This is a theme that is not well explored in the literature, or on the web. A recent discussion within our social media groups has prompted me to write down some principles that will help your partnership to deal with this rather common situation.
The weight of vulnerability
Vulnerability is an important consideration in almost all competitive decisions, but this particular case is one in which its role is even more crucial. Some facts must be kept in mind by all players when they are deciding how to react after 1NT is doubled for penalties:
Vulnerable overtricks are worth 200. Nonvulnerable, they are worth only 100. This means that the scale of rewards in your doubled 1NT contract can be either 180, 280, 380... (if nonvulnerable) or 180, 380, 580 (if vulnerable).
People are more familiar with the scale of rewards for defeating a doubled contract, but for completeness' sake, they can be either 100, 300, 500, 800, 1100... (nonvulnerable) or 200, 500, 800, 1100... (vulnerable). The two sequences are identical after 500 (identical if you keep in mind that a nonvulnerable player must go down one more trick to pay the same penalty as the vulnerable player).
When you want to escape from 1NT doubled, these numbers must be very clear in your mind. You have to judge which is the pathway that gives you the highest (or, the least worst) expected value.
Some defenses against a penalty double "turn the cube" and force the partnership to play 1NT redoubled . These scales become even more important in that case, because 1NT redoubled is game. The possible values are:
For making your contract:
560, 760, 960... (nonvulnerable)
760, 1160, 1560... (vulnerable)
For going down:
200, 600, 1000, 1600... (nonvulnerable)
400, 1000, 1600... (vulnerable)
Should I stay or should I go?
Keeping the above numbers in mind, there are some systemic considerations to be discussed now. The decision of sticking it out or running is heavily influenced by your agreements. If you cannot describe your hand appropriately, the situation becomes more problematic.
For example, suppose your partner's weak notrump opening is doubled and your escape methods are the very very simple "system on", meaning, 2 Clubs is Stayman, 2 of a red suit are transfers, 2 Spades are clubs, 2NT shows diamonds. (Although I am not endorsing this method, it has the very clear advantage of simplicity and of often placing the contract in your partner's hand, which is usually desirable. Doubler has the majority of power and partner must have some honors to be protected from the lead). The only "extra" is the redouble, which you use to escape to 2 of a minor (partner bids 2 Clubs and you correct to diamonds if you have long diamonds).
This scheme is ok for all hands with 5-card suits. It has some trouble in the weak 4432 and 4441 hands without both majors. (With both majors you bid 2 Clubs immediately). Imagine that you have a weak hand and 4 hearts and 4 diamonds, 2 spades and 3 clubs (i.e. a 2=4=4=3 hand). You will have to stick it out in 1NTx, or guess a red suit.
(By the way, especially if you are playing a weak notrump, guessing a red suit is not as bad as it seems. The opponents often have a game in this situation. And if you guess wrongly they sometimes don't have the right hand to double you there. That said, it is clearly better if you can ask partner's opinion about the reds).
The scheme of responses you chose basically forces you to (either guess or) stick it out in 1NTx with balanced hands. You have a minor escape hatch with 44 in the majors (you can stretch it with 43 there), but otherwise you will be committed to playing 1NTx with those hands. Not so bad with the right vulnerability, but worrisome if you are vulnerable.
In any case there are some clear principles emerging: don't be afraid of playing 1NTx if you have a balanced hand (sometimes this is your safest harbor in the storm. Sometimes there is no safe harbor in the storm). Take it out if you have a 5-card suit and a weak hand (the expectation of 2.5 cards with partner, which is probably close to the mark, shows that this is a reasonable trump suit -- and the weaker you are, the more important it is to get away from notrump when you actually have a reasonable trump suit).
The next post will look at this situation from the other side: your partner doubled their 1NT opening.