It is easy to see where Black went wrong. Prior to rolling Black yelled “gimme a four!” The dice snickered “tee hee!” and obliged him.
At first glance there are three obvious plays: 6/2, 6/3; 20/13, and 15/8. I would lean toward the latter, while Snowie would play 20/13. I don’t know who would play 6/2, 6/3, but I am sure someone would, and perhaps they would be right. Let’s see.
First, we’d better make sure we have found all the candidate plays. Since playing our three first leaves us fewer choices, why don’t we try each possible four, and then see what threes go with them?
If we play 6/2 we can play a three from our six-, five-, or four-point. I hope we agree that playing from the six-point is superior to those other two, so we can ignore them.
If we play 5/1 then the only three we would consider is 5/2. This leaves no blots, while moving the gap higher, so it is, as they say, a “real play.”
If we play 15/11 we know 11/8 must be better than 6/3, so we can ignore 15/11, 6/3. (It may or may not be better than some of the other plays under consideration, but we can be fairly sure that it cannot be better than 15/8 so we needn’t waste our time.) Similarly, we know that 20/13 must be better than 20/16, 6/3, and ignore that also. But perhaps we are overlooking a different play of the three? Yes! We also have available 20/16, 5/2 and 15/11, 5/2; neither of those plays may be rejected out of hand. To our original triad we have added: 20/16, 5/2; 15/11, 5/2; and 5/1, 5/2. Interesting that all three plays have in common covering the deucepoint.
A 3-ply evaluation by Snowie 4.1 ranks its candidate, 20/13, as best (of course) with 15/11, 5/2 as a close second, my 15/8 as third, and 20/16, 5/2 as fourth. It sees relatively little difference between the top choices. It does not care for 5/1, 52. which it sees as a blunder, ranking between 20/16, 6/3 and 15/11, 6/3. Snowie hates 6/2, 6/3.
Before we roll the choices out, let’s try to reason our way to the best choice.
What is wrong with 6/2, 6/3? Covering the deuce is good, but putting another builder lowdown is not (the main drawback to 5/1, 5/2 which piles another man on the ace). If White rolls a six we have no counterattack, greatly increasing her chance of anchoring on our six before we have brought our three blots around.
Between 20/13 and 15/8 I still prefer 15/8. Neither play is crushed by anti-jokers; if I knew I was going to roll 33 or 44 next I’d slightly prefer 20/13, and if I knew I was going to roll 55 or 66 I’d prefer 15/8, but it isn’t a big deal. I think 15/8 suffers more from multi-roll parlays, where after closing the board I am unable to extricate the last man back, but it has much better counterattack when White rolls an immediate deuce.
That leaves 20/16, 5/2 and 15/11, 5/2. Is there merit to shifting the slot? We do want to close White out if we can, but if we fail the deuce will be a problem for us. Keeping White behind a four-prime is good when we may have a lot of work still to do, but this late in the game the deucepoint (or even a phantom deucepoint game, where White stays on the bar, but we cannot close the deuce) is White’s best chance. Covering with the three is looking better when it works. How bad is it when White rolls an immediate five? Not bad at all. If we played 15/11 with our four we will have twenty-one shots back at White from the bar. We win nearly 80% after being hit and nearly 80% of our wins are gammons.
It is still too close for me to call, so we’ll do some rollouts. After a first pass 5/1, 5/2 proves, as predicted, to be a blunder, and 6/2, 6/3 is a triple whopper. The other four are still close, but we can finally and firmly eliminate those two candidates. Meanwhile, the top four have shifted positions. The new top choice is 15/11, 5/2, followed by 15/8, followed by 20/16, 5/2. Snowie’s original favorite, 20/13 is now looking like an error.
The first pass was 360 games, done at 3-ply, standard settings. After several extensions bringing the total games for each candidate up to 2160 these are the rankings.
15/11, 5/2 has an expected value of 1.550, winning 89.5% (80.9% gammons won, 1.5% gammons lost);
20/16, 5/2 has an expected value of 1.544, also winning 89.5% (80.0% gammons won, 1.2% gammons lost);
15/8 has an expected value of 1.535, winning 88.9% (80.9% gammons won, 1.6% gammons lost – and a smidgen more BGs won for those checking my math.
20/13 has an expected value of 1.524, winning 88.9% (79.5% gammons won, 1.2% gammons lost).
Interesting, isn’t it? This problem started out by rejecting covering, when it was done with the obvious four, only to embrace it, when it was done with the unobvious three.