One of the strangest phenomena of match-play backgammon is the idea that at certain scores you can be "not good enough to double" and "too good to double" at the same time. How can this be?
A Normal Match ScoreBefore we delve into the weirdness of strange match scores, let's look at a more common situation for comparison, which is illustrated in the chart below.
The x axis is your (cubeless) game winning chances. The colored lines on the chart represent your cubeful equity in terms of match winning chances: The red line shows your equity if you don't double this turn. The green line shows your equity if you double and your opponent takes. The blue line shows your equity if you double and your opponent drops.
Obviously, if your equity by not doubling is higher than your equity after double/take, you should not double (you're "not good enough"). If your equity after double/take is higher than your equity by not doubling, then you should double. Unless you are "too good to double" -- that happens if your equity by not doubling is higher than your equity after opponent drops your double.
If you know your game winning chances (gwc), you can figure out the proper cube action from this chart:
- If your gwc is less than 67% you are not good enough to double.
- If your gwc is between 67% and 74%, you should double and your opponent should take.
- If your gwc is between 74% and 83%, you should double and your opponent should drop.
- If your gwc is greater than 83%, you are too good to double -- you are better off playing on for a gammon.
Everything so far falls in line as you expect. As your game gets better, you go from being not good enough to double, to being good enough that you should double even though your opponent can take, to being good enough that your opponent must drop if you double, to being so good that you are better off playing on for a gammon. This is the way it is in money play, and it's the way it is at most match scores.
An Unusual Match ScoreBut at certain scores things can get weird. Look at the following chart. It shows the situation when you are leading 4-away/8-away, holding a 2-cube, and have good gammon chances.
As we saw before, you don't become good enough to double until your equity after double/take is higher than your equity by not doubling. In this case, that doesn't happen until your gwc reaches 89% (where the red line and green line intersect).
And, as before, you become too good to double when your equity by not doubling is higher than your equity if opponent drops your double. On this chart, that happens at 83% gwc (where the red line and blue line intersect).
Do you see the problem? You become too good to double before you're even good enough to double! Or, putting it another way: When your gwc is between 83% and 89%, you are too good to double and not good enough to double at the same time.
Why does this happen? There are two things that combine:
- With your opponent at 8-away and you holding a 2-cube, you have to be very cautious about redoubling -- much more cautious than normal. You're enjoying a nice lead in the match and don't want to squander it. If you redouble to 4, your opponent will flip the cube back at you so fast it will make your head spin. And suddenly the match is on the line. To guard against this, you must hold off doubling much longer than normal.
- You have good gammon chances (we've assumed 40% of your wins will be gammons). Since the cube is already on 2, winning a gammon gives you exactly the four points you need to win the match. You don't have to turn the cube to win; you can just cruise along and get your four points the easy (and safe) way, by winning a gammon. For that reason, you become too good to double earlier than normal.
These two factors produce a situation where you are too good to double and not good enough at the same time. Your doubling window has disappeared! As long as a substantial portion of your wins are gammons, you should never double at this score.
Strange but true.