Here's the latest example of my lack of understanding of post-contact probability. At an 84 (+9) pip disadvantage, I accepted the following cube with little reservation:



My thinking was that, over the course of 84 pips, my disadvantage could easily be overcome. A single double would do the trick, and I'm as likely to throw one as my opponent, so let's race.

As it turns out, GNU gives me an only 20.6% chance to win from this position! Clearly, I'm failing to appreciate the mathematics here. Why are my chances so low? Are there recognized rules of thumb for these kind of post-contact decisions? Relevant book recommendations would be welcome.

You're at a 9 pip disadvantage... BEFORE the roll. It will be greater after Red rolls, of course.

You say you will need a double to catch up. Well... that's only true if you roll a HIGH double. Double 1s and double 2s, and most likely double 3s, for example, won't cut it.

I suspect it wasn't a BLUNDER to accept the double... just a mistake.

If I remember correctly, a good rule of thumb is that in races of 70–120 pips or so, a lead of 10% plus means the leader will win about 75%. Red's lead is a higher percentage than this, so for that reason I would have dropped, assuming I remembered the rule of thumb correctly.

Pip count is the most important but not the only criterion. For instance black needs 4 crossovers and red only 3 before he can start to bear off. The distribution of red's homeboard checkers is also a little bit better. If you transfer one checker from 23 to 20 and in compensation the checker from 10 to 13 the pip count would be the same but it would be a take.

I would check out the Keith Count and Reichert Count on bkgm.com. Both counts put you just shy of a take (though they don’t always work).

Both authors original articles are here:

https://www.bkgm.com/articles/page04.html#races

Although the glossary might be offer a quicker summary:

https://www.bkgm.com/glossary.html#K

As for a book - Conquering Backgammon by Ed Rosenblum is a modern introductory text which covers both these counts (and much else).

It’s not just that you need a double to catch up. You need a high double AND you need your opponent to not get a high double to compensate. You are only considering the probability that you might get your double, but ignoring the fact that your opponent could get one too.

Good point about my disadvantage being BEFORE the role, soon to worsen. I'll remember that. By % lead, do you mean % difference?

Diff/Ave = 9/79.5 = 11.3%

That's a useful rule of thumb. Thank you.

Good point about crossovers, followed by a great illustrative example. Thanks for that.

Thanks for the resource leads. I'll definitely be looking into these.

Point well taken. I suppose there are three possible outcomes: I roll fewer doubles, I roll more doubles, and we roll the same number of doubles. I doubt these are weighted equally, but only one is desirable.

Here's how I would handle this position.

For races with checkers still in the outfield, I use the 8-9-12 rule. Initial double if the leader is up 8% in the pip count, redouble if up 9%+, and pass if the trailer is down more than 12%. The rule dates back to the 1970s and was the first computer-developed backgammon rule. It's simple yet highly reliable (though not perfect).

Here the leader's count is 75 and the lead is 9, so 9/75 = exactly 12%. At this point I'd look at the position to see if I can break a tie one way or another. Notice that Red's position is better than Black's in two ways. (1) He needs 3 crossovers to get all his men home while Black needs 4. (2) His structure is smooth, while Black has too many checkers on his 2-point and not enough on his 5-point. That breaks the tie, so it's double and pass.

That's some wonderful history - I love it. "Simple yet highly reliable" is just what the doctor ordered. It seems to put the greatest emphasis on pip count, and relegate the consideration of other factors like crossovers and structure to cases that are close to the 12% thresh-hold.

One question. If the % "lead" is 9/75 (at this 84-75 pip count), wouldn't that make 9/84 the % "trail"? Or is that being pedantic, and we just take % trail to mean trailing by the % lead?

Upon further reflection, @Robertie 8-9-12 rule seems commensurate with @EdwardCollins 10% lead=75% winning chances rule. Better yet, the pip difference required for a 10% lead is evident upon inspection (just move the decimal!). Fusing these rules, one can then compare the 10% pip difference to the actual pip difference and adjust accordingly for the 8% double and 12% drop.

In our 75-84 (+9) example, a pip difference of +7.5 is required for a 10% lead. Thus, the actual pip difference of +9 puts the trailer very close to the 12% drop.

Or in a 63-68 (+5) game, a pip difference of 6.3 is required for a 10% lead. Since the actual pip difference is +5, the leader has met the 8% doubling thresh-hold.

*Musings of a non-acrobatic mathematician.