Reply.
Please note that I said that �almost never worth it.� There are some very rare situations where it is optimal to double for less. Yes, they do come on doubles where a fourth card isn�t needed. (And possible against a 2 or 3 where a 4th card is unlikely)
However Money Management is NOT one of the reasons to do this, absent special circumstances like a tournament.
The reason one might do this is Risk Aversion. However, on these hands where you don�t need a fourth card, the doubling EV is huge. So the cirtical fraction is quite high.
Here is a slightly simplified analysis. As an example, let me look at A6 v 4. The EV for hitting is about 6%, the EV for doubling is about 12%. This is off the top of the deck.
This means that you optimally want about 6% of KEB in the betting square. If you had originally bet as much as 4% of KEB, you would want to add another 2% on your double.
However, there are rare situations where we would bet less than 3% of the top of the deck. In the other situations, you double for the full amount.
For other doubles, like 11 v 4, the EV is MUCH higher, making these fractions much higher.