Playing for Gammon

Maximizing Gammon Chances
Max Levenstein
Amalgamated from a thread on the BGOnline Forums, August 2011

Bearoff Checker Play
I had little understanding of this position, and therefore blundered this roll very nicely.

White (4 points)
7 point match
Blue (5 points)
Blue to play 3-1

"What are your thoughts about this position — what is going on or what might occur?"

My gammon value is huge here so I wanted to increase my chance of winning a gammon. I have an even number of checkers left, so taking one checker off will likely not reduce the number of rolls I need to bear off. I would like to avoid leaving a shot, although I may still be a favorite to win a game (not a gammon) if I do get hit and come around again.

With these thoughts in mind, I played 4/1, 4/3. This helps clear the last point on the next roll and reduces my chances of getting hit.

Casper van der Tak
You are not in danger of losing the game, because you are very far ahead in the race and have access to the cube. You would like to win a gammon. If you count crossovers for both sides, or crossovers and pips in the gammon race, you will see that you are an underdog to win the gammon. So you look for plays that increase your chances of winning a gammon.

4/3, 4/1 is a slow play. You will need four or five rolls to bear off; your opponent is on roll and likely needs only three rolls to avoid agammon. Among the safe plays, 4/off looks better, because you'd need 4 rolls only after that (you can afford a miss). White is not going to hang around waiting for a shot.

You should also look at the unsafe plays. Two of them:

  • 3/off, 1/off creates a 3-roll position with lots of misses (likely 4 roll).

  • 3/off, 3/2* creates a 4-roll position (you can afford a miss), and slows your opponent down, adding a crossover and 2 pips to the gammon-save finish line. Both the risky plays risk getting hit and have a reduction in gammon chances, but that should not worry you much as you are somewhat unlikely to win a gammon to start with.

I think with this analysis you'd need to be able to see that one of the risky plays is best, followed by 4/off. Among the two best plays, probably you want to hit, because hit and dance is big.

Neil Kazaross
"I have an even number of checkers left, so taking one checker off will likely not reduce the number of rolls I need to bear off."

This is one of the causes for your blunder and a common misconception. The even-vs-odd rule applies when you are very close to a pure roll-oriented speed-board bearoff. After playing 4/3, 4/1 you have eight checkers left and a persistant gap on the 2 point. You're quite likely to waste a turn by rolling a missing 2.

Compare with playing the clearly speedier 4/off. Now you still have that 2 gap, but you have 7 checkers left and can afford to miss once with a 2, which currently fills the gap from the 4 pt and likely prevents further misses.

4/off is probably close to half a roll quicker than 4/3, 4/1. So if we ignore shot potential we see that 4/off is much better than 4/3, 4/1.

[Editor's question:  But isn't it dangerous to leave an odd number of checkers on your highest point?]

Leaving a shot here is not nearly as bad as it usually is when bearing off since your opponent has only a two-point board and other blots lying around. You have many checkers off and cube access and can basically cash at will for a couple turns after being hit. If you get hit, you can probably still play for a gammon for a turn or two.

Even if you play 3/off, 1/off, leaving an ace shot, (hoping to not miss later and thus gain a full turn) and get hit with a joker 3-1 or 1-1 allowing opponent to make his 6 point, you still may be too good to double, since you can still fluke him with a gammon and cash later.

The same applies after 3/0, 3/2* if he jokers with 3-2. I think you're still too-good and, of course, if you cautiously cash, you have only lost your gammon chances by getting hit, not the game. However, most of the opponent's hits leave at least one blot on his side of the board and you may be able to rehit at some point and then gammon him.

Safety in this position is only a minor consideration. Speed of bearoff is most important as he has only a two-point board and you have many checkers off and you can cash if you get scared after getting hit.

"Therefore, I played 4/1, 4/3. This helps clear the last point on the next roll."

Clearing the point doesn't matter here. Speed to bearoff does. As we have seen, nearly any other play is superior to 4/1, 4/3 here. 4/off clearly is better and 3/off, 1/off and 3/off, 3/2* are probably still better. Taking two checkers off is fastest and runs no risk of losing the game, due to cube access.

Hitting is somewhat slower than 4/off but sets the opponent back two pips and one crossover. Hitting gains huge if he dances with 4-4 which otherwise usually saves gammon, and gains big on any of the other three dances. I think it is slightly correct to hit.

Daniel Murphy
Using GnuBG's bearoff analyzer, we can see how often Blue bears off in x rolls (when not hit, of course).

Rolls 3/2*, 3/off 3/off, 1/off 4/off 4/3, 4/1
2 .01621 .12963 .01158 .00926
3 .21929 .24810 .24833 .18190
4 .58035 .56670 .66238 .43220
5 .18291 .05000 .07721 .37510
6 .00125 .00000 .00050 .00154

Number of Rolls to Bear Off (if not hit)
3/off, 1/off
2 rolls
3 rolls
4 rolls
3 rolls
4 rolls
5 rolls
3/2*, 3/off
3 rolls
4 rolls
5 rolls
4/3, 4/1
3 rolls
4 rolls
5 rolls

From the table it's clear that 4/3, 4/1 is much slower than 4/off, but not clear that one of the other two plays is better, since the numbers for the aggressive plays don't account for White's fans and hits and loss of 2 pips if Blue hits.

The table below shows GnuBG's 2-ply evaluation for Blue's play followed by White rolling 4-4, 4-1, or 1-1 (numbers that fan if Blue plays 3/2* 3/off), and for 6-1 and 6-2 (numbers that hit or don't hit if Blue leaves a blot).

Roll 3/2*, 3/off 3/off, 1/off 4/off 4/3, 4/1
4-4 1.517 1.071 1.010 1.008
4-1 1.517 1.018 1.281 1.211
1-1 1.517 0.097 1.303 1.234
6-2 1.108 1.280 1.180 1.125
6-1 1.275 1.048 1.214 1.147

No surprise that 3/2*, 3/off gains a lot on the fanning numbers, but look at the 6-2 and 6-1 lines — getting hit with a 2 or a 1 isn't nearly as bad as one might have thought.

"I am still amiss as to why hitting with the 3 is preferred over taking a checker off the 4 point."

The table below should give an idea of when and by how much one play is better than the other. For each of White's rolls, the table gives Blue's match equity having played 4/off or 3/off, 3/*2 and the weighted difference (nondoublets being counted twice) between the two.

Match equities are from 40 GnuBG 2-ply 7776 trial rollouts.

Match Eq Weighted
for 3/off, 3/2*
4/off 3/off, 3/2*
4-4 75.351 87.777 12.426 Fan with 4-4 is very big.
4-1 82.524 87.777 10.506 Fan is big.
3-1 83.064 86.319 06.510 3-1 is barely better for White than fanning.
1-1 83.064 87.777 04.713 Fan is big but tiny 11 is bad anyway.
5-1 81.824 83.582 03.516 From gaining ~0.5 rolls and ~1 crossover.
4-3 80.588 82.342 03.508 (" ")
6-4 77.652 79.335 03.366 (" ")
5-3 79.482 81.142 03.320 (" ")
6-1 80.686 82.342 03.312 (" ")
6-3 78.470 80.089 03.238 (" ")
5-4 78.667 80.176 03.018 (" ")
6-5 76.880 78.383 03.006 (" ")
3-3 76.245 77.418 01.173 (" ")
6-2 79.746 79.834 00.176 Average racing number; hits but leaves 18 return hits from the bar.
5-5 75.118 75.174 00.056 Still some slight gammons chances.
6-6 75.118 75.140 00.022 2 doublets vs. 5-4 still wins a gammon.
2-1 83.017 80.955 (04.124) Worst racing number; hits but leaves 18 return hits from the bar.
2-2 82.506 77.047 (05.459) Numbers that hit and safety mean fewer gammons lost.
5-2 80.686 77.486 (06.400) (" ")
4-2 81.824 77.480 (08.688) (" ")
3-2 83.130 75.549 (15.162) The small racing number 3-2 hits and efficiently covers.

Here is a rollout for money play and with our opponent owning the cube. We see what we only rarely lose this game no matter what we play.

White is Player 2

score: 0
pip: 53
Unlimited Game
Jacoby Beaver
pip: 24
score: 0

Blue is Player 1
Blue to play 13

1.Rollout13/off 3/2* eq: +1.2064
98.95% (G:22.73% B:0.30%)
1.05% (G:0.00% B:0.00%)
Conf.: 0.0007 (+1.2057...+1.2072) - [100.0%]
Duration: 2 minutes 14 seconds
2.Rollout14/offeq: +1.1994 (-0.0071)
99.88% (G:20.11% B:0.10%)
0.12% (G:0.00% B:0.00%)
Conf.: 0.0006 (+1.1988...+1.2000) - [0.0%]
Duration: 22.8 seconds
3.Rollout13/off 1/offeq: +1.1887 (-0.0178)
98.88% (G:21.18% B:0.17%)
1.12% (G:0.00% B:0.00%)
Conf.: 0.0010 (+1.1877...+1.1896) - [0.0%]
Duration: 3 minutes 24 seconds
4.Rollout14/3 4/1eq: +1.1498 (-0.0566)
99.86% (G:15.24% B:0.04%)
0.14% (G:0.00% B:0.00%)
Conf.: 0.0005 (+1.1493...+1.1503) - [0.0%]
Duration: 16.4 seconds
1 5184 Games rolled with Variance Reduction.
Dice Seed: 67389374
Moves and cube decisions: XG Roller+

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