A Few Pips Can Make a Big Difference

Gerryby Gerry Tansey

The following position came up the other night during the weekly tournament.  I was leading 5-2 in a match to 7 when White doubled me.  I remember most of the details, but there was one White checker whose location I couldn’t quite remember later.  It was either Position 1, Position 2, or Position 3, which I have listed below.

Position 1

Tansey 1

Position 2

Tansey 2

Position 3

Tansey 3

White clearly has a strong double in all three positions.  White’s 13 hitting numbers usually win the game, and often lead to a gammon.  White has extra incentive to turn the cube at the score, since Black will never redouble, and a doubled gammon gets White to the Crawford game.

But what happens when White misses?  Black is then heavily favored to cover his blot on the 4 point, resulting in a prime-vs-prime position that is fairly symmetric.  When White’s miss contains a 1, he may be a bit better off since he can split his back checkers and get to the edge of the prime.  But White may miss by rolling a big set of doubles instead, after which he will be an underdog when Black makes his own 5-prime.

A very crude way to analyze Position 1 is to let each of White’s 36 rolls represent a game, and then to estimate how many games each side will win.  So in this example White hits in 13 games.  I’ll estimate that White wins 12 of those games (it is rare when we can assign 100 percent of the games to one side in a position with contact – this is backgammon we’re talking about, after all!).  For the remaining 23 games, when White fails to hit, I’m going to split these right down the middle, 11.5 to each side.  It’s crude, yes, and definitely not 100 percent accurate, but it is what I’m capable of doing over the board.  So we end up with 23.5 wins for White, 12.5 wins for Black.

What about gammons?  Here I’m going to estimate even more crudely than before.  I’ll just say that White wins a gammon in 9 of the 13 games in which he hits, and maybe one or two of the games in which he doesn’t.  So 10-11 games out of 36 are gammon wins, perhaps 30 percent, maybe a smidge below.

That’s about as far as I got over the board.  Below is Position 1M, which shows a rollout of the cube action in Position 1 as a money game.  As you can see this is a huge money take (and also a huge double).  Black should be a little afraid of getting hit here, but that doesn’t mean Black should pass this in a money game.  Black should relish the prospect of owning the cube in the majority of positions when he is missed, since he can make White pay when the prime-vs-prime game goes Black’s way.

Position 1M

Tansey 1

 

Analyzed in Rollout No double Double/Take
Player Winning Chances: 64.09% (G:25.42% B:2.49%) 63.78% (G:26.02% B:2.80%)
Opponent Winning Chances: 35.91% (G:10.07% B:0.63%) 36.22% (G:10.27% B:0.63%)
Cubeless Equities +0.454 +0.910
Cubeful Equities
No double: +0.492 (-0.143) ±0.014 (+0.478..+0.507)
Double/Take: +0.635 ±0.022 (+0.614..+0.657)
Double/Pass: +1.000 (+0.365)
Best Cube action: Double / Take

You can see that our estimates were a little off, but close enough to get the right decision.  It seems I overestimated White’s gammon chances, and completely ignored his backgammon wins (which are small but not trivial).  Of course, during the match, I did not care about my own gammon wins, as they are worthless on a 2-cube at the score, but in a money game, Black does win some gammons of his own.  It all adds up to a very easy money take, but a much more difficult decision at the match score.

Now, if we compare Positions 1, 2, and 3, we should note that Black should be even more eager to take the cube as we advance White’s spare checker.  This is due to the fact that when White misses Black’s blot, the game mostly turns into a prime-vs-prime battle.  It is better to be behind in the pip count in a symmetric prime-vs-prime game, because the player who is behind has more time to play rolls without busting his front position.  By the way, all three positions are still doubles for White in a money game – the position is just so volatile that the cube has to be turned immediately.  But now I’m going to show you the rollouts for the three positions at the match score of 5-2 to 7, just to show you how tough a game backgammon can be:

Position 1

Tansey 1

 

Analyzed in Rollout No double Double/Take
Player Winning Chances: 64.31% (G:25.68% B:2.30%) 64.11% (G:25.82% B:2.29%)
Opponent Winning Chances: 35.69% (G:10.06% B:1.67%) 35.89% (G:10.24% B:1.91%)
Cubeless Equities +0.334 +1.050
Cubeful Equities
No double: +0.664 (-0.336) ±0.012 (+0.652..+0.676)
Double/Take: +1.050 (+0.050) ±0.017 (+1.034..+1.067)
Double/Pass: +1.000
Best Cube action: Double / Pass

Position 2

Tansey 2

Analyzed in Rollout No double Double/Take
Player Winning Chances: 62.94% (G:25.56% B:2.40%) 63.26% (G:25.01% B:2.16%)
Opponent Winning Chances: 37.06% (G:10.66% B:1.46%) 36.74% (G:10.19% B:1.70%)
Cubeless Equities +0.296 +0.980
Cubeful Equities
No double: +0.619 (-0.361) ±0.013 (+0.606..+0.633)
Double/Take: +0.980 ±0.018 (+0.962..+0.998)
Double/Pass: +1.000 (+0.020)
Best Cube action: Double / Take

Position 3

Tansey 3

Analyzed in Rollout No double Double/Take
Player Winning Chances: 60.18% (G:25.04% B:2.49%) 60.28% (G:24.71% B:2.54%)
Opponent Winning Chances: 39.82% (G:10.78% B:1.54%) 39.72% (G:10.64% B:1.63%)
Cubeless Equities +0.235 +0.847
Cubeful Equities
No double: +0.539 (-0.308) ±0.013 (+0.526..+0.552)
Double/Take: +0.847 ±0.017 (+0.830..+0.864)
Double/Pass: +1.000 (+0.153)
Best Cube action: Double / Take

In the course of moving the spare checker 6 pips, we go from a medium-small pass, to a borderline take, to a huge take.  The gammon wins don’t change much – gammons happen mostly when White hits on the first roll.  But Black’s winning chances go up over 4 percentage points just by moving one of White’s checkers six pips, with no other changes in the structure of the position.

Like I said, I don’t remember which position I was actually facing over the board.  I do remember that I got hit… and then backgammoned for the match.  Tough game, this.

 

 

 

 

 

3 thoughts on “A Few Pips Can Make a Big Difference

  1. You were probably still flustered by your opponent’s monumental cube error in the previous game (!)… Yet despite this you still made the right decision to take here, as I’m 99% certain the position was #2.
    I’m curious about one other factor that your stellar (as usual) analysis ignores — strength of opponent. The rollouts assume an equal opponent, no? But given that the rollout was such a close decision, should you, the vastly superior player in this case, PASS anyway and “wait for a better spot” as we say in poker — given the relatively high gammon chances you’re currently facing?
    The answer in this specific case may easily be no, and probably is, so I guess the better question is, are there times when strength of opponent *should* factor in??? I have made a lot of money in poker by passing on slightly +EV decisions simply to fade the variance against a weaker player and “get him later” so to speak in a much higher +EV situation. I’m curious if the same logic applies to BG sometimes (although I realize there’s not much “later” in a match to 7…).

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    • This is a good question. I’m not really sure what the answer is. One thing that crossed my mind in this particular instance is that if I pass at 2-away/5-away, I put you at 2-away/4-away, which is a score fraught with its own volatility issues. I think it’s possible I’m supposed to pass because of the skill difference, but I’m not sure where to draw the line. A strong player told us he would not have given the cube I gave in the previous game (at 5-0/7), but my gut tells me this is giving away too much (since it was something like a 0.400 pass).

      There is a book out there by Trice and Jacobs called “Can a Fish Taste Twice As Good?” which gets into these issues. I haven’t read it, but I think one of the conclusions is that the stronger player should pass races but not necessarily pass gammonish positions (especially if there is some skill left in the position if things go wrong for the weaker player). But since I don’t really know what adjustments to make, I don’t tend to make many. Maybe I’ll pass a very close take in a racing game, but that’s about it. XG still beats me like a drum, and it makes no adjustments whatsoever.

      I did once see MCG pass a backgame against one of the weaker open division players that was not even technically a double, so there may be something to this. There was also this discussion on the BGOnline forum recently. http://www.bgonline.org/forums/webbbs_config.pl?read=171753

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