To bid the grand or not
I see a lot of comments on reputable bridge sites like "With a Grand Slam, the maths shows that you should bid it when you have a 75% chance of making it."
I have, however, been unable to find the maths justifying it.
So I did my own.
Consider Bidding 7H NV:
If you make 13 tricks you get 1510
If you make 12 tricks you get -50
If instead you had stopped in 6H:
If you make 13 tricks you get 1010
If you make 12 tricks you get 980
So bidding 7 gains you 500 (1510 - 1010) if you make it
And loses you 1030 (980 - (-50)) if you don't
I assume this 2 to 1 ratio is where the received wisdom comes from.
But...
If you're playing at Matchpoints, then it's always go for it if it's 50% or better.
What about IMPs?
A difference of 500 equates to 11 IMPS, whereas -1030 equates to -14 IMPs.
For a given probability P of making the grand slam, your expected gain in IMPs is: 11P - 14(1-P)
The breakeven point is when 11P-14(1-P) = 0. Solving this gives 25P=14, or P = 14/25 = 56%.
This is a long way from the 75% I've seen suggested.
Where have I gone wrong?
Comments
OK I know this isn't about the laws or directing, but the people on here are those I would trust to explain my error!
If you are playing aggregate scoring (e.g. Hubert Phillips, but nobody uses that method anywhere else), then your first calculation is probably correct, at least assuming that 12 tricks are always available.
Your Matchpoint argument assumes that at every other table (a) they all bid to a slam (& in the same denomination), and (b) they always make the same number of tricks.
Quite often, you will find that (a) many are only in game (or worse) and therefore simply bidding the small slam will gain you most of the matchpoints - going for the grand doesn't gain much, but could lose what you gained just by bidding the slam; (b) if you do make 13 tricks you will still beat many that only made 12, even if you are only in the small slam.
Similar arguments apply to IMPs scoring - you are assuming that the slam is bid at the other table. You should also calculate the IMPs gained / lost if the score at the other table is 480/510, or even other possible scores
I entered your question into Grok3 (Elon Musk's AI). Here's the answer it came up with:
https://grok.com/share/c2hhcmQtNA==_649adfe0-ff1a-4b4d-98ba-6024c44b874e
Hope it helps.
At the bottom of the very impressive Grok answer is a reference to website bridge-tips.co.il and link to document http://bridge-tips.co.il/wp-content/uploads/2015/08/Percentages-for-Bidding-Games-Small-Slams-and-Grand-Slams-at-Duplicate-Teams.pdf
So the real credit for this comprehensive analysis (and for thinkng just like you) goes to Glen Meeden and G. M. Prabhu.
Is it right though? Looks like yes - the maths aligns with another analysis I've randomly searhed for: https://www.bridgewebs.com/stamford/150719ScoringDocument.pdf
I wonder why the received wisdom for bidding games in teams is to do it on 33% probability? Meeden and Prabhu say it's 45% non-vul (much closer to 50% than you might expect) and 38% vul. People tend to think in terms of pure points (170 v 420 or 620) but that doesn't explain it.
There is one addtitional variable - the play. The analyses assume perfect defence (and less significantly, perfect declarer play). There is little opportunity for defensive mistakes in a grand slam - i.e. usually it either makes or it doesn't. There is greater chance of defensive mistakes in a small slam and greater still in game. Therefore some impossible games and a few small slams will make, meaning you should bid them a little more often than their odds suggest. Corresponding that reduces the gains for grand slam and lengthens the grand slam odds.
"I wonder why the received wisdom for bidding games in teams is to do it on 33% probability?"
Is it? I've never heard that.
As Jeremy noted, there is not a good place on these forums for this discussion.
Should we just leave it to the parallel discussion on bridgewinners should-i-bid-the-grand?