2 away 2 away strategy
Bernhard Kaiser
: I've ready several of the 'proofs' about -2:-2 matches and I remain
: unconvinced. The proofs I've read all seem to imply that nothing can be
: gained by not doubling, I think that this is incorrect.
: Clearly there is no point to doubling if one has no market losers, since
: the cube can be passed on the next roll. So, in order to prove doubling
: is correct it must be proven that there is no situation where the equity
: is higher without doubling. I believe such situations exist.
: For example positions can be reached using Jellyfish that have X on roll with
: 40+% gammons, and 30%+ losses without, I believe, either side having a
: previous market losing sequence. These positions are most easily reached
: with X rolling a 44 or 55 to make a 4 point board and O dancing. This is
: X's best sequence and O still has a take.
: In these positions X appears to be better off playing for a gammon, in
: which case he doesn't double. O has no market losers until he gets a shot
: so he definitely shouldn't double. When O is later able to run off a gammon
: without ever getting a shot, I claim X then has the first optimal double
: to cash the game.
: Anyway I have yet to see a proof that this type of position is impossible
: and without that the proofs seem incomplete. If I missed this from someone's
: proofs I apologize.
: Comments?
: BadDice
Hi BadDice!
You fell into the same trap as i did in my article a while before.
It is just not right, that a position could exist, where O has a take and
X should play for a gammon! Since O has to take, X will win all the matches
after doubling he would have won gammon without double. You have really
nothing to gain by not doubling, since O will double all the time he could
lose his market!
onepointer
l
Kit Woolsey
: I've ready several of the 'proofs' about -2:-2 matches and I remain
: unconvinced. The proofs I've read all seem to imply that nothing can be
: gained by not doubling, I think that this is incorrect.
: Clearly there is no point to doubling if one has no market losers, since
: the cube can be passed on the next roll. So, in order to prove doubling
: is correct it must be proven that there is no situation where the equity
: is higher without doubling. I believe such situations exist.
: For example positions can be reached using Jellyfish that have X on roll with
: 40+% gammons, and 30%+ losses without, I believe, either side having a
: previous market losing sequence. These positions are most easily reached
: with X rolling a 44 or 55 to make a 4 point board and O dancing. This is
: X's best sequence and O still has a take.
: In these positions X appears to be better off playing for a gammon, in
: which case he doesn't double. O has no market losers until he gets a shot
: so he definitely shouldn't double. When O is later able to run off a gammon
: without ever getting a shot, I claim X then has the first optimal double
: to cash the game.
: Anyway I have yet to see a proof that this type of position is impossible
: and without that the proofs seem incomplete. If I missed this from someone's
: proofs I apologize.
: Comments?
: BadDice
Of course such a position can be reached. However you are incorrect when
you say that X is better off playing for a gammon in such a position.
The key is that, although X may not have lost his market yet, from such a
position there certainly will be market losers (i.e. sequences after
which X will be better than 70% to win). Granted these may be positions
which X has a high gammon winning probability, but that won gammon is not
certain. Since O will of course make sure that he doesn't lose his
market (O being the perfect player he is), there are no scenarios where X
gains from not doubling -- if X loses the game he will never lose only 1
point. Consequently, by not doubling now X costs himself in the
variations where he wins the game but doesn't win a gammon -- if he ever
gets to a position where he doubles and O correctly drops then X will have
wished he had doubled earlier. As I have shown X has no corresponding
gain from not doubling now, so faililng to double does cost equity.
It is true that there are positions where, at this score, it would be
correct for X to play for a gammon, since his equity in playing on is
higher than if he doubles and O correctly passes. However, for such a
position to have been reached it would mean that X had at some previous
point failed to double when O still had a take but X had market losers,
such as the position you describe. As I have shown, not doubling in this
position is an error by X. Therefore, a position where it is correct to
play for a gammon cannot be reached without somebody having made a cube
error previously.
Hope this explains it.
Kit
Michael Klein
[discussion of an early game gammonish position where X has yet to
have a market losing sequence]
>Of course such a position can be reached. However you are incorrect when
>you say that X is better off playing for a gammon in such a position.
[explanation of why it is correct to double in gammonish positions
with market losers at 2 away/ 2 away]
> Since O will of course make sure that he doesn't lose his
>market (O being the perfect player he is), there are no scenarios where X
>gains from not doubling -- if X loses the game he will never lose only 1
>point.
[ explanation of why positions where it is correct to play on for the
gammon, while constructable, should never arise with correct doubling
by both players]
Just a minor quibble: even a perfect O will not be able to double in
advance of market losers which come on joker sequences. Say that X is
playing for the gammon with 50% Gs, 40% plain game wins, and 10%
losses with all losses happening by fly shot on the next roll. O
can't double because the equity gain from possibly saving a G is
greater than that from being able to win on a 2 cube -- even if
hitting the shot would be a market loser for O.
While this scenario does give some benefit to X from not doubling
(earlier) if the position comes up just right, the only reason that it
works it that he has already lost his market, and if he could have
predicted it in advance, he would have been better off doubling. The
only cases where X, having lost his market, should be happy to be
playing with a 1 cube instead of a 2 cube are those where he wins
enough gammons to have a match equity above .7 and loses more plain
games than he wins. Of course, in these cases, O can adopt the
strategy of doubling at first market losing sequence. I suspect that
this may not be O's optimal strategy if there are few immediate market
losers and later ones will be reliably predictable, but if this
strategy is good enough to defeat the benefit to X of playing on for
the G, then any superior strategy is.
Since X's equity, predicted in advance, is never higher by not
doubling and then playing for the gammon, I don't think that these
occasional single game losses for inferior doubling strategies by X do
anything to undermine the proof.
Michael Klein
mklein on FIBS
James Eibisch
>Therefore, a position where it is correct to
>play for a gammon cannot be reached without somebody having made a cube
>error previously.
Only if we think in terms of two-ply market losers - either X loses his
market by rolling well or O loses X's market for him by rolling badly on
his turn. A lot of people, including myself, often think in terms of
one-ply market losers - that roll by X that loses his market. I've heard
people ask why X doubled at -2 -2 when he had no market losers. For
those people (and I'm trying to teach myself here as well), it's
important to understand why a position like this is a double.
1 2 3 4 5 6 7 8 9 10 11 12
+------------------------------------------+ O: - score: 0
| O X | O | X X O |
| O X | | X O |
| X | | X O |
| X | | O |
| X | | |
| |BAR| |v 2-point match
| | | |
| O | | X |
| O | | O X |
| O | | O X |
| X X O | | O O X |
+------------------------------------------+ X: - score: 0
24 23 22 21 20 19 18 17 16 15 14 13
X on roll.
I'm not good at contriving positions, so this isn't a good example. It
may even be wrong, but it serves only as an illustration.
X might not have any direct market losers (although 2 2 and 6 6 are
excellent - as I said, it's not a good example). So he doesn't double.
But what if he rolls something fairly neutral, maybe missing the second
man, and O then rolls 6 6?
X will have a better position after his move almost regardless of what
he rolled. O rolled so badly that suddenly X's market is gone. Not
doubling here would be a cube error on X's part even though there might
not be an obvious market loser. Thinking two-ply is crucial here.
This will be obvious to many, but I have witnessed confusion over such
cube handling, including a direct mail from an r.g.b. reader. And I find
writing is among the best ways to teach myself, so please forgive my
indulgence :-)
Hope this is of help to somebody.
--
_
James Eibisch ('v') N : E : T : A : D : E : L : I : C : A
Reading, U.K. (,_,) http://metro.turnpike.net/J/jeibisch/
=======
Walter G Trice
[snip]
>It is true that there are positions where, at this score, it would be
>correct for X to play for a gammon, since his equity in playing on is
>higher than if he doubles and O correctly passes. However, for such a
>position to have been reached it would mean that X had at some previous
>point failed to double when O still had a take but X had market losers,
>such as the position you describe. As I have shown, not doubling in this
>position is an error by X. Therefore, a position where it is correct to
>play for a gammon cannot be reached without somebody having made a cube
>error previously.
I have one small quibble with Kit's very lucid explanation. I believe
that it is possible for X to become too good to double without having
made any error. This can happen if the first player to have market
losers is O and he incorrectly fails to double and then rolls pretty
well. Now X has a rather bad position -- perhaps he would have to
drop after most of his rolls. But he gets lucky and rolls a super
joker. Since this is a 'legitimate' way to lose your market at this
score I don't see why you can't become too good in the process.
-- Walter Trice