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Optimal non-contact bearoff distribution

Posted By: Timothy Chow
Date: Wednesday, 14 January 2015, at 10:01 p.m.

In Response To: Optimal non-contact bearoff distribution (leobueno)

It may be worth pointing out that it does not follow from the optimality of the 7-5-3 position that one should, while bearing in, strive to arrive at that position.

1. First of all, the goal, in the end, is to the win the race, and this is not exactly the same as minimizing the expected number of rolls to bear off. Perhaps someone with access to a two-sided database can answer the following question: Suppose the opponent has the 7-5-3 position, and I have not borne off any checkers yet. Of all 79-pip positions that I could have, which one gives me the highest equity? Does the answer change if I give my opponent an "optimal" position with 59, 69, 89, or 99 pips?

2. Note that it's not always possible to move from one optimal position to another. For example, Mike's table says that the arrangement below is optimal for 86 pips.

Money Game
Blue on roll. Cube action?
White86

Blue86
GNUBg Id: 2PdTAADY91MAAA:cAkAAAAAAAAA
XGID=----BEF-AA------aa-feb----:0:0:1:00:0:0:0:0:10

If I roll 43 or 52 then I can arrive at the 7-5-3 position, but not if I roll 61. More generally, I suspect that the strategy of "maximize my probability of getting the 7-5-3 position if I happen to have exactly 79 pips at some point" is not the same as the strategy of "minimize the expected number of rolls I have to bear off." The 7-5-3 position is pretty inflexible-looking, and it might be the case that if you aim for the 7-5-3 position, usually you won't hit it, and instead you'll reach a position that is actually fairly wasteful. I suspect it may be better to aim for a more flexible position that is less likely to hit the 7-5-3 position specifically, but is also less likely to lead to an awkward position as a result of a bad roll.

3. The 7-5-3 position is sometimes extrapolated to a general principle of "bring your checkers in to the 6pt exactly if you can so as not to waste pips." This principle is certainly not correct in general.

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