Rebidding after an overcall

Here's an area of bridge theory that seems to me hasn't garnered a lot of attention: rebids when your RHO has overcalled. As far as I know, there are just two conventions cater specifically to this situation: Larry Cohen's so-called Good-Bad 2NT and Eric Rodwell's support double. Both calls are conventional only when made by opener.

The second is the better known convention and we can summarize it as follows (this may not correspond to Rodwell's original, but it's how I think it should be played):

you open 1 grape, LHO passes, doubles, or bids 1 diamond, partner bids 1 plum (such that he only promises four plums), and RHO makes a simple overcall (or 1NT showing two suits). A raise to two plums promises four (or more) plums and double promises exactly three plums. Note that you must have the option of bidding two plums to show four (that's to say RHO has overcalled 1♠, 2♣, 2♦, or 2♥). By extension, if RHO doubles, a redouble also shows exactly three plums.

A support double doesn't say anything about strength but most of the time it shows a balanced hand, which must therefore be in the range of a 1NT rebid (whatever that might be) or possibly a 2NT rebid.

Now let's look at the Good-Bad 2NT, a kind of Lebensohl extension. In the same auction as described above (although I believe it only applies if RHO bids one of 2♣, 2♦, 2♥, or 2♠). A bid of 2NT is not natural, but shows a two-suited hand that is not strong enough to bid again with equanimity. Responder now bids 3♣, assuming pass follows 2NT, and opener either passes or bids his second suit.

While I like the idea of the convention, it does seem to have at least one flaw. Suppose your hand is ♠KQ53 ♥T97 ♦J543 ♣K2. Partner opens 1♦ and you bid 1♠, and now your LHO bids 2♥. Partner rebids 2NT (Good-Bad). It's looking like partner has the minors. Clearly, we can't bid 3♣ because when partner passes it would be disastrous. So we should bid 3♦ instead (as long as partner doesn't assume this shows extras). Presumably this is how it's played though I've never seen that written up before. I'd certainly like to hear comments on this aspect.

So, let's assume that we are playing the GB2NT and the support double as described. Let's assume we have the same hand as before and the auction is the same up to the point of LHO's 2♥. We can now summarize the calls that partner might make:

• 2♠: a minimum four-card raise of spades, perhaps ♠J964 ♥8 ♦AQ96 ♣KQ93;
• 2NT: a minimum two-suited hand, in this case the minors, possibly ♠J4 ♥8 ♦AQ986 ♣AQT93;
• 3♣: a good two-suited hand, in this case the minors, perhaps something like ♠A4 ♥8 ♦AQ986 ♣AQT923;
• 3♦: a good rebiddable diamond suit, possibly ♠A4 ♥82 ♦AKJ953 ♣Q93
• 3♥: a good hand that might make 3NT if we have hearts well stopped, perhaps ♠A4 ♥82 ♦AKJT953 ♣K3
• pass: relatively balanced hand with fewer than three spades and indeterminate strength. We might make a BOP double with our hand, although 3♦ would also be a reasonable call (we know partner has at least four diamonds amongst his 11 or more non-spades).
• double: relatively balanced hand with exactly three spades and indeterminate strength -- all calls by our hand, except a cue-bid, are non-forcing.

Now let's look at a slightly different auction: 1♣ p 1♥ 2♦. Now, the fourth suit, spades, if bid by partner, would be a reverse. It is possible to swap the strengths of 2♠ and 2NT. Why not let 2S be a minimum reverse (not a bad hand, but not up to full strength) and therefore passable and let 2NT (forcing) show the fourth (higher-ranking) suit with a good hand? In the case of 2♠, opener might have only four clubs and something like a good 14-16. In the case of 2NT, opener promises 17+ hcp and 5 or more clubs. After 2NT, responder bids 3♣ or 3♠ "to play" while a red suit would be game-forcing.

Suitable hands for partner in this sequence might be:

• 2♠: ♠KQ53 ♥Q5 ♦83 ♣AK652
• 2NT: ♠KQ53 ♥Q5 ♦A3 ♣AK652

Comments welcome.