll v. (n+1)-roll
> position is a take?
>
n = 18.
25 v. 26 is an initial double but not a redouble.
30 v. 31 isn't even an initial double.
193 v. 194 is a beaver.
Of course anyone who plays backgammon with 387 chequers per side deserves
everything they get.
Tom
Stephen Turner wrote:
>
> Samuel Pottle wrote:
> >
> > Bonus questions:
> >
> > -- What is the smallest number n for which an n-roll v. n-roll
> > position is not a proper initial double?
> >
>
> It's n = 9 (the smallest n which can't occur in real backgammon).
> 33 v. 33 is a beaver.
>
> > -- What is the smallest number n for which an n-roll v. (n+1)-roll
> > position is a take?
> >
>
> n = 18.
> 25 v. 26 is an initial double but not a redouble.
> 30 v. 31 isn't even an initial double.
> 193 v. 194 is a beaver.
>
> Of course anyone who plays backgammon with 387 chequers per side
> deserves everything they get.
>
Enjoyed this immensely, but can you provide some insight as to the
methodology you used to arrive at these figures, ie. how did you come up
with the numbers?
Tom
Stephen Turner
Tom wrote:
>
> Enjoyed this immensely, but can you provide some insight as to the
> methodology you used to arrive at these figures, ie. how did you come up
> with the numbers?
>
I just wrote a little program to calculate them:
#include <stdio.h>
#define N (500)
int main(){
int i, j, t;
double eq1[N][N], eq2[N][N], eq0[N][N];
for (t = 0; t < N; t++) {
for (i = 0; i <= t; i++) {
j = t - i;
if (j == 0) {
eq0[i][j] = -1.0;
eq1[i][j] = -1.0;
eq2[i][j] = -1.0;
}
else if (i <= 1) {
eq0[i][j] = 1.0;
eq1[i][j] = 1.0;
eq2[i][j] = 1.0;
}
else {
eq0[i][j] = -5 * eq0[j][i - 1] / 6 - eq0[j][i - 2] / 6;
eq1[i][j] = -5 * eq2[j][i - 1] / 6 - eq2[j][i - 2] / 6;
eq2[i][j] = -5 * eq1[j][i - 1] / 6 - eq1[j][i - 2] / 6;
}
if (2 * eq2[i][j] > eq1[i][j]) {
if (2 * eq2[i][j] > 1.0) {
printf("%3d%4d [%7.4f:%7.4f:%7.4f] redouble/drop\n", i, j, eq0[i][j],
eq1[i][j], eq2[i][j]);
eq1[i][j] = 1.0;
eq0[i][j] = 1.0;
}
else {
printf("%3d%4d [%7.4f:%7.4f:%7.4f] redouble/take\n", i, j, eq0[i][j],
eq1[i][j], eq2[i][j]);
eq1[i][j] = 2 * eq2[i][j];
eq0[i][j] = 2 * eq2[i][j];
}
}
else if (2 * eq2[i][j] > eq0[i][j]) {
printf("%3d%4d [%7.4f:%7.4f:%7.4f] initial double/take\n", i, j,
eq0[i][j], eq1[i][j], eq2[i][j]);
eq0[i][j] = 2 * eq2[i][j];
}
else if (eq2[i][j] >= 0)
printf("%3d%4d [%7.4f:%7.4f:%7.4f] no double/take\n", i, j, eq0[i][j],
eq1[i][j], eq2[i][j]);
else
printf("%3d%4d [%7.4f:%7.4f:%7.4f] no double/beaver\n", i, j,
eq0[i][j], eq1[i][j], eq2[i][j]);
}
printf("\n");
}
return(0);
Chuck Bower
In article <6794ig$avi$1...@syseca.syseca.fr>,
Philippe Michel <mic...@syseca.fr> wrote:
>In article <678vn9$iao$1...@dismay.ucs.indiana.edu>,
>Chuck Bower <bo...@bigbang.astro.indiana.edu> wrote:
>>In article <6789g4$fa5$1...@syseca.syseca.fr>,
>>Philippe Michel <mic...@syseca.fr> wrote:
>>
>> (snip)
>>>For what it's worth, I've had Jellyfish play out the -2:-1 Crawford
>>>score about 200 times and the results were:
>>>
>>>Leader wins: 55%
>>>Trailer wins single game: 34%
>>>Trailer wins gammon: 11%
>>>
>>>giving the trailer 28% match winning chances.
>>>
>>>This is an extreme case, and the sample is too small for these numbers
>>>to be considered accurate, but it is clear to me that if Woolsey's table
>>>(based on a sample of roughly similar size as far as I know) may not be
>>>accurate even to 1%, computing a table from scratch starting with simply
>>>a "precise" gammon rate isn't likely to be better.
>> (snip)
>I started a 2pt match with initial score -2:-1 Crawford, saved the initial
>position, had Jellyfish complete it (with its "running man" button), reloaded
>the saved position, etc... So it played at level 5 only, but used the right
>gammon prices (or so I think - the result seems to suggest it anyway).
>
>> Based on these numbers (which seem to be statistically lacking, as
>>you point out) trailer has upped his/her gammon fraction from near
>>1 out of 4 games to 1 out of 3, but traded some wins in the process.
>
>Huh ? 11 out of 45 is still near 1 out of 4. What surprised me was that
>the leader won so many games. 55% wins instead of 50 is a huge plus.
Yes, I took the wrong denominator, but now I'm beginning to get
incredulous. See comments below...
>> One interesting observation that can be taken from your numbers is
>>that the player trying to avoid gammons (leader) picks up ground relatively
>>speaking compared to the player going for extra gammons. To see this,
>>note that when JFv3.0 level-6 plays against itself using money strategy
>>and at this match score, then the trailer wins the match about 32% of
>>the time.
>
>The trouble is that using money strategy is inadequate at this score.
I think we can all agree on that...
>In his original article about his equity table, Kit Woolsey points
>that in his sample, the leader won a little more than 50% of the games
>(but not significantly more, his sample was small too) in the
>Crawford-even-number-to-go games. Summing all these results tend to
>lessen the influence of the importance of gammons compared to the -2:-1
>score only.
>Moreover, I'm inclined to think that Jellyfish is at least as good at
>finding a good balance between winning gammons and avoiding losses or
>vice-versa at these scores with lopsided gammon prices than even the best
>players in the 80s.
I'm sure this is debatable, particularly at level-5, but I for
one am not on the debating team...
>Regarding the fact that the leader picks up ground compared to the trailer,
>it seems to me that in this situation, the onus is on the trailer. The fact
>that the leader has to avoid gammons more than usual is a handicap, but
>it is more than compensated by the fact that he can disregard his own gammons
>and concentrate on the wins.
>Maximizing his winning chances is an excellent way to avoid gammons that
>he can apply all the time. On the other hand, the situations where the
>leader has to sacrify winning chances to save the gammon are more sporadic
>IMHO.
Let me again summarize your result, based now on the (proper) calculation
of gammon fraction at 11/45 = 24%. At cubeless money play, the match trailer
here should win 50% of all games and 13% of all games will be gammon wins.
Here s/he wins only 45% of all games, and 11% of all games are gammon wins.
The trailer has lost ground in BOTH categories. SO, if this is due to a
change in strategy by the match leader, then:
WHY NOT USE THIS STRATEGY AT MONEY PLAY???? You can see that if a
person were to adopt this strategy (and Phillepe's logic holds) then assuming
only 1% gammon wins s/he will break even. Anything more than 1% gammon
wins makes him/her a NET WINNER. Here is a JF-like table (ignoring BG's)
which shows what I just said:
total gammon
wins wins
Player 1 55% 1%
Player 2 45% 11%
Player 1's cubeless equity is 0.55 + 0.01 - 0.45 - 0.11 = 0
Now you COULD argue that by adopting this strategy, a player gives
up almost all of his/her gammon win vig. But this is hard to believe.
Sometimes your best play to win the game also garners some gammons.
Note that at this kind of play (level-5), the standard deviation
on the total win/loss numbers is 3.5% for a 200 game trial. That is, the
95% confidence interval (2 s.d.) for player 1's wins is 48% < WIN < 62%.
In other words, 200 trials just isn't enough to draw the conclusions we've
been drawing (that is, the player working on avoiding a gammon has such
a huge advantage). Note that hand rollouts and the more sporting "props"
(for money) usually don't give statistically significant results, unless
the two sides are VERY far apart in reality. I think possibly the same
principle is being illustrated here.
Philippe Michel
In article <67cbll$cpu$1...@dismay.ucs.indiana.edu>,
Chuck Bower <bo...@bigbang.astro.indiana.edu> wrote:
>[...]
> Let me again summarize your result, based now on the (proper) calculation
>of gammon fraction at 11/45 = 24%. At cubeless money play, the match trailer
>here should win 50% of all games and 13% of all games will be gammon wins.
>Here s/he wins only 45% of all games, and 11% of all games are gammon wins.
>The trailer has lost ground in BOTH categories. SO, if this is due to a
>change in strategy by the match leader, then:
The strategy of the trailer certainly plays a role too. For instance, he
may try blitzes that he wouldn't consider if his gammons would be merely
normally valuable, costing him some wins.
Regarding the fact that he loses ground in gammons, it seems to me that
you just can't increase your gammon rate very much; maybe push it to 30%
(of your wins), and it is offset by the fact that you don't win as many
games.
> WHY NOT USE THIS STRATEGY AT MONEY PLAY???? You can see that if a
Isn't this what robots (especially TD-Gammon) tend to do, at least compared
to the style of play that was standard among human experts in 1990 ? Of
course, their tactical accuracy in opening- to middle-game play makes them
win quite a few gammons too...
> [...]
> Note that at this kind of play (level-5), the standard deviation
>on the total win/loss numbers is 3.5% for a 200 game trial. That is, the
>95% confidence interval (2 s.d.) for player 1's wins is 48% < WIN < 62%.
>In other words, 200 trials just isn't enough to draw the conclusions we've
>been drawing
Your conclusions, please :-). I merely pointed that at the distribution
of wins and gammons must vary depending on the score, and -2:-1 Crawford
is an extreme case. A chi2 test of the assumption that the trailer should
win 50% of the games (and even a little more "because he can take all his
chances whereas the leader has to avoid gammons" (isn't that conventional
wisdom ?)) shows that it has 80% chances or so to be wrong. This is not
satisfactory for a statistician but enough for a backgammon player to
consider dropping it, no ?
>(that is, the player working on avoiding a gammon has such
>a huge advantage). ^^^^^^^^^^^^^^^^^^^^^^^^^^^^
I think this is a red herring. The main goal of the leader is to win
the game. He can work at this at every roll. On the other hand, the
situations where he has to go out of his way to avoid the gammon
(say the aces point games, where he will tend to run early whereas
the trailer could stay util the end) are more rare.
For instance, take a 54 opening roll. According to Jellyfish level 7,
the trailer at -2:-1C should play 13/8 13/9 and the leader should split,
winning 0.5% more games than he would with the former play. These small
gains happen all the time and they add up.
I don't know how huge an advantage the leader has in this situation
(the 55% and 28% figures don't mean much), but I'm conviced that
(contrary to conventional wisdom) he is more comfortable than usual
and as a consequence that using 50/50 wins (and whatever gammon fraction)
to estimate the trailer equity at Crawford games (and especially -2:-1)
will tend to overvalue it.
Chuck Bower
In article <67dmn5$gm8$1...@syseca.syseca.fr>,
Philippe Michel <mic...@syseca.fr> wrote:
>In article <67cbll$cpu$1...@dismay.ucs.indiana.edu>,
>Chuck Bower <bo...@bigbang.astro.indiana.edu> wrote:
>
>>[...]
>> Let me again summarize your result, based now on the (proper) calculation
>>of gammon fraction at 11/45 = 24%. At cubeless money play, the match trailer
>>here should win 50% of all games and 13% of all games will be gammon wins.
>>Here s/he wins only 45% of all games, and 11% of all games are gammon wins.
>>The trailer has lost ground in BOTH categories. SO, if this is due to a
>>change in strategy by the match leader, then:
>
>The strategy of the trailer certainly plays a role too. For instance, he
>may try blitzes that he wouldn't consider if his gammons would be merely
>normally valuable, costing him some wins.
True, the trailer can (and does?) change strategy. But it looks like
here (i.e. the 200 game JF rollouts) that if we are to believe the numbers
then that change in strategy (by the trailer) was a mistake. S/he should
have just played standard (gammons count twice as much for BOTH sides).
That would lead to better results. (But I still doubt the statistics; see
below...)
(snip)
>
>> WHY NOT USE THIS STRATEGY AT MONEY PLAY???? You can see that if a
>
>Isn't this what robots (especially TD-Gammon) tend to do, at least compared
>to the style of play that was standard among human experts in 1990 ? Of
>course, their tactical accuracy in opening- to middle-game play makes them
>win quite a few gammons too...
I don't know how you came to this conclusion (that robots just try
and win games and don't worry about gammons--if that is what you are saying).
In fact, Kit is of the opposite opinion: he told me in an e-mail that he
felt that the BOT's are much more efficient (my words...) at extracting
gammons than the humans whose matches he used to create his table. That
is why the table is consistent with a lower gammon fraction (22-23%?) than
the numbers the BOTs come up with (at money play--26-27%). I'm sure this
is open to debate; I was just wondering what evidence you have for saying
that they concentrate on wins. (Please excuse me if I'm mis-interpreting.)
>> Note that at this kind of play (level-5), the standard deviation
>>on the total win/loss numbers is 3.5% for a 200 game trial. That is, the
>>95% confidence interval (2 s.d.) for player 1's wins is 48% < WIN < 62%.
>>In other words, 200 trials just isn't enough to draw the conclusions we've
>>been drawing
>
>Your conclusions, please :-). I merely pointed that at the distribution
>of wins and gammons must vary depending on the score, and -2:-1 Crawford
>is an extreme case. A chi2 test of the assumption that the trailer should
>win 50% of the games (and even a little more "because he can take all his
>chances whereas the leader has to avoid gammons" (isn't that conventional
>wisdom ?)) shows that it has 80% chances or so to be wrong. This is not
>satisfactory for a statistician but enough for a backgammon player to
>consider dropping it, no ?
I don't see the distinction between what a statistician believes is
sufficient and what a (non-statiscian) BG player believes (or should)
believe. I'm not a statistician, and 80% confidence doesn't carry much
weight, especially when the conclusion is counter-intuitive.
>>(that is, the player working on avoiding a gammon has such
>>a huge advantage). ^^^^^^^^^^^^^^^^^^^^^^^^^^^^
>
>I think this is a red herring. The main goal of the leader is to win
>the game. He can work at this at every roll. On the other hand, the
>situations where he has to go out of his way to avoid the gammon
>(say the aces point games, where he will tend to run early whereas
>the trailer could stay util the end) are more rare.
>
Consider this fact: Gammon losses are as bad as wins are good (assuming
simple loss is the status quo.) Here of course I am referring to post-
Crawford matches with the cube on 2 and trailer needing more than 2.
I do agree that much of the time the best play to win the game is also
the best play to avoid gammon. But when there is a conflict the leader
should compare the risk vs. gain (what else is new) and make the play
based on both wins and gammon losses being treated with EQUAL weight.
My conclusion is the same (am I hardheaded?) as last post. The
number of trials is insufficient to reach a conclusion that the match
leader has a 55-45 game winning advantage AND the match trailer has a
25% gammon fraction in this situation. Furthermore, these same numbers
are not consistent with what I consider optimal strategy by both sides
at these scores. I may not have convinced you of that, but maybe I'm
just a bad convincer...
Kevin Bastian
Thanks Stephen. This cleared it up for me ;-) LOL
Stephen Turner <sr...@cam.ac.uk> wrote in article
<34990569...@cam.ac.uk>...