When to bid a slam

I often hear players saying something like "if it were IMPs, I'd bid the slam" or "if it were matchpoints, I'd bid it".  In fact, Marty Bergen's September column in the Bridge Bulletin quotes people saying the former sentiment.  This reasoning doesn't directly follow from the scoring table, but I think I can understand the logic behind it.  Let's do the math.

In theory at matchpoints, that's to say when you are playing against your peers, making any bid will be right if it gains 51% or more of the time and wrong if it gains in 49% or fewer cases.  If you make the bid and the resulting contract is on a pure finesse, then you'll break even in the long run.  Therefore, you want your slam to be better than 50% to be right.

At IMPs, again assuming you are playing against your peers, a slam is theoretically also a 50-50 proposition.  If you're right (bidding/not bidding as appropriate), you gain 11 imps not vulnerable and 13 imps vulnerable.  If you're wrong, you lose 11 or 13 imps respectively.

So there's no difference in the form of scoring as far as bidding slams, right?

Well, not quite.  If you're not playing against your peers (let's say you're a good player at a daytime bridge game), you may get a good board just by making 12 tricks when the others only gather 10 or 11 tricks.  You don't need to risk going down in slam under such circumstances.  Let's say you're contemplating bidding 6NT (no good fit having been found).  Weaker players are notoriously bad at taking all their tricks in notrump contracts, so just making 12 tricks is likely to get you 8 (or 9) out of 11 points.  That's a decent result.  Thus bidding the slam risks 8 to gain only 3.  The argument is weaker when there is a good major suit fit and lots of points because many pairs are likely to bid slam and if it's there, most will make 12 tricks.

What about at IMPs?  Going back to our hypothetical 6NT.  You will likely gain only 1 or, at most, 2 imps by your superior declarer play.  Not nearly so compelling.  Thus there's a little more incentive to actually bid the slam.  Even so, against weaker declarers, you are risking 11 (13) to gain only 10 (12).

And what about grand slams?  At matchpoints, we theoretically need the same 50% when playing against our peers.  But again, just taking 13 tricks may be good enough against weaker players.  Especially since some of those players won't even be in a small slam and so bidding and making 6 with an overtrick is likely to be a very good score, perhaps 10 out of 11.  In other words, you will be risking 10 mps to gain 1.  Not good odds!

The gain for bidding and making a grand at IMPs is essentially the same as for a small slam: 13 or 11 imps, depending on vulnerability.  However, the risks are greater.  If you're wrong (and the small slam is making), you are losing 17 or 14 imps depending on vulnerability.  Thus the probability you need is either 56.7% (vul) or 56% (n-v).  These are the figures when you are playing your peers.  The odds are even less in favor of bidding when you are playing a weaker team.  They might not even get to the small slam at the other table!

In this case, by stopping in six, you were already gaining 11 or 13 imps.  The extra grand slam bonus would amount to only a net gain of 3 or 4.  The risk is admittedly slightly less now: 11 or 13, as in the case where you were deciding on slam versus game.  So, playing the weaker team, you risk 11 (or 13) to gain 3 (or 4).  Now you need cards that give you a probability of at least 78.6% (n-v) or 76.5% (vul).  That's why you shouldn't bid a grand unless you can count 13 tricks.  And why the late great Barry Crane expressly forbade his teammates to bid grand slams. 

So, in conclusion, I suppose there are hands where, at the club, you'd bid a slam at IMPs where you wouldn't at MPs.  But at the Life Master Pairs (or equivalent), the small slam odds really are 50-50 at either form of scoring.