Match Play

 3-away/4-away: tricky cube decision

 From: Kit Woolsey Address: kwoolsey@netcom.com Date: 18 July 1994 Subject: Re: cube decision Forum: rec.games.backgammon Google: kwoolseyCt5tB3.MMz@netcom.com

```Wayne King wrote:
>  13              18      19            24
> +-----------------------------------------+ X
> | .  X  .  .  .  O |   | X  X  X  X  X  . |
> |    X           O |   | X  X  X  X       |
> |                  |   | X     X  X       |
> |                  |   |                  |  ---
> |                  |   |                  | | 1 |
> |                  |BAR|                  |  ---
> |                  |   |                  |
> |                  |   |                  |
> |                  |   |             O    |
> |                  |   | O  O  O  O  O    |
> | X  .  .  .  .  O |   | O  O  O  O  O  O |
> +-----------------------------------------+ O
>  12              7       6              1
>
> Pipcounts: O = 86, X = 86
>
> Match Score O-3 away
>             X-4 away
>
> O on roll. Cube action?
>
> The following position came up in my 2nd round match of Snoopy's
> tournament.  As O this position puzzles me.  O has many market losers
> which would seem to point to doubling, yet X has a very efficient
> re-double if O misses.  I would appreciate any thoughts as to (1) what is
> the correct cube action and (2) how one in general would take into
> account an opponents re-double when deciding whether or not to double.

Very interesting problem.  I could write a book about match cube
decisions like this (come to think of it, I *have* written such a book).
The score 3 away, 4 away is perhaps the trickiest match score of all.

The first step toward attacking any decision on whether or not to double
should always be to put yourself in the other guy's shoes and ask if he
should take.  Sometimes this solves the problem quickly.  If the answer
is no, or maybe not, then the double is automatic.  If he has a monster
take then probably you shouldn't be doubling unless the position is
super-volatile (which admittedly this one is).  In this sort of position
if your opponent has a Big take you probably should not be doubling if
you are ahead in the match, since he has more cube leverage.  However if
you are behind in the match then you should be inclined to double if
there is a sufficient chance you will lose your market.  However if his
take is fairly close, then it is usually correct to double even if you

Now, to the actual position.  In order to avoid overly complex math, I'm
going to make some rough assumptions -- the way I would do at the table.
The idea is to make these assumptions about equally favorable for both
players, so you can get a good approximation of the equity in the
position.  Let's suppose that if O hits the shot he always wins a single
game (he might not, but that may be compensated for by a possible
gammon).  If O misses the shot X does have huge cube leverage, since O's
take point is 40%.  Let's assume that X will redouble, O will take, and
that X will win 60% of the time.  O has 16 hitting numbers out of 36.
Therefore: 4/9 of the time O will be ahead 1 away, 4 away, for 83%
equity.  5/9 of the time they will play for the match, with O's equity
40%.  Averaging these out, O's equity is about 64%.  Since if X passed
O's equity at 4 away, 2 away is 68%, it looks like X has a fairly close
decision, so it might be right for O to double since a lot rides on the
next roll.

Let's see if this turns out to be correct.  Suppose O doesn't double.  If
he hits by our assumptions he will win and be ahead 2 away, 4 away for
68% equity -- this happens 4/9 of the time.  If he misses X won't be
flinging the cube over automatically, so let's just assume that X will
win a 1-game 60% of the time and O will win a 1-game 40% of the time.
Since O is missing 5/9 of the time, he will win 2/5 of these, so he will
win 2/3 of the time for 68% equity and lose 1/3 of the time for 50%
equity.  This averages to 62% equity.  Since we already estimated O's
equity in the match at 64% if he doubles, it looks like it is correct to
double.

I make no claims that this is the correct answer.  I made several rough
and questionable assumptions, and if you change some of them it might
well change the final result.  Also I give no guarantees that my
arithmetic is correct.  However, this is the approach one should take at
the table.

Kit
```

 Robert Koca  writes: ```As part of his analysis kitwoolsey wrote: > Therefore: 4/9 of the time O will be ahead 1 away, 4 away, for 83% > equity. 5/9 of the time they will play for the match, with O's equity > 40%. Averaging these out, O's equity is about 64%. should be 59%. and wrote: "If O misses the shot X does have huge cube leverage, since O's take point is 40%. Let's assume that X will redouble, O will take, and that X will win 60% of the time." and in next paragraph, "Suppose O doesn't double. ... If he misses X won't be flinging the cube over automatically, so let's just assume that X will win a 1-game 60% of the time and O will win a 1-game 40% of the time." Surely if X wins 60% of the time in the first case (after doubling) X will win significantly more than that in the second case (with cube access). As a rough measure of this importance consider a pure race in which pipcount is tied 80-80 (this position may very well become close to this). In a money game, equity to player on roll without cube access is .02 if cube is neutral then equity is +.25. I would think that if X wins 60% cubeless after a miss by O, a good approximation to equity with neutral cube is that X wins 70% single game and loses 30% single game. This changes result by about 1% (5/9*10%) of time, X goes to 50% instead of 68% My conclusion is that not a double. 59% < 61% ,Robert Koca bobk on FIBS koca@orie.cornell.edu ```

 Kit Woolsey  writes: ```> should be 59%. Of course Bob is correct here. I mentally did my 4/9 backwards. Note my disclaimer about my math! > Surely if X wins 60% of the time in the first case (after doubling) > X will win significantly more than that in the second case (with > cube access). As a rough measure of this importance consider a pure > race in which pipcount is tied 80-80 (this position may very well > become close to this). In a money game, equity to player on roll > without cube access is .02 if cube is neutral then equity is +.25. > I would think that if X wins 60% cubeless after a miss by O, a good > approximation to equity with neutral cube is that X wins 70% single > game and loses 30% single game. This changes result by about 1% > (5/9*10%) of time, X goes to 50% instead of 68% Not taking the cube access into account was one of the rough approximations I made. Bob's analysis does a better job of fine tuning things. > My conclusion is that not a double. 59% < 61% And quite likely a correct conclusion. This just shows how difficult these match equities can be. If I can't get it right doing the calculations in the comfort of my home (and I think I can immodestly claim pretty good expertise on the subject), imagine the pressure one is under at the table, perhaps in the finals of a big tournament with big \$\$ at stake. Nobody promised things would be easy! However the more adept one is at doing these calculations, the more likely one is to make the right decision at the table, or at the very least avoid some catastrophic cube action which dumps gobs of equity down the drain. Kit ```

### Match Play

1-away/1-away: advice from Bernhard Kaiser  (Darse Billings, July 1995)
1-away/1-away: advice from Stick  (Stick+, Mar 2007)
1-away/1-away: and similar scores  (Lou Poppler, Aug 1995)
2-away/3-away: playing for gammon  (Tom Keith, Feb 1996)
2-away/4-away: Neil's rule of 80  (Neil Kazaross, June 2004)
2-away/4-away: cube strategy  (Tom Keith, Dec 1996)
2-away/4-away: practical issues  (Mark Damish, Jan 1996)
2-away/4-away: trailer's initial double  (Kit Woolsey, Jan 1996)
3-away/4-away: opponent's recube  (William C. Bitting+, Feb 1997)
3-away/4-away: racing cube  (Bill Calton+, Nov 2012)
3-away/4-away: tricky cube decision  (Kit Woolsey+, July 1994)
3-away/4-away: what's the correct equity?  (Tom Keith, Sept 1997)
4-away/4-away: take/drop point  (Gary Wong, Oct 1997)
5-away/11-away: redouble to 8  (Gavin Anderson, Oct 1998)
7-away/11-away: volatile recube decision  (Kit Woolsey, May 1997)
Both too good and not good enough to double  (Paul Epstein+, Sept 2007)
Comparing 2-away/3-away and 2-away/4-away  (Douglas Zare, Mar 2002)
Crawford rule  (Chuck Bower, May 1998)
Crawford rule  (Kit Woolsey, Mar 1997)
Crawford rule--Why just one game?  (Walter Trice, Jan 2000)
Crawford rule--history  (Michael Strato, Jan 2001)
Delayed mandatory double  (tem_sat+, Oct 2010)
Delayed mandatory double  (Donald Kahn+, Dec 1997)
Doubling when facing a gammon loss  (Kit Woolsey, Jan 1999)
Doubling when opponent is 2-away  (David Montgomery, Dec 1997)
Doubling when you're an underdog  (Stein Kulseth, Dec 1997)
Doubling window with gammons  (Jason Lee+, Jan 2009)
Free drop  (Ian Shaw, May 1999)
Free drop  (Willis Elias+, Oct 1994)
Gammonless takepoint formula  (Adam Stocks, June 2002)
Going for gammon when opp has free drop  (Kit Woolsey, Jan 1998)
Going for gammon when opp has free drop  (Kit Woolsey, Apr 1995)
Holland rule  (Neil Kazaross, Apr 2010)
Holland rule  (Kit Woolsey, Dec 1994)
Leading 2-away with good gammon chances  (Douglas Zare, Feb 2004)
Match play 101  (Max Urban+, Oct 2009)
Matches to a set number of games  (Tom Keith+, Oct 1998)
Playing when opponent has free drop  (Gilles Baudrillard+, Dec 1996)
Post-crawford doubling  (Scott Steiner+, Feb 2004)
Post-crawford doubling  (Maik Stiebler+, Dec 2002)
Post-crawford doubling  (Gus+, Sept 2002)
Post-crawford mistakes  (Rob Adams, Sept 2007)
Post-crawford/2-away: too good to double  (Robert-Jan Veldhuizen, July 2004)
Slotting when opponent has free drop  (onur alan+, Apr 2013)
Take points  (fiore+, Feb 2005)
Tips to improve cube handling  (Lucky Jim+, Jan 2010)
When to free drop  (Dan Pelton+, Oct 2006)
When to free drop  (Tom Keith+, July 2005)
When to free drop  (Gregg Cattanach, Dec 2004)
When to free drop  (Kit Woolsey, Feb 1998)
When to free drop  (Chuck Bower, Jan 1998)
Which format most favors the favorite?  (Daniel Murphy+, Jan 2006)