Only the inviter should stretch

Introduction

Only the inviter should stretch. This nugget of sound advice comes from Howard Piltch, formerly one of New England's great players and teachers and now living in Indiana.

First, let's think about it for a moment. Most bridge players want to play higher-scoring contracts if they can so that there is a tendency to stretch a bit, often by both sides of a partnership. This is especially likely when we are vulnerable, although presumably everyone can see the vulnerability. Unfortunately, the result of both partners stretching is often an unsuccessful contract. Normally, the inviter knows that the partnership is within two (or three) points (high cards and distribution) of being able to bid game when issuing the invitation. Often, but not always, the invitee has limited their hand to about a three-card range. [The exact numbers are not all that important.]

Here's a hand that came up recently (in fact there were two very similar hands, one at the club, one on BBO, where the auctions were identical and the result the same – the only difference was that the club hand was at teams).

Here, South ("Invitee" – moi, I am ashamed to say) thought they had prime values and accepted the game try. Unfortunately, the contract was une chute, as the French would say. There were two diamond losers and two spade losers. The "Inviter" was a BBO robot (GIB) and, although minimum, I think most people would consider this a "three-card limit raise" and bid the same. Was this pair unlucky? Yes, a little. But neither player had any useful distributional features and South knew that they had exactly eight trumps between them. Having eight trumps, as opposed to nine or more, typically leaves declarer with no recourse in the event of a bad trump split or honors in the wrong place. This is especially true of a 5–3 fit. In fact, 3NT is unassailable and perhaps I should have offered to play there, if accepting the game try.

The other similar hand went down because of a 4–1 trump split – again, no margin of error when things don't go right. On that occasion, I was the inviter and my partner the optimistic invitee. Actually, a speculative double by the holder of ♥QJTx offside would have resulted in a 9-imp loss (somehow it went down only one at the other table).

Often it is the invitation itself that carries significant risk. It is frequently the case that an invitation is made when the bidding is two levels below game. Going to the next higher level (and stopping there) is risky because we get no additional reward for being one level higher, yet there is now a finite probability that we will not make any contract at all. Then we feel like the dog in Aesop's fables who, carrying a bone in its mouth and then crossing a bridge, looked into the water and saw another dog with a bone. He attacks the "other dog" and of course loses the bone in the process.

The invitee, on the other hand, while also taking a risk by going to game, has a much better risk-reward ratio because the game bonus now comes into play directly. Indeed, the partnership may already be too high (on account of the invitation) in which case the bump to game has very little downside (the extra undertrick is unlikely to be a big factor – unless doubled).

Having said all that, it may seem that it is the inviter that should not stretch since most of the risk, that is the un-hedged risk, is taken in that step. But if the inviter never stretched, there would inevitably be a lot of games missed. Feel free to skip the next section and go straight to the conclusion.

Detailed analysis

Let's look at some examples in no-trump contracts (because these are so much simpler). The simplest is when opener bids 1NT (let's say this shows 15–17 hcp) and partner invites by bidding 2NT. What are the risks? The following table shows the (approximate) average number of tricks you can take with the points shown in opener's or responder's hand. For the average tricks, I have used Matthew Ginsberg's analysis. I haven't adjusted for the fact that, for example, 10 opposite 15 will generally do slightly better than 8 opposite 17 because the hands are more balanced and therefore tend to offer more transportation. I have also assumed that if the average number of tricks is x, then the probability P(n) of taking n tricks is n+1–x (80% where n = 8 and x = 8.2) and P(n+1) = x–n (the 20% zone in this case). For simplicity, I have eliminated the possibility of taking n–1 or n+2 tricks.

	opener	15	16	17
resp
8		7.6	8.2	8.7
9		8.2	8.7	9.1

Below, we show the expected IMP losses playing against par and when not-vulnerable (figures in red are for overbidding and in green for underbidding). To explain some of the values, we can look at an example: we see that in the minimum case (15+8), we can take 8 tricks approximately 60% of the time and only 7 tricks the other 40%. So, with 8 hcp, responder takes a risk by inviting – scoring against par, he will lose 4 or 5 imps (depending on vulnerability) 40% of the time and break even the rest of the time. Net expected loss: 0.34 or 0.43, that's to say 1.6 or 2 imps multiplied by the probability of opener having exactly 15 hcp (44%) and by the probability of having 8 (48.5%) rather than 9 hcp (51.5%). If opener, knowing of his partner's taste for inviting with only 8 points, raises to game only with a full 17, then he will miss a making game 20% of the time (6 or 10 imps). If he also "stretches" and accepts with any 16, he bids too high 30% of the time (10 or 13 imps). When combined with the probabilities of the various holdings (15, 16, 17) we get the expectations of loss shown in the table.

Not-vulnerable	opener:	accept on 17	accept on 16 or 17
responder:
conservative	0.27	0.19+0.55+0.71	0.55+0.51
aggressive	0.27+0.34	0.20+0.19+0.34	1.29+0.51+0.34

And the following table is the equivalent for vulnerable contracts.

Vulnerable	opener:	accept on 17	accept on 16 or 17
responder:
conservative	0.45	0.32+0.91+1.18	1.18+0.66
aggressive	0.45+0.43	0.32+0.34+0.44	1.67+0.66+0.44

Conclusion

The following table shows the summary using approximations (for vulnerable contracts – the non-vulnerable numbers are lower but the relative values remain similar).

Vulnerable	opener:	conservative	aggressive
responder:
conservative		3	2
aggressive		2	32/3

So, we can conclude that the best a pair can do is to have a conservative player sitting opposite an aggressive player. It doesn't really matter which way around they are. "Normal" bridge is to invite with a (non-stretchy) 9 and accept with an (optimistic) 16 – minimizing losses at approximately 1.33 (not vul) or 2 (vul) IMPs. This is the time-honored strategy. Note that these expected losses are against a team of robots that peek, not only during the play but during the auction! How would we do against flesh and blood teams? Assuming that they are using their optimum strategy too, i.e. the same strategy, then we should push these kinds of boards, assuming best play and defense at both tables. If the other team has their optimist and pessimist sitting the other way, there may be swings, but they should even out in the long run. Note that the numbers have already take into account the fact that it's worth pushing for games, especially when vulnerable.

So, how does this relate to the subject of this blog? Well, we've shown one way of doing things: the inviter is conservative and invitee aggressive. This reduces the incidence of invitations but is reasonably accurate. How about when it's the other way around, following the rule "only the inviter may stretch"? It would work just as well, though more invitations means that the opponents have a better idea of what's going on. And, according to Howard, this is the right way to do it in all of those other less clear cut situations. I've always followed his advice. I hope my partners feel the same way.

There is an obvious point – that if inviter keeps quiet, it doesn't give the invitee even the opportunity of an acceptance. No doubt there are other scenarios, for example major suit game tries, where it does make sense that only the inviter should stretch. But the main moral of the story is simple: don't be aggressive – or conservative – on both sides of the invitation!