5-point Holding Games : Poker Player

5-point Holding Games

One of the most common formations in backgammon is the 5-point holding game. You anchor on your opponent's 5-point, but are hit several times in the game. You make your board but trail in the race.  Often there will be an exchange of hits that leave a marked imbalance in the race.

In his excellent book "Advanced Backgammon," Bill Robertie expostulated the following treatment of the "normal" 5-point holding game. The "normal" holding game is one where the holding side has 2 checkers on the holding point and all other checkers past the midpoint, and the attacking side has the midpoint and 8-point. The suggested treatment was:

a) The attacking side needs a slightly larger advantage than in a straight race to double
b) The position is always a take - as racing chances decrease, shot chances increase
c) If the attacking side has any additional outfield points as landing spots, the position should be evaluated as a straight race.

Early on, I added a corrolary to Robertie's rules, which is that the position is a pass if there is any reasonable chance of being outraced to a gammon. This tipped positions to a pass once the racing deficit gets into the 55-60 pip range.

Let's look at a few positions and see how these rules hold:

Leading by 10 pips
Leading by 10 pips, it is no double and a trivial take. We are leading by about 9% in the race, so it would be a double and take if there were no contact.

Here we have an extra landing space. Our winning chances are up a bit, but it's still not quite a double. We don't lose a lot by doubling, because in most sequences both sides will move forward a bit and we only need to gain a pip or two to get into the doubling window, but a pass would be a grievous error. White can play constructively for several turns, and 44, 55, or 66 would make him a favorite.

Leading by 20 pips
This is a close double and a close take. Although we have a solid advantage, the volatility is fairly low. The only market losers are large doubles (33 and above). If we had only 2 checkers on the midpoint, 22 would be a market loser also, as we could play 13-9(2) and have a fairly safe bearin with a nice racing lead. Still, it is a comfortable take even though we trail by 18% in the race.

When we add the bar point as a landing spot, things change. It is now a close pass, with winning chances over under 78%.

Leading by 30 pips
This is the first position where we get a different result than Robertie. A Snowie 3-ply evaluation makes this a comfortable take (by about 0.050) but it is a slight pass on a rollout. Any number of factors could make it correct to take - the difference of .009 could easily be outweighed by considerations of match score, strength of opponent, or even just if you feel lucky or not! Because of the low volatility of these positions, failing to double is not terrible, as White is unlikely to have a big pass next shake. Still, there is no reason to give him a free chance to roll something like 66 (which would tighten up the race) or 22 (which lets him step up to the bar point and directly pressure the checkers on the midpoint).

Adding the bar point makes this a very solid pass - not a surprising result at all. Failing to double costs only a little, but only because there is very little White can do to get back in the game. A small number followed by 66 for White will leave Black with a solid but not quite doubling advantage, and if Black shakes 65, 44, or 22 he will move 2 off the midpoint, and a hit will give White the edge. But that is still only 20/1296 really bad sequences.

We Lead by 40 pips
Our win percentage here is almost the same as in the 30-pip position, although we lose a handful more gammons. It is still a pass, of course. But something interesting happens if we move one checker.

If we strip the midpoint, we win an extra percent and a half more games and gammons, and it becomes a very solid pass. The extra value we get from 22 and 33 (they come much closer to truly clearing) and our being able to hold the 8-point after 62 are not insignificant.

We Lead by 50 pips
I found this a bit surprising. Our gammons go up, but wins drop significantly, from 77.3% to 75.2%. I did 1296 rollouts on each position, so this is not an insigificant sample, and the range of statistical variation is relatively small. Black is completely out of time and will have to roll well or start leaving shots.

However, when we move the spare on the midpoint up to the 8point, it is now a solid pass. In this position we have an extra roll before things start to come to a head, and the same numbers play better that we looked at in the earlier position. The message - it's not all about the race, it's about what Black has to do to clear.

We Lead by 60 pips
Now, even with the 8-point stripped and the spare on the midpoint, the gammons are high enough that it's a solid pass. We win more games in this variation than in some others, but the gammon losses are just too many. We are one big set of doubles away from being favored to win a gammon, and that's too much for White to risk.

The Bottom Line

Most of the conventional wisdom is correct. However, there appears to be a "sweet spot" where the attacking side has an almost insurmountable racing lead but still has enough time to have several chances to clear the midpoint. In close cases, the outfield structure appears to be fairly important. Having a stripped midpoint can add as much as 0.10 to the attacking side's equity.

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