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hit or bear off?

Posted By: Timothy Chow
Date: Friday, 11 November 2011, at 3:33 p.m.

In Response To: hit or bear off? (DanielP)

I would hit, too, but I want to mention a few things that others have not explicitly stated.

First of all, of course, either 6/5* or 1/off leaves your opponent 11 shots. As far as winning the game is concerned, you'll win if you're not hit, and if you are hit...well, then it gets a little complicated to analyze, but for a first approximation you can say that the decision is close, and make a mental note to come back to it if other considerations are also close.

The next thing to consider are gammons, which are especially important since your gammon value is 1 at this score. The key questions are:

1. What is the minimum number of pips your opponent needs to get off the gammon? Here it's 47 pips if you don't hit. That's about 6 rolls on average.

2. How many rolls will you need to bear all your checkers off? Note that bearing off two checkers here will leave you with 13 checkers, and 13 is odd, so bearing two does not help you as much in the race as it would if you had an even number of checkers. That's already a hint that you probably don't gain much by bearing two. Anyway, with 13 or 14 checkers, it will probably take you 6 or 7 rolls to bear off (you might roll doublets but you might also miss).

The gammon race is therefore close, so hitting should gain quite a lot. It gains 5 pips and also puts your opponent on the bar against a four-point board, so she might dance and lose even more pips. It's enough of a factor that you shouldn't need to go back and worry about the second-order effects that we set aside earlier. If you were far ahead or far behind in the gammon race then you would probably need to think harder.

By the way, the crude assessment of the gammon race I gave above (number of pips of your opponent vs. number of your own checkers) is, I think, good enough for this problem, but I'd be interested in hearing if others have more sophisticated methods. Doug Zare recently published an interesting article on Gammon Village on this subject that I plan to study further.

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