Free drop
VSG
my opponent, being a switched on cookie, doubles me immediately. I drop.
Correct?
Free dropping is something I have known about but not really implemented until
recently. I have a few questions regarding it if i may.
1. Does the above strategy apply when the score is 4-3 or 4-2?
2. Does the concept apply to 3 point matches?
3. Is there anywhere a list of which opening rolls qualify for free dropping.
ie. 5-1, 4-1 or opponents opening 6-1, 3-1 etc.I've been guessing up until now
on what I consider to be poor openings.
4. Are there any exceptions to the rule so to speak? ie. Opponent is far
stronger/weaker.
General free drop info would be appreciated.
Alan Webb
Webby's Backgammon Site
Daniel Murphy
>5 point match. i'm leading 4-1 (post crawford) I have an opening roll of 5-1..
>my opponent, being a switched on cookie, doubles me immediately. I drop.
>Correct?
>
>Free dropping is something I have known about but not really implemented until
>recently. I have a few questions regarding it if i may.
See Backgammon Galore r.g.bg archive articles by Messrs, Bower, Kahn,
Karr and Woolsey:
http://www.bkgm.com/rgb/rgb.cgi?view+437
http://www.bkgm.com/rgb/rgb.cgi?view+84
http://www.bkgm.com/rgb/rgb.cgi?view+474
http://www.bkgm.com/rgb/rgb.cgi?view+451
________________________________________________
Daniel Murphy www.cityraccoon.com/
Humlebæk Backgammon Klub www.hbgk.dk/
Raccoon on FIBS www.fibs.com/
Raccoon on GamesGrid too
VSG
Daniel Murphy <rac...@cityraccoon.com> schrieb in im Newsbeitrag:
3742aa7a...@news.inet.tele.dk...
> On Wed, 19 May 1999 13:01:23 +0200, vsg...@t-online.de (VSG) wrote:
>
> >5 point match. i'm leading 4-1 (post crawford) I have an opening roll of
5-1..
> >my opponent, being a switched on cookie, doubles me immediately. I drop.
> >Correct?
> >
> >Free dropping is something I have known about but not really implemented
until
> >recently. I have a few questions regarding it if i may.
>
> See Backgammon Galore r.g.bg archive articles by Messrs, Bower, Kahn,
> Karr and Woolsey:
>
> http://www.bkgm.com/rgb/rgb.cgi?view+437
> http://www.bkgm.com/rgb/rgb.cgi?view+84
> http://www.bkgm.com/rgb/rgb.cgi?view+474
> http://www.bkgm.com/rgb/rgb.cgi?view+451
>
>
Oops, knew I should have checked there first. I apologise.
The general consencus was the decision is based more on who wins the opening
roll rather than the numbers on the dice. Interesting. The last 2 in the above
list, should anyone else wish to have a look, were the most relevant. Not as
complicated as I thought although I'm still not so sure about the odd away even
away business. ie 4-3 and 4-2 in a 5 pointer.
regards
Alan Webb
Webby's Backgammon Site
http://marina.fortunecity.com/frog/303/BGHome.htm
Ian Shaw
VSG wrote in message <7huf1a$81t$1...@news08.btx.dtag.de>...
>Not as
>complicated as I thought although I'm still not so sure about the odd away
even
>away business. ie 4-3 and 4-2 in a 5 pointer.
>
>
Ask yourself, "How many doubled games does my opponent need to win at this
match score?" You must assume that your opponent will always double
immediately.
If you are leading 1-away 3-away then your opponent needs to win two games:
one automatically doubled and then at 1-away 1-away. If, however, he doubles
and you drop, the score becomes 1-away 2-away. Your opponent now only needs
to win one automatically doubled. You have given away a whole game!!
If you are leading 1-away 4-away then your opponent also needs to win two
games: both automatically doubled. If the first game starts off badly for
you, you can drop to 1-away 3-away and your opponent still has to win two
games as above. This is why the drop is "free"; you have lost a game but
your opponent still needs to win the same number.
LESSON: YOU ONLY HAVE A FREE DROP YOUR OPPONENT IS AN EVEN NUMBER OF GAMES
FROM WINNING.
Hope this helps. If not, sit down with a pencil and paper and mark off
imaginary results from 1-away 8-away until it clicks - that's what I had to
do!
ian
VSG
Ian Shaw <ian....@riverauto.co.uk> schrieb in im Newsbeitrag:
6TA03.6408$%x.5057@wards...
Thanks, Ian that helped a lot.
Íve been dropping every poor start post crawford up to now :)
It is logical really. I wonder if Spock would be any good at Backgammon? :)
Robert-Jan Veldhuizen
>The general consencus was the decision is based more on who wins the opening
>roll rather than the numbers on the dice. Interesting. The last 2 in the above
>list, should anyone else wish to have a look, were the most relevant. Not as
>complicated as I thought although I'm still not so sure about the odd away even
>away business. ie 4-3 and 4-2 in a 5 pointer.
Maybe it helps to think "backwards". It all starts at 2-away 1-away
post-Crawford. It's easy to see that you have a free drop at that score;
if you take, it's for the match, if you drop, the next game is for the
match. So whenever you're less than 50%, drop.
Now consider 3-away 1-away (P-C). The trailer doubles. If you drop,
you'll get at the above score, with just ONE game left to play (either
immediately or after an immediate double/free drop). If you take, the
trailer will still have to win THIS game AND the next (except for
gammons). So, you don't have a free drop at this score. Only when the
trailer doubles you very late with big winning/gammon chances, you
should still drop.
Now 4-away 1-away (P-C), the trailer doubles. If you drop, you'll get at
the above score; so normally with two games left for the trailer to win.
If you take, he has to win THIS game AND the next (again barring
gammons), so that's again two games. Conclusion: you have a free drop!
The pattern goes on... free drops at trailer even away, not at odd away.
Related to this is the Crawford game itself, let's say the score got to
4-0 in a 5pt match, so 1-away 5-away Crawford. Should the leader care
about losing a gammon in this game?
Losing single: 1-away 4-away P-C, trailer has to win two games, leader
has ONE free drop.
Losing gammon: 1-away 3-away P-C, trailer has to win two games, leaders
doesn't have a free drop anymore.
So, the only difference is the free drop! Now that's not worth much, so
in practice the leader can make moves that give you higher winning
percentages at the cost of losing more gammons, if the choice pops up!
Sometimes very useful, and easy to forget... Always look at the
matchscore at the start of the Crawford game; is the trailer odd away
you don't care much about losing a gammon. If he's even away, you care a
lot about losing a gammon!
--
Robert-Jan/Zorba
** New email address: R.J.Vel...@cable.A2000.nl **
Don Woods
> Ask yourself, "How many doubled games does my opponent need to win at this
> match score?" You must assume that your opponent will always double
> immediately.
>
> If you are leading 1-away 3-away then your opponent needs to win two games:
> one automatically doubled and then at 1-away 1-away. If, however, he doubles
> and you drop, the score becomes 1-away 2-away. Your opponent now only needs
> to win one automatically doubled. You have given away a whole game!!
>
> If you are leading 1-away 4-away then your opponent also needs to win two
> games: both automatically doubled. If the first game starts off badly for
> you, you can drop to 1-away 3-away and your opponent still has to win two
> games as above. This is why the drop is "free"; you have lost a game but
> your opponent still needs to win the same number.
>
> LESSON: YOU ONLY HAVE A FREE DROP YOUR OPPONENT IS AN EVEN NUMBER OF GAMES
> FROM WINNING.
I understand the logic of this explanation, but I always find one thing
lacking from it.
Presumably the "free drop" is worth something. I.e., you're happier if
you're (say) 1-away 6-away than if you're 1-away 5-away, because you
have a free drop as described above.
But that means that if your opponent is an ODD number of points from
winning, e.g. 1-away 7-away, then you DO gain something by dropping
compared to playing out a losing position, because you end up 1-away
6-away instead of 1-away 5-away, and you've gained a free drop.
So, to take an extreme case, if your opponent waited a long time to
double in a post-Crawford game, 1-away 7-away, such that your chances
of winning are very small (for some definition of "very small"), you
should drop. E.g., if he's down to two men on his 3 and 1 and you
have four men on your 6, and he finally offers the double, you should
(probably?) drop even though you have a 1 in 650 chance of winning.
I _assume_ that the value of the free drop is small enough that there
is no combination of opening rolls where you should drop at odd-away
just to get the new free drop. But I don't actually know the value of
the free drop, so I'm not totally sure. If the opponent opens 3-1 and
you respond 5-2 or some such, is that still worth taking? How bad does
a position have to be before it turns into a post-Crawford drop?
-- Don.
-------------------------------------------------------------------------------
--
-- Don Woods (d...@navilinks.com) NaviLinks provides real-time linking.
-- http://www.navilinks.com/~don I provide personal opinions.
--
Sander van Rijnswou
> > LESSON: YOU ONLY HAVE A FREE DROP YOUR OPPONENT IS AN EVEN NUMBER OF GAMES
> > FROM WINNING.
>
> Thanks, Ian that helped a lot.
>
> Íve been dropping every poor start post crawford up to now :)
There is one more thing about that the free drop: You get at most
one in every game. Suppose you are say 1a8a and
you use your free drop the score becomes 1a7a. Since your
opponent will always win an even number of points his
score will always stay odd. You never have a chance to
free drop again.
So when a score of 1a8a comes up you'll have to ask yourself
when am I going to use my free drop. Well, when
you either get off to a really bad start. OR when the score
is 1a2a, in that particular case the free drop is worth
a little bit more to boot because this game is much
more important than a game at 1a8a.
Sander
Julian Hayward
writes
>5 point match. i'm leading 4-1 (post crawford) I have an opening roll of 5-1..
>my opponent, being a switched on cookie, doubles me immediately. I drop.
>Correct?
Yes.
>1. Does the above strategy apply when the score is 4-3 or 4-2?
At 4-3 yes, at 4-2 no. If the trailer needs an even number of points to
win (2n, say), he must win n games with the cube on 2. Giving up one
point means he still needs to win n games.
>2. Does the concept apply to 3 point matches?
Yes, at 2-1 post-Crawford.
>3. Is there anywhere a list of which opening rolls qualify for free dropping.
>ie. 5-1, 4-1 or opponents opening 6-1, 3-1 etc.I've been guessing up until now
>on what I consider to be poor openings.
If you get the opening roll, there are only three rolls which justify
dropping: 2-1, 4-1, 5-1. All others leave you at least a slight
favourite. If your opponent makes the opening move you get doubled after
two moves, so a bit too long for an exhaustive list but it should be
fairly easy to guess who has had the better start.
>4. Are there any exceptions to the rule so to speak? ie. Opponent is far
>stronger/weaker.
The opponent's strength really doesn't come into it. The free drop is
simply a choice of - would you rather play this game out as it stands,
or start again with negligible cost? Drop if you feel your position is
worse than average, and you expect to do better given another chance.
(The cost is that if you drop, you won't have a free drop available in
subsequent games, but it's only worth about 1%).
--
Julian Hayward 'Booles' on FIBS jul...@ratbag.demon.co.uk
+44-1344-640656 http://www.ratbag.demon.co.uk/
---------------------------------------------------------------------------
"Doh! Stupid poetic justice!" - Homer Simpson
---------------------------------------------------------------------------
Julian Hayward
>I _assume_ that the value of the free drop is small enough that there
>is no combination of opening rolls where you should drop at odd-away
>just to get the new free drop. But I don't actually know the value of
>the free drop, so I'm not totally sure. If the opponent opens 3-1 and
>you respond 5-2 or some such, is that still worth taking? How bad does
>a position have to be before it turns into a post-Crawford drop?
The free drop is, IIRR worth about 1% match-winning chance (from expert
comment on old threads). So, to be worth giving the opponent a whole
game towards his target means the opponent must have about 1% MWC and
after being dropped has about 2% - assuming opponents are equal that
means you must be winning by something colossal, I think it comes to
about 1-away 16-away?
Christopher D. Yep
>
> In article <q490akd...@clari.net>, Don Woods <d...@clari.net> writes
>
> >I _assume_ that the value of the free drop is small enough that there
> >is no combination of opening rolls where you should drop at odd-away
> >just to get the new free drop. But I don't actually know the value of
> >the free drop, so I'm not totally sure. If the opponent opens 3-1 and
> >you respond 5-2 or some such, is that still worth taking? How bad does
> >a position have to be before it turns into a post-Crawford drop?
>
> The free drop is, IIRR worth about 1% match-winning chance (from expert
> comment on old threads). So, to be worth giving the opponent a whole
> game towards his target means the opponent must have about 1% MWC and
> after being dropped has about 2% - assuming opponents are equal that
> means you must be winning by something colossal, I think it comes to
> about 1-away 16-away?
>
I assume you mean that the leader should drop post-crawford at
1-away/17-away (instead of 1-away/16-away) if he's an underdog after the
first roll - thus conceding 1 pt. but gaining a free drop.
Actually the free drop would be worth about 1% (actually I think closer
to 1.5%) if the leader had the privilege of restarting every game
(*without* conceding any points) post-crawford in which he found himself
the underdog after one roll. However, at 1-away/even-away
post-crawford, the leader only has one free drop to use. If the free
drop were really worth 1% in absolute match equity across all match
scores, then at 1a/17a post-crawford, it would actually be correct for
the leader to always drop after the first roll [since at best he would
be a slight favorite, e.g. 58% in the current game], even if he is the
favorite!
One way to think about the free drop is that it allows the leader to
restart one game in which he has started out badly. However, he can
only restart one game. At 1-away/2k-1 away, the probability of the
trailer winning the match is, *without* considering the possibility of
gammons [considering gammons doesn't really change any of the argument
below], (1/2) * (1/2) * ... * (1/2) = (1/2)^k. If we look at a
particular match that the trailer wins and look at his probability of
winning each game at the *moment* of each of his doubles [after the
first roll], his winning chances in each game [after the first roll]
might be something like 0.5, 0.45, 0.55, 0.48, 0.52, 0.53, 0.47, etc.
which means that his chance of winning the match is (0.5) * (0.45) *
(0.55) * (0.48) * (0.52) * (0.53) * (0.47) ... etc. [this is not exact
but close enough - I don't mean that the number of games above 50% will
exactly equal the number of games below 50%; the reason why this product
is less than (0.5)^k is because we didn't allow for random variation,
which is of course one reason why the trailer has a chance to win the
match in the first place]. From this we see that at best, the free drop
allows the leader to change one of these probabilities from slightly
above 50% to exactly 50%. So in absolute terms, the value of the free
drop goes to zero as k goes to infinity.
It's actually not correct to drop at 1-away/2k-1 away [i.e. 1a/odd-away]
post-crawford after any sequence of opening rolls. See another post in
this thread that I'll (try to) write later tonight.
Chris
Christopher D. Yep
> Actually the free drop would be worth about 1% (actually I think closer
> to 1.5%) if the leader had the privilege of restarting every game
> (*without* conceding any points) post-crawford in which he found himself
> the underdog after one roll.
Please ignore this last statement - it's incorrect. For 1a/2k-away
post-crawford, the value of the leader having the option of restarting
(*without* conceding any points) *every* game in which he is the
underdog after the first move is worth less and less as k increases. In
fact, it's value (in absolute match equity) also approaches 0 as k ->
infinity, just as the value (in absolute match equity) of the free drop
in normal backgammon approaches 0 as k -> infinity.
I'm not sure why I wrote what I did - I posted shortly after waking up
this morning. Now that I've had time to review it, I think everything
else in the original article is correct.
Chris
Julian Hayward
writes
>
>I assume you mean that the leader should drop post-crawford at
>1-away/17-away (instead of 1-away/16-away) if he's an underdog after the
>first roll - thus conceding 1 pt. but gaining a free drop.
>
Here's another thought. Suppose you are at some score like 1-away
16-away. You have a free drop, but the match has potentially quite a
long way to go. How bad does your start have to be to want to use the
free drop rather than saving it for a future start that is even worse?
Do you always use it at the earliest opportunity? My gut feeling is that
you use it whenever you start the underdog, but I'll have to sit down
now and work it out properly...
Christopher D. Yep
this will be formatted correctly. If not, I hope it's at least readable.]
Sander van Rijnswou wrote:
> So when a score of 1a8a comes up you'll have to ask yourself
> when am I going to use my free drop. Well, when
> you either get off to a really bad start. OR when the score
> is 1a2a, in that particular case the free drop is worth
> a little bit more to boot because this game is much
> more important than a game at 1a8a.
I think it's not always clear which games are "worth" more. For positions
reached in a match in which the volatility of the game *result* (measured in
absolute points, not just points on a 1-cube) is high, it can be said, in
general, that the rest of the game is "worth" more than other games. Examples
of such positions are those in which
the cube has reached (or has good
potential to reach) a high level (e.g. >= 4) or in which there are a lot of
gammons. In post-crawford play, however, I would argue that the 1a/1a game is
actually *less* important than earlier post-crawford games. Here's why:
Suppose it is 1a/(2k-1)-away p.c. Without accounting for the possibility of
gammons, the trailer's probability of winning the match (call it MWC, i.e.
match winning chances) is (1/2)^k. Now look at two scenarios:
Scen 1: The trailer always has a lapse of concentration in the 2k-1/1a p.c.
game. However, if he wins the 2k-1/1a game, he plays well the rest of the
match.
Scen 2: The trailer plays well in all games, except when he reaches 1a/1a. If
he gets this far, then he will have a lapse of concentration in the 1a/1a
game. Let's define "lapse of concentration" (LOC) to mean that the trailer
only wins 40% of the time. Assume that he wins 50% of the time if he doesn't
have a LOC. In both scenarios, the trailer's MWC is (0.4)*(0.5)^(k-1)! It
doesn't matter when he has a LOC as long as he only has one LOC! [Similarly,
if the leader has one LOC game while the trailer plays perfectly, then the
trailer's MWC is (0.6)*(0.5)^(k-1), regardless of which game is the leader's
LOC game.] Thus, I claim that all p.c. games have equal value (when there is
no possibility of gammons). Intuitively then, when we consider gammons, the
1a/1a game becomes worth less than previous p.c. games. Gammons don't count
in the 1a/1a game, whereas they do count for the trailer in previous p.c.
games. Let's try to quantify this with an example:
Suppose that 20% of all wins are gammons. Define a LOC game to mean that a
player only has a 40% chance to win the game, i.e. 8% gammon wins, 32% single
wins, 48% single losses, and 12% gammon losses. Assume that if not a LOC game
that each player wins 10% gammon wins and 40% single wins.
Now, suppose that the score is 3a/1a p.c.
Scen A: The trailer (T) has a LOC at the 3a/1a game, but not in the 1a/1a
game. His MWC is 8%(100%) + 32%(50%) = 24%.
Scen B: T plays perfectly at 3a/1a, but has a LOC in the 1a/1a game. His MWC
is 10%(100%) + 40%(40%) = 26%.
Scen C: T never has a LOC. The leader (L) has a LOC at 3a/1a, but not in the
1a/1a game. T's MWC is 12%(100%) + 48%(50%) = 36%. Thus, L's MWC is 64%.
Scen D: T never has a LOC. The leader (L) plays perfectly at 3a/1a, but has a
LOC at 1a/1a. T's MWC is 10%(100%) + 40%(60%) = 34%. Thus, L's MWC is 66%.
We see that scenario B is better than scenario A and scenario D is better than
scenario C from the point of view of the player having the LOC. Thus, I
conclude that the 1a/1a game is "worth" less than the 3a/1a game. With similar
calculations, we can show that the 1a/1a game is worth less than the
(2k-1)-away/1a game p.c. for any k > 1.
Regarding the free drop available at 1a/2k-away:
Although I didn't do the math, each post-crawford game is "worth"
approximately the same except the 1a/1a game which is worth less than
the previous p.c. games. The free drop can be thought of as a mini-LOC for the
trailer. Suppose that it is 1a/8a p.c. and that after the first roll the
leader is a very slight underdog (say 49.9% to win with a constant 20%
gammon fraction for both sides). Then it's probably (actually almost
surely) correct for the leader to "save" his free drop. The reason is that
the gain if he uses it now is tiny (changing the current game from 49.9%
to 50%) whereas if he saves it, he has a good chance of using it on
another game when he will probably be an even bigger underdog. Since the
1a/6a and 1a/4a games are just as "important" as the 1a/8a game (and
the 1a/1a game is just slightly less important), if/when the leader does
use his free drop, it will be of more value.
Theoretical interest:
If the score is 2k-away/1-away post-crawford and k is extremely large, then
the leader's free drop strategy should be to save his free drop for the worst
possible opening sequence. This is because when k is extremely large there
is almost a 100% chance this this opening sequence will occur before
the trailer has won 2k points. Since each p.c. game before 1a/1a is
"worth" approximately the same, the leader gets his best value by waiting
for this opening sequence (i.e. probably Trailer 3-1 followed by a bad roll
for the leader, at which point the leader is perhaps a 42%
underdog).
Practical interest:
Suppose the match score is 2k-away/1-away post-crawford. For k = 1, one
should always use the free drop if an underdog. For small k > 1, it's
usually correct to use the free drop if an underdog since it's far from
certain that the leader will get another chance to use it (i.e. he may be
the favorite (after one roll) in all of his successive games). Furthermore, as
discussed above, the 1a/1a game is 'worth" less than previous p.c. games. Thus
if the leader were *psychic* and knew that he would only get two chances to
use the free drop, once at 8a/1a and once at 1a/1a, and both as a 45%
underdog (assume a constant 20% gammon rate for both sides), then he
would prefer to use it at 8a/1a since the 8a/1a game is more important than
the 1a/1a game. Thus, in general, it's usually correct to use the free drop
as an underdog before the 1a/1a game.
The exception is that if the leader is only a very slight underdog, he may
wish to save his free drop (as previously disucssed in the example of
the leader being only a 49.9% underdog in the 8a/1a game). As k gets
larger, the situation approaches more and more that described
in the "theoretical interest" paragraph. Specifically, as k gets
larger, the leader is more and more willing to "save" his free drop (e.g.
for some k he will only use his free drop if worse than a 49% underdog,
for some larger k he will use it if worse than 48%, for some even larger
k he will only use it if worse than 47%, etc.) until finally for
extremely large k he should save it until the *worst* opening
roll sequence occurs.
Snowie error reports:
There was a suggestion awhile ago that perhaps Snowie should report match
error rates. Specifically if a player made a move that lowered his MWC by 10%
versus the correct move, then he should be penalized .10 in the match error
sum (presumably the sum of the errors are then divided by the number of
moves made and then multiplied by some factor based on the match length
in order to get a "match error rate" that could be used across all match
lengths). However, based on the discussion in this article, this error
method is not accurate. For example, suppose the score is 3a/1a p.c.
Again consider two scenarios:
Scenario E: The trailer (T) makes a LOC-type move near the beginning of the
3a/1a game. His MWC drops from 30% to 24%.
Scenario F: T makes a LOC-type move near the beginning of the 1a/1a game. His
MWC drops from 50% to 40%.
In the first scenario, Snowie would only penalize T .06, while in the second
scenario T is penalized .10. However, as discussed in this article, the error
in the first scenario is more severe and has more of an impact on the match
outcome!
Summary:
Without the possibility of gammons, each post-crawford game is equally
valuable/important. With the possibility of gammons, the 1a/1a game is
slightly less valuable/important than the p.c. games before it. All p.c.
games which occur before the 1a/1a game have approximately the
same value/importance.
It's usually correct to use the free drop if an underdog. In some cases as
a slight underdog, it's better to save it to potentially use it in a later
game. When the trailer is a large number of games away, the leader has
more incentive to save the free drop since there's a high probability that
he will be able to use it later when he is an even bigger underdog. When
the trailer is an extremely large number of games away (e.g. 2,000), the
leader should only use the free drop if the worst possible opening
sequence occurs.
Errors in the 1a/1a game (or 2a/1a with the double having been accepted)
are slightly less "severe" than errors made in earlier p.c. games.
The absolute error in MWC is often not the right measure of an error. A
better error measure is the relative change in MWC for the underdog (even
this is not perfect; e.g. in scenarios E and F above the trailer's MWC
falls 20% in relative terms [0.80 * 30% = 24%; 0.80 * 50% = 40%], yet we
know that the LOC in scenario E is more severe than the LOC in
scenario F).
Chris