Strategy--Checker play

Forum Archive : Strategy--Checker play

Bearing off with contact

From:   Daniel Murphy
Date:   25 March 1998
Subject:   Re: Two problems for Jellyfish analyzer.
Google: wrote:
>    Please if anyone has got Jellyfish analyzer, tell me the match
>  equities for all the possible moves to these 2 problems:
>     PROBLEM 1:
>     24:2w, 23:2w, 22:2w, 21:2w, 20:2w, 19:3w, 13:1w,
>      6:2b,  5:2b,  4:3b,  3:1w,  2:2b,  1:1b.
>       SCORE:   black:11 white:12 CRAWFORD game
>       BLACK to play:  4 - 2
>     PROBLEM 2:
>     24:3w, 23:2w, 22:2w, 21:2w, 20:2w, 19:2w, 13:1w,
>      6:2b,  5:2b,  4:2b,  3:1w,  2:1b,  1:1b.
>       SCORE:   black:11  white:12 CRAWFORD game
>       BLACK to play:   6 - 3.

Drawing a couple of diagrams revealed something worth talking about
here. I'll say this up front, though:

JellyFish is a useful analyzing tool, but I don't think asking "what
does JellyFish say?" is the best way to approach a backgammon problem.
JellyFish won't be helping you during your next tournament match.

If I've gleaned anything from Kit Woolsey and Hal Heinrich's excellent
-- and difficult -- book "New Ideas in Backgammon," it's this:

Becoming a better backgammon player is mainly a process of enhancing
your understanding of fundamental backgammon principles, and
continually refining your application of these principles to problems
you meet in play. While mathematical skills and knowledge of technical
backgammon details are certainly helpful, that's not what backgammon
is all about. With a good understanding of fundamental principles,
difficult problems can become easy -- without much calculation and
even without much thinking. And even if applying fundamental
principles don't lead you to the best play, you won't often be wrong
by much.  And that's good -- and good enough -- for most backgammon

With that in mind, let's go to the positions.


  | O  O  O  O  O  O |   |                O |
  | O  O  O  O  O  O |   |                  |
  |                O |   |                  |
  |                  |   |                  |
  |                  |   |                  |
  |                  |   |                  |
  |                  |   |                  |
  |          X       |   |                  |
  |    X     X  X  X |   |                  |
  | X  X  O  X  X  X |   |                  |

  SCORE:   black:11 white:12 CRAWFORD game
  BLACK to play:  4 - 2

X already has 5 checkers off and is well ahead in the race. X is not
likely to lose unless X leaves a shot soon, but since O has a closed
board, getting hit soon will hurt.  On the other hand, X doesn't have
to worry too much about getting a second checker hit.

At this score, X can certainly use a gammon. Gammons win the match for
X, but after a single win X will have to win the next game also, so
winning a single game now only gives X a 50% chance to win the match.
But O need roll only 23 pips -- 3 average rolls -- to get off the
gammon, so gammon seems unlikely even if X plays aggressively. Even if
X takes two checkers off now, X isn't likely to bear off in only four
more rolls because of the gap on the 3 point and the need to play
remaining numbers safely.

Three possible plays -- one safe, one aggressive, and one rather wild
-- stand out:

(1) 6/4 6/2  The safe play clears the highest point, leaves only 62
and 52 as bad numbers on the next roll, makes later shots very
unlikely (especially since O cannot wait long), but takes no checkers

(2) 4/off 2/off  The aggressive play takes two checkers off, saving a
whole roll in the bearoff, but appears to be a little more dangerous
since it leaves X with 3 points to clear and no spare checkers on the
4, 5 and 6 points. While only 63 leaves a shot next time, it will be a
double shot, and numbers like 61, 51, 53, 41, 43 and 31 leave X with
an odd checker on the highest point or with a dangerous "interior gap"
(an empty point between two made points).

(3) 5/3* 5/1  The wild play takes no checkers off, leaves a blot, but
puts O on the bar.  If O misses, this play may win a few more gammons
-- who knows whether putting O on the bar is worth not taking any
checkers off -- but it looks wrong to leave an immediate blot,
especially an "interior blot" which may be difficult to clear even if
missed the first time.

Principle: It's wrong to leave unnecessary shots unless the resulting
position is clearly better than alternative plays. Leaving a blot
doesn't seem clearly better here.

And my over the board analysis ends here. I'd estimate that the gain
from playing 4/off 2/off, saving a whole roll in the bearoff, is
probably worth the extra risk of leaving a shot. But I'd also estimate
that since X is a huge favorite to win after either play, with only
small gammon chances after either play, the gain, if any, from playing
4/off 2/off is likely to be small.

(Mathematically inclined players who keep up with current backgammon
literature will be aware of some formulas and reference positions that
might help them estimate how likely O is to get off the gammon and how
likely X is to leave a shot. If you know them and can use them, go for

Other plays seem much worse.

(4) 6/off takes a checker off but leaves like play (3) leaves an
unnecessary blot. In 6/off's favor (compared to worse plays) it leaves
the blot where it will be easiest to clear.

Principle: If you must leave a blot during the bearoff, you should
leave it on a point above your made points, where it will be easier to
clear if missed than an "interior blot" between two made points.

(5) 5/1 2/off leaves an interior blot on the 5 point. This should be
worse than 6/off.

(6) 4/off 4/2 leaves an interior blot on the 4 point. This should be
worse than 5/1 2/off.

(7) 6/2 4/2 takes no checkers off and leaves a blot on the six point.

(8) 5/1 4/2 takes no checkers off and leaves an blot on the 5 point.
This should be worse than 6/2 4/2.

(9) 5/1 6/4 leaves two blots.

(10) 4/off 5/3* takes a checker off but leaves 3 blots.

(11) 6/2 5/3* is very imaginative, leaving 4 blots.

And JellyFish says ... well, yes, but first I'd like to talk a bit
about how much a gammon is worth to X in this position at this match

At most match scores and in many positions, weighing the gains and
losses of "safe" and "gammonish" plays can be exceedingly difficult.

For one thing, the value of a gammon depends on the match score. For
example, if you're way ahead, gammons aren't so important, since you
still have good match winning chances after winning only a single
game. If you're way behind, gammons become more attractive, since your
match winning chances after winning only a single game are still low.

Another complication is whether both sides can fully use the extra
points for a gammon. Another is whether the resulting score following
a single win or gammon brings the trailer to an even or odd number of
points from victory. This makes a difference at most match scores, but
is easiest to see in the Crawford game. For example, at 1-away 4-away
Crawford, the trailer would very much like to win a gammon, because
then the trailer can double for the match in the next game. But at
1-away 3-away Crawford, gammons hardly matter at all, because
(ignoring the leader's small advantage of having a "free drop") the
next game will decide the match whether or not the trailer wins a
gammon this game.

Further, if both sides have gammon chances, trying to weigh two or
more plays and calculate the answer becomes impossible for most
players. There are a couple of formulas that will give you an answer
-- see, e.g., Chuck Bower, Ron Karr, David Montgomery and Michael
Zehr's articles archived at Backgammon Galore -- but
only if you plug in the right numbers. Few players can do that over
the board.

If you're used to playing money games, you probably know that the
gammonish play must generate more than two additional gammons for each
additional loss to be better than the safe play. That's because --
with the cube on two, losing instead of winning costs 4 points -- the
2 that aren't won plus the 2 that are lost -- while winning a gammon
instead of winning a single game gains only 2 points -- winning 4
instead of winning 2.

But at 2-away 1-away Crawford, the math is nothing like money play
and, compared to other match scores, is simple to compute.

At this score a gammon wins the match 100% of the time, and a single
win only gives yields 50% match winning chances. That means that the
gammonish play need generate only one additional gammon for each
additional loss to break even (not, as in money play, two additional

Some examples:

Play A: 90% wins (10% gammons and 80% single wins) = 50% match equity.

Play B: 85% wins (15% gammons and 70% singles wins) = 50% match

Play A and Play B give identical match winning chances for the trailer
at 2-away 1-away Crawford.

Play C: 70% wins (25% gammons and 45% single wins) = 47.5% match

Play C is worse, trading 20% fewer wins for only 15% more gammons.

Play D: 60% wins (45% gammons and 15% single wins) = 52.5% match

Play D is the winner, trading 30% fewer wins for 35% more gammons.

The math here should be pretty clear. But what should also be clear is
how difficult it would be, in most positions, to make decisions over
the board by trying to calculate exactly how many single and gammon
wins and losses two or more plays will get you!

Back to Problem 1:

Here are JellyFish's Level 7 evaluations and Level 6 rollouts (1296
rollouts, seed 99):

               L7      L6      L6    L6     L6     Match
               Equity  Equity  W     Single Gammon Equity
(1)  4/o 2/o   0.950   1.005   96.6  89.3   7.3    51.95%
(2)  6/2 6/4   0.979   0.982   97.5  94.2   3.3    50.40%

JellyFish L7 prefers the safe play, but the rollout gives a small edge
to taking two checkers off.

As expected, the plays that leave blots are clearly worse.

(3)  6/o       0.508   0.527   74.7  71.3   3.4    39.05%
(4)  5/1 2/o   0.432   0.449   70.7  67.3   3.4    37.05%
(5)  5/1 5/3*  0.494   0.400   65.9  57.8   8.1    36.90%

Note that JF L7 grossly overvalues play 5. With an interior gap on the
5 point and an interior blot on the 3 point, the position will be
difficult to clear.

(6)  6/2 4/2   0.367   0.378   68.0  66.2   1.8    34.90%
(7)  4/o 4/2   0.372   0.362   66.9  64.6   2.3    34.70%
(8)  5/1 4/2   0.360   0.345   66.2  64.0   2.2    34.40%
(9)  5/1 6/4   0.030   0.038   51.4  50.4   1.0    26.20%
(10) 4/o 5/3  -0.419  -0.406   27.5  23.2   4.3    15.90%
(11) 6/2 5/3  -0.630  -0.686   14.9  13.4   1.5     8.20%

Now let's look at Problem 2, which is easily answered by applying a
backgammon fundamental.


  | O  O  O  0  O  O |   |                0 |
  | O  O  0  0  O  O |   |                  |
  | 0                |   |                  |
  |                  |   |                  |
  |                  |   |                  |
  |                  |   |                  |
  |                  |   |                  |
  |                  |   |                  |
  |          X  X  X |   |                  |
  | X  X  0  X  X  X |   |                  |
  SCORE:   black:11  white:12 CRAWFORD game
  BLACK to play:   6 - 3.

Same match, one roll later. X has played 4/off 2/off and O has rolled
a 5, played to the acepoint (which was wrong, by the way: playing the
outfield checker to the barpoint is best, still keeping a spare 6 to
play, avoiding some gammons and keeping a better bearoff board in case
O gets and hits a shot).

63! How unfortunate.

6/off is forced, of course. Of the three possible 3's, 6/3 leaves a
triple shot. Triple shots are worse than double shots. 5/2 and 4/1
both leave a double shot. Remembering our principles, we should expect
that 5/2, which leaves the blots on the easiest points to clear, is
better than 4/1, which leaves an interior blot on the 4 point.

JellyFish agrees:

               L7      L6      L6    L6     L6     Match
               Equity  Equity  W     Single Gammon Equity
(1)  6/o 5/2   0.269   0.323   63.2  57.4   5.8    34.50%
(2)  6/o 4/1   0.209   0.265   60.8  56.9   4.8    33.25%
(3)  6/o 6/3* -0.190  -0.100   42.4  37.3   5.1    23.75%

In these and many backgammon positions, remembering and correctly
applying fundamental backgammon principles can lead to good decisions
without much, if any, calculating. The trick is to learn, through
experience and study, which principles are most important in a given
position. That's not easy, but it sure is a lot easier, usually, than
trying to calculate your way to the best play.

In Position 1, saving a full roll in the bearoff seemed likely win
more gammons, but given X's large winning chances and small gammon
chances after either play, saving a full roll did not appear to be
likely to gain much, either. Other possible plays in Position 1
violated the principle of not leaving unnecessary blots unless the
resulting position is clearly better than other alternatives. In
position 2, the principles of avoiding interior blots -- and
unnecessary blots -- ranked the three possible plays perfectly.

Daniel Murphy
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Strategy--Checker play

Avoiding major oversights  (Chuck Bower+, Mar 2008) 
Bearing off with contact  (Walter Trice, Dec 1999) 
Bearing off with contact  (Daniel Murphy, Mar 1998)  [Long message]
Blitzing strategy  (Michael J. Zehr, July 1997) 
Blitzing strategy  (Fredrik Dahl, July 1997) 
Blitzing technique  (Albert Silver+, July 2003)  [GammOnLine forum]
Breaking anchor  (abc, Mar 2004) 
Breaking contact  (Alan Webb+, Oct 1999) 
Coming under the gun  (Kit Woolsey, July 1996) 
Common errors  (David Levy, Oct 2009) 
Containment positions  (Brian Sheppard, July 1998) 
Coup Classique  (Paul Epstein+, Dec 2006) 
Cube ownership considerations  (Kit Woolsey, Apr 1996) 
Cube-influenced checker play  (Rew Francis+, Apr 2003)  [GammOnLine forum]
Defending against a blitz  (Michael J. Zehr, Jan 1995) 
Estimating in volatile situations  (Kit Woolsey, Mar 1997) 
Gammonish positions  (Michael Manolios, Nov 1999) 
Golden point  (Henry Logan+, Nov 2002) 
Hitting loose in your home board  (Douglas Zare, June 2000) 
Holding games  (Casual_Observer, Jan 1999)  [Long message]
How to trap an anchor  (Timothy Chow+, Apr 2010) 
Jacoby rule consideration  (Ron Karr, Nov 1996) 
Kamikaze plays  (christian munk-christensen+, Nov 2010) 
Kleinman Count for bringing checkers home  (Øystein Johansen, Feb 2001) 
Late loose hits  (Douglas Zare+, Aug 2007)  [GammOnLine forum]
Mutual holding game  (Ron Karr, Dec 1996) 
Pay now or pay later?  (Stuart Katz, MD, Nov 1997) 
Pay now or pay later?  (Stephen Turner, Mar 1997) 
Pay now or play later?  (Hank Youngerman+, Sept 1998) 
Play versus a novice  (Courtney S Foster+, Apr 2004)  [GammOnLine forum]
Playing doublets  (Grunty, Jan 2008) 
Playing when opponent has one man back  (Kit Woolsey, May 1995) 
Prime versus prime  (Albert Silver+, Aug 2006)  [GammOnLine forum]
Prime versus prime  (Michael J. Zehr, Mar 1996) 
Saving gammon  (Bill Riles, Oct 2009) 
Saving gammon  (Ron Karr, Dec 1997) 
Splitting your back men  (KL Gerber+, Nov 2002) 
Splitting your back men  (David Montgomery, June 1995) 
Trap play problem  (Brian Sheppard, Feb 1997) 
When in doubt  (Stick+, Apr 2011) 
When to run the last checker  (Stick Rice+, Jan 2009) 
When you can't decide  (John O'Hagan, Oct 2009) 

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