A sensible gamble?

11-01-24

At the cointrée table, sitting to left of dealer (so bidding third), I recently received the following rather appealing hand:

This looks great for a potential Capot in Sans Atout (No Trumps)! Assuming I get a favourable lead from partner (who will be leading to the first trick), let's look at the potential issues, and how we can go about getting there.

The hand in question

The Play

As long as one of the opponents isn't holding all three missing Clubs that suit is safe. At the table I don't know this probability off-hand, but feel it's small enough for it to be a reasonable punt (for the record the a priori value is 14/253 ~= 5.5% — obviously one could also take into account inferences from the auction).

Hearts is a little trickier. If it breaks 2-2-1 (the most likely distribution: around 45%) there is no issue. If the split is worse, I have a potential loser in the Eight, if an opponent is holding Kxx(xx). In such a case, are they likely to retain this suit until the King becomes a winner (i.e. the last trick)?

Let's think about the discard strategy of a player defending against a Capot contract. If they have either missing Ace they will probably get rid of the corresponding suit, reasoning that I am unlikely to attempt Capot holding cards in a suit which I don't have the Ace in. Obviously any suit holding only losers (78, 789 etc.) will be thrown, and singletons, again assuming they will at best fall to my Ace, although these will probably come last to avoid giving information away.

After discarding such 'bad' suits (if any), what then? Well, they are quite likely to discard also those with probable losers, such as cards that are unlikely to be cashable (79, 89, maybe 7Q). Generally short suits, and prioritising retaining suits with non-Ace higher cards (Ten, King, possibly Queen).

However, after some cards have come down they will have some clues from the play. If their partner discards a suit with an Ace, once revealed, they will probably ditch it as well. Otherwise they may try to provide cover for other suits. If 987 of some suit is discarded, their partner may hang on to, say, TJ in case declarer is holding AK. But if instead it is A87 discarded, their partner will probably ditch TJ.

Of course, if my partner holds one of the missing Aces I should be able to discard my possibly Heart loser on that, in which case this consideration is void.

Altogether, it initially seems likely that a Capot will come good then. I'm willing to take the risk (depending on what happens in the auction) — this was an early hand, with the match score at 249 (us) vs 305 (them) in a game to 3000. At a later stage I might have wanted some safer points, but no need to worry about that now. The next question is how to reach Capot in the bidding?

The Auction

Partner and LHO (Left-hand-opponent, i.e. non-dealer) both passed, leaving me first to speak. I'm happy to go all the way to Capot even without knowing anything about partner, but this will only work if they leads me a Club or Heart. If they have no Aces, this is a priori a 50-50 guess on their part (two 'right answers' out of the four possible suits). But in practice it likely won't be - they probably will not have so many cards of these suits (as I already have a bunch), and therefore should assume I'm more likely to have length in one of these. If they hold either missing Ace (or even both!) then the chance of them leading a wrong suit is reduced (or eliminated!). They could only have Spades and Diamonds, although this is not massively likely. So a not-completely-crazy idea is for me to open Capot Sans and pray.

Another option is that I try and clue partner in on which suit to lead. I bid one of my suits, hope someone else bids so that it gets round to me again, and then jump to Capot Sans. Partner will then know at least one suit I hold cards in, and thus the chance of a favourable lead increases. If I did go for such an option then, which suit would be best to offer?

Just from the number of each in my hand, I am more likely to find partner with a Heart than a Club (which is generally how I prefer partners anyhow). They didn't open Hearts, but this doesn't tell us a huge amount — at best we can mark holdings like J9KQ as unlikely (but not impossible - partner may still pass this with very poor remaining suits). The trouble with Hearts, though, is:

  1. that's where my weakness (the '8') is, so I'd rather play out my Clubs before moving to Hearts to keep the defenders guessing,
  2. I don't want to play in Hearts! I might get lucky with how the remaining cards lay, but I'm probably going down.

With a bad break there's a good chance I'll get coinched, but even if not there's a chance noöne else bids and I'm stuck in the contract, which would be quite irritating.

In contrast, if I end up in Clubs I'm happy — I'll probably make 120 or 130, and would have to be pretty unlucky to miss 80 or 90. The greater issue here, compared to Hearts, is that partner is more likely to have no Clubs, in which case they're back to guessing between their three suits for which to lead. What are the odds they have no Clubs? Again I don't have the figure to mind at the table, but can make a quick over-the-thumb calculation — if I say each missing Club has a 2/3 chance of being with an opponent, and I pretend that each one is independent, the chance of all three being with the defenders (i.e. partner has none) is around (2/3)^3 = 8/27 ~= 1/3 (for reference, the actual figure is a little lower: 506/1785 ~= 28%). Combined with the fact partner might figure out to lead Hearts anyhow, this feels like a more appealing choice than bidding Hearts.

There remains the question of the level to bid. It doesn't really matter in this instance in terms of communicating to partner, but I want to give myself the best chance that:

  1. someone bids again (opponent or partner, don't mind too much, just as long as the action will get back round to me)
  2. if it turns out that noöne else bids I'd prefer to end up in a contract I can make, so that we at least end up with something

As discussed, with the latter I'm probably okay to open up to 100 without much worry (and potentially more). So let's look at the possibilities of opponent or partner bidding.

It's unlikely LHO will bid again if only I speak - they didn't open, and so will be unlikely to overcall. But RHO is a different story. There are four Jacks, three Nines, and two Aces unaccounted for, and with only me speaking, it is reasonable to believe that a few of these will lie with RHO. As such, there's a decent chance they have a decent enough hand to overcall, provided I don't come in too high.

What about partner? If I open 80 they will only bid if they hold the Jack of Clubs - about 1/3 chance. If I bid 90 (or more) they will only bid if they have either of the missing Aces. And if I bid 90 and they also hold the Jack of Clubs, what will they make of my bid? Would they assume I'd made a mistake and try to correct? Would they pass to keep the bidding low? I'm not really sure what I would do as partner in such a situation.

Given this, it seems like I'm probably best maximising my chance of RHO overcalling, and as such my best will be the lowest option, 80. Any higher bid would probably decrease the chance of RHO bidding, without enough of a compensating increase in the chance of partner bidding. So I open the bidding at 80 Clubs.

The Reality

At the table, I indeed opened 80 Clubs, RHO overcalled 90 Tout Atout (indeed they did have three of those missing Jacks!), which was followed by a couple of passes, at which point I jumped to Capot Sans (and unsurprisingly was not outbid). Partner led the Ace of Diamonds (allowing me to discard Eight of Hearts), and then led the Seven of Clubs, which I won with the Ten. RHO also followed with the Jack, meaning their king fell under my Ace, and thus the rest of mine were certain winners. Capot made.

Here are the hands as they appeared at the table: 

                (partner; first to bid and lead)
                C: 7
                H: 7
                S: KQ9
                D: AKJ
                           (dealer)
        C: -               C: KJ
        H: KQ9             H: J
        S: AT7             S: J8
        D: TQ              D: 987
                C: ATQ98
                H: AT8
                S: -
                D: -

I definitely had some good fortune - my partner had an Ace and a Club, meaning I could dump my (potential) loser, and we got to my hand for the winners. But just because it worked out on the day doesn't necessarily mean that it was the right thing to do. What other possibilities could there have been? Obviously there is a little discussion of some relevant probabilities above, but that largely excludes the inferences from the bidding.

As a quick-and-not-very-scientific test, I dealt out a few sets of hands for the other three players, just to get a sense of some 'typical' holdings. I ketp going until I had five valid deals — many of these in practice I had to reject as partner or LHO would almost certainly have bid first (so they could not have been holdings in the situation I was in). In fact, for five valid deals (some of which may even be a stretch, as you will see), I had to reject sixteen other deals where partner and/or LHO would have bid, which gives some sense of how much information we could infer from the lack of bids by the time it got to me (i.e. that their silence was indeed quite informative). This obviously was not the most efficient way to analyse things, but works for getting a rough feel for things. At some point I might circle round and do a proper simulation (with some crude-ish rule for rejecting biddable hands), but until then, let's look at how these other deals would have played out.

Alternate deal #1

                (partner; first to bid and lead)
                C: J7
                H: K9
                S: AJT8
                D: -
                           (dealer)
        C: K               C: -
        H: 7               H: QJ
        S: 97              S: KQ
        D: ATQ7            D: KJ98
                C: ATQ98
                H: AT8
                S: -
                D: -
Partner may well have opened this 80 Spades, but assuming they're conservative we'll count it. Both suits split fine, and partner has a Club to lead, so we are all cushty play-wise. In the auction, even if RHO doesn't overcall 90 Diamonds, partner, with Jack of Clubs and an Ace, will certainly bid to allow us to get to Capot.

Alternate deal #2

                (partner; first to bid and lead)
                C: -
                H: J
                S: Q98
                D: K987
                           (dealer)
        C: 7               C: KJ
        H: K9              H: Q7
        S: TK              S: AJ7
        D: ATJ             D: Q
                C: ATQ98
                H: AT8
                S: -
                D: -
LHO might stretch to 80 Diamonds? But I'm not sure I would, quite. Suits split fine, but partner doesn't have a Club to lead — however based on my large jump bid I would assume they would lead their short suit, Hearts, under the assumption that I would probably have a couple of long good suits. However there's a decent chunk in the auction I end up stuck in 80 Clubs. RHO may hope that we end up in Clubs and coinche, so stay quiet (and only really has a stretch of 90 Spades), and LHO probably doesn't want to overcall if only I have spoken. Biggest hope is probably partner who may try the change to Diamonds due to their Clubs void, but with a fairly poor suit and nothing on the side, they will probably pass also. 80 Clubs is at least makeable, but a bit disappointing.

Alternate deal #3

                (partner; first to bid and lead)
                C: J
                H: KQ
                S: KJ
                D: Q97
                           (dealer)
        C: K7              C: -
        H: J7              H: 9
        S: QT7             S: A98
        D: A               D: TKJ8
                C: ATQ98
                H: AT8
                S: -
                D: -
Play-wise it's fine - a Club lead, and splits fine in both my suits. In the bidding partner at least will raise Clubs, and RHO may even overcall Diamonds — either way we get to Capot.

Alternate deal #4

                (partner; first to bid and lead)
                C: J7
                H: 7
                S: TK7
                D: 97
                           (dealer)
        C: K               C: -
        H: QJ9             H: K
        S: Q               S: AJ98
        D: QJ8             D: ATK
                C: ATQ98
                H: AT8
                S: -
                D: -
Another slight stretch (in terms of plausibility) — LHO might go for 80 Hearts, but would be a bit wild to do so. In the auction RHO is almost certain to overcall Spades (most likely, perhaps 100 or even 110), in which case we have no issue. They might (but probably won't) overcall 90 Sans Atout - in which case what should I do? I know partner has no Aces, but they will raise Clubs, so I know the lead is safe. I can probably guess that RHO has some length in Spades and Diamonds, which then increases the chance of a bad split. So I might settle into (say) 110 Clubs. If instead I go for the riskier Capot Sans, my Clubs are fine, but my 8 of Hearts is trouble. It all depends on how LHO discards. They will keep Hearts or Diamonds for after I have finished with Clubs, as neither singleton has any hope. RHO has one free discard in the first trick to give some information - I would imagine they would chuck King of Hearts, which their partner would probably interpret as indicating lack of strength (I guess?). In which case we lose out, as they hang on to Hearts, and win the final trick with the Queen.

Alternate deal #5

                (partner; first to bid and lead)
                C: J
                H: Q9
                S: QJ8
                D: AJ
                           (dealer)
        C: K               C: 7
        H: KJ              H: 7
        S: AT              S: K97
        D: KQ8             D: T97
                C: ATQ98
                H: AT8
                S: -
                D: -
Maybe partner bids 80 Spades, or 90 Tout Atout, but let's assume they are more conservative than that. Well the auction will certainly stay alive with partner raising Clubs, and with the lay of the cards we are again safe as houses.

Overall then I am fairly happy that my choice at the table was the right plan of action. This random sample gives a probable 80% success rate. If we take a huge leap of logic and say that this is an accurate estimate of the true success rate, our 'expected score' is 500x0.8 - 500x0.2 = 300, so unless we are bidding up to 150 in Clubs with 100% success, this is the better option. At any rate, I certainly didn't analyse things in this level of detail in the moment, but this does reflect the broad shape of my thinking. I think there are a few points however in retrospect that will be useful to bear in mind in future hands.

Epilogue: A bonus (half-)deal

Here's a bonus holding of partner I got when trying out deals: 

                (partner; first to bid and lead)
                C: -
                H: J9
                S: K
                D: J9AK7
Now I excluded this from the above analysis as partner, with such a hand, would open the bidding, probably for 100 Diamonds. But I thought this was interesting for me - I know that if trumps are locked up and partner has four safe tricks I've certainly got the rest covered if partner leads correctly. So what could I bid in this situation? Some options:

Normally I play a discouraging / indirect signalling system — i.e. I discard from suits I don't want partner to lead. But here I only have suits I want! So am I just scuppered in this scenario? Well, assuming partner plays off J9 the first two tricks, I can discard both of my Aces. How would partner interpret this? If we are playing Capot, points are irrelevant, so why would I give up top cards which are sure-fire winners? I would only do so if holding tens of the suit! So with these first two discards, partner should know that I have tens in both suits, and switch to one of these once trumps are secured (in this case after at most three tricks, and obviously Hearts in practice).

That's fine and dandy if partner has two losers only, but I have thrown away two winners, and after cashing my tens will in a spot of bother in most situations (including this one, where partner has three losers). I can probably afford, in reality, to throw away at most one winner. This is still a trade-off - my hope is that the information I transmit with this to help partner avoid leading a Spade is, on average, more valuable than keeping the winner. I guess then my best bet is to throw Ace of Hearts first (as partner is more likely to have Hearts than Clubs, and I am more likely to promote my lower Clubs, and thus need to keep my Club winners). Next I'll throw the eight (which I don't need), which means I won't discard a Club until the third trick - I'm hoping partner will take this as an indication that I'm not really trying to discourage Clubs (otherwise I would've discarded earlier), to allow for the eventuality that they are void in Hearts. With that in mind, it probably makes sense to just go straight to 140 or Capot in Diamonds, depending on how risky I'm feeling.

This probably comes off in practice — assume that Diamonds split 2-1, and I discard A8 Hearts on the first two rounds, which exhaust trumps. I think partner will then make the Heart switch (correctly assuming I have the Ten), and can ditch their losers other two losers on my Ace Ten of Clubs, leaving only the remaining trump tricks to cash.