In Part 1 of this post I noted that the play given in 501 of 13/12* 7/3 is actually not the best play. For easy reference the position is:

Red to play 14

Pip counts: Blue 121, Red 83

Position ID: `m22BgEG2cxsEAA` Match ID: `QYkQAAAAAAAA`

The better play as noted in Part 1 is 13/12* 8/4, but the real question is why is this play better?

Certainly having an extra spare on the 4 point as opposed to the 3 point does not seem to gain much in this position as you would normally prefer to diversify the spares.

Perhaps the best question to ask here is what loses the game for red, and that answer is pretty easy, blue must get a hit. Not only must blue get a hit, but that hit needs to come pretty quickly because blue does not have the timing to play out an extended backgame or ace point game. Their board is going to crunch soon, assuming they get the two checkers off the bar.

So red needs to make sure that blue does not get a quick shot. With either of the two plays under consideration there is not chance of blue hitting the red blot on either the 7 or 8 the next time around. So the answer must lie deeper.

There are several possible scenarios on blues next roll, these are:

16 Rolls get 1 checker off the bar (split evenly between coming in on the 1 and 5)

16 Rolls dance

4 Rolls get both checkers in 2 rolls bring checkers to the 1 and 5 point, 1 roll makes the 1 and one stacks the 5. (yes technically these are blues 24 and 20 points, but we are looking at this from red’s perspective).

Let’s look at the case were blue brings in one checker on the ace point.

After red plays 13/12* 7/3 and blue plays bar/24 we have the following position:

Red on roll

Pip counts: Blue 132, Red 78

Position ID: `m20BwFB25xoCAA` Match ID: `QQkAAAAAAAAA`

There are a lot of good rolls for red in this position, 66, 62, 63, 65, 52, 53, and 32 all make the ace and put blues checker back on the bar. Of course that was the whole reason for playing 7/3 was to maximize the rolls that made the ace point.

But since we are trying to figure out why this is the weaker of the two plays we are considering we need to look not only for the good rolls for red but also for the not so good rolls. Seeing the bad rolls in this case is not so easy, at least for me. To help out with the analysis I used the temperature map feature on gnubg, so here are the equities and best moves based on a 2ply evaluation. (I used 2ply because both 3ply and 4ply give 7/3 as best while 2ply gives 8/4 as best in a basic evaluation) for red on this roll.

Now let’s compare this to the temperature map after moving 13/12* 8/4 and blue moving bar/24.

This is beginning to give us some hints as to why 8/4 is better than 7/3. Now we have to use a little caution here as the gnubg evaluation is not necessarily correct, but this is a good tool to help us see what might be going on.

Note that after 8/4 the worst roll for red is 31, and playing this correctly gives red an equity of +1.011. But after 7/3 31, 42, and 43 are all worse for red than the worse roll after 8/4. There are also 3 rolls for red after 8/4 that are better than the best roll for red after 7/3. Of course we are not looking at the whole picture here, just the case where blue enters one checker on their 24, but we are beginning to see a way to attack this analysis in detail. In part 3 of this post we will look at little more closely at why some of these rolls are worse after 7/3.