Inviting game after single major raise

This post focuses on the hands that are borderline between passing and making a game try after you open 1H/S and partner raises to 2. I'm making the following assumptions:

1. Partner will have three card support only.
2. Partner's HCP range is about 4-8 (but not 4 HCP if 4333). If you play 2/1 GF then you probably respond 1NT on very weak hands with support so your range is more like 6-9. I'm assuming 2/1 responses can be only around 9+ and will include the hands with three card support and 9 HCP. If not, adjust the strength of the opening hand accordingly.
3. Partner will always decline the game try with 4-5 HCP and always accept with 8. With 6-7 he will choose the correct action 70% of the time (because your game try will enable him to evaluate more accurately).
4. For the IMP calculations, I assume they pass at the other table.
5. I'm also disregarding the possibility that they will balance if you pass. This is obviously quite likely and argues in favour of bidding, not so much to shut them out, but because you no longer have to worry about losing IMPS in the case where the other table just make the two level contract your way.

5431 Hands

	Tricks	7 or less	8	9	10 or more
AK873	HCP=4	2.01%	3.81%	2.78%	1.10%
KJ65	HCP=5	2.88%	6.28%	6.79%	3.20%
AT5	HCP=6	1.33%	5.37%	8.61%	5.98%
4	HCP=7	0.68%	3.72%	10.19%	10.52%
	HCP=8	0.29%	1.77%	7.45%	15.26%

Expected IMP gain from inviting: Vul 0.52, Non Vul -0.08

The numbers in the table are the percentage of hands in each category according to the simulation. Partner's HCP are on the rows and the number of tricks you take on the columns.

So you show a slight loss by bidding on nonvul but it's worth continuing vul. This was a little surprising to me because I would never consider passing that hand, regardless of vulnerability. Possibly I have not increased my standards for making game tries while I have been progressively lowering my standards for single raises. Let's try with a slightly weaker hand:

	Tricks	7 or less	8	9	10 or more
AK873	HCP=4	3.04%	4.03%	1.97%	0.37%
KT65	HCP=5	3.16%	6.95%	5.49%	1.62%
AT5	HCP=6	2.37%	7.35%	8.49%	3.90%
4	HCP=7	0.73%	5.75%	10.66%	7.76%
	HCP=8	0.36%	3.36%	9.77%	12.87%

Expected IMP gain from inviting: Vul -0.66, Non Vul -0.89

So this one is a clear pass. A rule of thumb is that 5431 shapes need a good 15 to invite (or 39 on the 11-7-4-2-1 scale).


6322 Hands

	Tricks	7 or less	8	9	10 or more
AK8732	HCP=4	1.64%	3.60%	3.08%	0.83%
K6	HCP=5	2.50%	7.15%	6.14%	2.20%
AT5	HCP=6	1.42%	6.52%	9.24%	5.38%
43	HCP=7	0.56%	4.78%	9.72%	9.38%
	HCP=8	0.30%	2.63%	9.38%	13.55%

Expected IMP gain from inviting: Vul -0.08, Non Vul -0.5

This one has a good 14 HCP and 37 11-7-4-2-1 points. It's a tossup vulnerable and a pass nonvul.

	Tricks	7 or less	8	9	10 or more
AK8732	HCP=4	1.03%	3.34%	3.64%	1.40%
K6	HCP=5	1.79%	6.04%	7.37%	3.68%
AJ5	HCP=6	0.74%	4.78%	9.36%	7.55%
43	HCP=7	0.33%	3.08%	8.85%	12.04%
	HCP=8	0.12%	1.75%	7.09%	16.01%

Expected IMP gain from inviting: Vul 1.06, Non Vul 0.27

Changing the D10 into a jack makes it worth another move. It's now about an average 15 and has 38 11-7-4-2-1 points.


5332 Hands

	Tricks	7 or less	8	9	10 or more
AK873	HCP=4	3.17%	3.89%	2.44%	0.56%
KT2	HCP=5	3.85%	7.04%	5.38%	1.99%
AQ5	HCP=6	2.58%	7.67%	8.89%	4.28%
43	HCP=7	1.23%	5.06%	9.60%	8.58%
	HCP=8	0.45%	2.81%	8.17%	12.37%

Expected IMP gain from inviting: Vul -0.51, Non Vul -0.77

This above average 16 HCP hand is not worth inviting, though you wouldn't find many players passing it. Let's make it a bit better:

	Tricks	7 or less	8	9	10 or more
AK873	HCP=4	2.38%	4.04%	3.69%	1.26%
KJ2	HCP=5	2.65%	6.40%	7.34%	3.33%
AQ5	HCP=6	1.42%	4.85%	8.51%	6.81%
43	HCP=7	0.66%	3.40%	8.73%	11.36%
	HCP=8	0.31%	2.07%	6.40%	14.39%

Expected IMP gain from inviting: Vul 0.65, Non Vul 0.01

It's now a typical 17 and is worth a move, but nonvul it is still right on the margin. So 5332 hands need quite a lot to invite. There's also less chance they will balance so you may get to play 2S when it makes just eight tricks. However you will also be able to make 3NT sometimes when 4S goes down so a 2NT game try if available could still be worth a shot.

The minimum to invite with 5332 is a very good 16 or 42 on the 11-7-4-2-1 scale.


5422 Hands

	Tricks	7 or less	8	9	10 or more
AK873	HCP=4	1.98%	3.42%	3.77%	0.67%
A2	HCP=5	2.37%	6.09%	8.76%	2.48%
KQT5	HCP=6	1.35%	4.71%	10.66%	5.55%
43	HCP=7	0.81%	3.23%	9.53%	10.61%
	HCP=8	0.31%	1.71%	7.24%	14.73%

Expected IMP gain from inviting: Vul 0.54, Non Vul -0.06

This is a good 16 and has 41 11-7-4-2-1 points. It's just barely worth a move.


6421 Hands

	Tricks	7 or less	8	9	10 or more
AK8732	HCP=4	2.11%	3.67%	3.03%	1.01%
K2	HCP=5	2.71%	6.08%	5.90%	3.34%
QJT5	HCP=6	1.96%	5.73%	7.68%	6.06%
4	HCP=7	1.25%	4.80%	9.17%	9.65%
	HCP=8	0.71%	3.18%	8.34%	13.64%

Expected IMP gain from inviting: Vul 0.06, Non Vul -0.38

This is borderline with 13 HCP and 33 11-7-4-2-1 points.


6331 Hands

	Tricks	7 or less	8	9	10 or more
AK8732	HCP=4	0.96%	4.19%	3.42%	1.44%
A32	HCP=5	1.87%	6.20%	6.77%	3.46%
QJ5	HCP=6	1.05%	5.18%	9.37%	6.40%
4	HCP=7	0.51%	3.68%	9.39%	10.87%
	HCP=8	0.23%	2.12%	7.95%	14.94%

Expected IMP gain from inviting: Vul 0.57, Non Vul -0.06

Here you need 14 HCP or 35 11-7-4-2-1 points.


5521 Hands

	Tricks	7 or less	8	9	10 or more
AK873	HCP=4	1.48%	4.24%	3.87%	0.74%
A2	HCP=5	1.78%	6.24%	6.97%	2.59%
QJT52	HCP=6	1.14%	5.96%	9.53%	5.52%
3	HCP=7	0.41%	3.89%	9.67%	10.59%
	HCP=8	0.22%	2.34%	8.18%	14.64%

Expected IMP gain from inviting: Vul 0.35, Non Vul -0.21

This is right on the margin, again a hand that few players would consider passing. This is possibly the most striking example, partly because people routinely overbid 5-5 hands. It has a good 14 HCP and 36 11-7-4-2-1 points.


Summary

The following is the minimum strength to invite for each hand shape:

Shape	HCP	11-7-4-2-1
6421	13	33
6331	14	35
5521	Good 14	36
6322	Average 15	38
5431	Good 15	39
5422	Good 16	41
5332	Very good 16	42

A further interesting point is how little the vulnerability affects the decision. Even on the marginal hands above, there are only a couple where bidding is right if vulnerable but not if non-vulnerable. Certainly the difference is small, maybe less than a ten. This needs further investigation.