Sky asked (and rightly so) why I accepted the invitation last board with a pretty mediocre hand. I did have a 5th diamond and a full 13 points so I don't think my hand was the worst it could have been for the bidding but it was still pretty bad. The reason is I've been told you should stretch to bid vulnerable games at IMPs. I want to math it all out though, to justify following that advice in the future. Maybe it's hocum.
At any rate, assume the same number of tricks will be made at each table. (Not a guarantee, look to board 106 for an example.) Assume you don't get doubled in any situation, since it should be at least close if you're talking about accepting an invitation. Then we have the following pay-out table, with 4 possible cases. (We bid 3 or 4, they bid 3 or 4.)
| 3-3 | 3-4 | 4-3 | 4-4 |
| Make 2 | 0 | +3 | -3 | 0 |
| Make 3 | 0 | +6 | -6 | 0 |
| Make 4 | 0 | +10 | -10 | 0 |
| Make 5 | 0 | +10 | -10 | 0 |
So now to determine our expected IMP gain with each action (pass or accept) we need to work out the odds of the opponents going on and the odds of each result.
Assume they always wimp out and stay in 3. Then by passing we get 0 IMPs no matter the result. By going on we lose 3 IMPs when the board makes 2. We lose 6 IMPs when the board makes 3. We gain 10 IMPs whenever the board makes 4 or more. Let's abstract things even more and assume the board never makes 2 or 5. We make 3 p of the time and 4 1-p of the time. Then the equilibrium point is when -6*p + 10*(1-p) = 0, or when p = 62.5%. So if game is 37.5% or better and the opponents pass then we need to bid on.
Assume they always blast on to 4. Then by bidding on we get 0 IMPs no matter the result. By passing we gain the 6 IMPs when it fails and lose the 10 when it makes. Again, the equilibrium point is the same 62.5%.
So, the question then becomes, does game make 37.5% of the time or better when holding QJTx - xx - AKTxx - Kx and partner invites? Partner should have 10-12 points for an invite here I think, so can we build an optimal 10 count fitting his bidding to make? AKxx - xxxx - Qx - xxx makes on a club lead. AKxx - xxxx - Jx - Qxx makes whenever you can pick up the Q of diamonds, which is better than 37.5% for sure. 9xxx - AKxx - Jxx - Qx is pretty much the same, making whenever you can pick up the Q of diamonds. Or maybe you should be ruffing a diamond to try to set them up whenever they split 3-3 (36% itself, plus you make on the 4-2 splits with QJ doubleton). Hands where this is true are hands like Kxxx - AKxx - xx - xxx when you actually make 5 on a club lead when diamonds are 3-3 and can also ruff twice on any 4-2 diamond split when trumps are 3-2. Kxxx - AKxx - Qx - xxx is pretty much cold for 4, but that's probably not an invite at all but a blast to game.
But there are plenty of invites with no play. Kxxx - KQxx - Qx - xxx loses 4 tricks off the top whenever they don't lead a club and East has an entry.
Was this hand an accept? No, probably not, but I think it's pretty close. Turn the K of clubs into points in any other suit (even QJ tight in hearts) and I think it is an accept.
Really it comes down to partnership agreement. If partner is inviting with a bad 8 count and would blast on with the above 12 count then clearly passing is right. But if I can trust partner to not be a moron with his invites and if he's a little conservative with his 11-12 counts then accepting is going to be right.
But why is this important when vulnerable and not as much when non-vulnerable? Well, instead of the numbers being -3, -6, +10, and +10 they're -2, -5, +6, and +6. Then the equilibrium point is 54.5%, so you should only stretch to bid 45% games instead of 38% games. That doesn't seem like enough of a difference to really warrant a truism, but it is still a difference. Working to include the making 2s and making 5s should widen that gap as well, but not by a whole lot.
