Non-linearity in Bridge

I apologize in advance for the length of this article. I could perhaps make it shorter, if only I had the time. But if you don't want to wade through a lot of preamble, then skip to the last few paragraphs.

We live in a world where the observed relationships of quantities, at least at the macroscopic level that we normally experience, are either linear, quasi-linear (or, more formally, monotonic), non-linear, or unrelated.  We take linearity (or at least quasi-linearity) for granted – for example, we press a little harder on the gas pedal and the car goes a little faster.  Of course we learn from experience that this is not a purely linear system – pushing the pedal twice as far down doesn't make the car go twice as fast.  But there are other times when non-linearity rules, for instance when a microphone is placed in front of the speakers at a wedding reception or similar gathering and we experience the dreaded squealing of the audio system.  Non-linearity is one of the key factors in chaos theory.

Because of the integral nature of the various scoring tables at bridge, scoring shares some similarities with quantum theory – there is a finite set of states that any particular deal can take on.

Indeed, there are several different scoring tables at bridge, depending on which phase and/or form of the game we are playing at the time.  None of them is purely linear.  And that is perhaps the essence of bridge – why we all find it such a fascinating game and part of why it takes so long to learn to play well.

Let's take as our first example a contract which makes eight tricks in spades, nine in notrump or clubs.  For simplicity, we will leave out the red-suit contracts.  We are not vulnerable and we'll assume that our opponents will double when we are more than one trick short of our contract.  Starting then with 1♣ and going up to the five level, here are the scores we will receive:

Potential scores for black-suit or NT contracts:

level	♣	♠	NT
1	110	110	150
2	110	110	150
3	110	-50	400
4	-50	-300	-50
5	-300	-500	-300

This is so non-linear, it's almost chaotic.

The next way of looking at things, is to compare, for a given contract, the score for each trick we take. For example, the contract of 1NT, doubled but not vulnerable.  When we compare our score with tricks, we find that it is quasi-linear.  Score monotonically increases with tricks, but the increment varies (it's either 300, 280, 200 or 100).  Here are the scores for taking 0-13 tricks:

-1700, -1400, -1100, -800, -500, -300, -100, 180, 280, 380, 480, 580, 680, 780. In practical terms, though, it isn't enough to know how the scoring table behaves. Duplicate Bridge isn't normally played at total points. In some ways the most complex situation is matchpoints because there are typically many other tables in play and the complexity of estimating your matchpoints based on your actual score is way beyond the scope of this blog.  The best you can do at matchpoints is to guess whether the call you are contemplating will have a better than even chance of improving the number of matchpoints you will receive. The scoring at teams however is more tractable and, as usual, quasi-linear.

The reason that it's easier to predict outcomes at teams is that there is only one other table and the IMP table is fixed and monotonic (order-preserving).  Normally, at any stage of the game you will be choosing between one of two options, each of which has a predictable outcome.  Let's take as an example a decision as to whether or not to bid a vulnerable game.  If you bid it and it makes (for now, we assume perfect play at your table), you will score 0 or 10 imps, assuming that the opponents at the other table are making a similar choice.  If you stop in a making part-score (no game available), you will score 0 or 6 according to the decision at the other table.  To simplify the decision, we temporarily ignore the other table and think as follows: bidding game risks losing 6 to gain 10.  These are reasonable odds and account for the fact that players like to bid vulnerable games at teams.  Or another way to look at it is this: if the game contract depends only on one finesse, then our expectation of gain for bidding the game is 5 – 3 = 2 imps.  Of course, this calculation ignores the fact that trumps may be stacked against you and that if you bid the game, an opponent might double.  Thus, if you make such (normal) games three times out of every eight (37.5%), you will break even.

Now, let's assume that we've bid the vulnerable game and there are two lines of play from which to choose.  One is successful, the other is not.  Assuming for now that the other declarer is in the same contract (our outcomes will be different if that is not the case), we will score either 13, 0 or -13 IMPs, depending on the other declarer's actions.  Again, we will ignore the other table and consider that our play will either win or lose 13 imps.  As an extreme example, let's say that we have a sure line to make and an alternative line that will make an overtrick.  Again, we assume that our counterpart is facing the same decision.  Taking the alternative line risks 13 to gain 1.  Such a gamble would be crazy -- unless of course you're playing the last board of a KO and you strongly suspect that the current net score is zero or plus/minus one. Knockouts are the most non-linear scoring system of all (they involve a mathematical function called the Heaviside Step Function).

There's one more important non-linearity to consider with IMPs, which arises when the two tables are not in the same contract. If there's nothing to the play, the IMPs changing hands will be simply based on the differences in the contract. But suppose that there is a difference in the play: now, the total IMPs available on the board is greater than in either of the other two cases (contract the same, play the same). You're in game, you have three inevitable losers outside of trumps and you take a finesse for the trump queen (missing five). It loses and you are -100. At the other table, declarer is in a part score: he can afford to lose to the trump queen but cannot risk a ruff so plays trumps from the top picking up the doubleton queen. At that table, you are -170. You lose 7. But if you too had dropped the queen, you'd have won 10 instead. So there were 17 IMPs available on that board and you lost them all!

So much for the non-linearity of IMPs in general and knockout matches in particular.  How about a Swiss (or Round-Robin) where we are playing for victory points?  The VP scale is a mix of quasi-linearity (in the middle) and non-linearity (at the extremes).  This is where the ability to estimate is so important.  You must forget all about those odds of 37.5% for a vulnerable game as you get closer to the end of a match.  Let's say that things have been going well for you in this set.  You bid an iffy vulnerable game earlier and made it.  The opponents had a misunderstanding with a slam auction and went down non-vulnerable.  You've made a couple of good part-score decisions and the other boards were flat.  You estimate five for the game you bid (there's a 50% chance the opponents got there too) and 11 for the slam (your teammates never make that sort of error).  The part-score decisions have you up by approximately another four IMPs.  So, you estimate that you are up by 20.  If the last board is flat, you will win the match by 18-2 victory points (assuming the 20 point scale*).  Bidding a game will gain 10 imps (but only 2 VPs) if you're right, but could lose 6 imps (2 VPs) if you're wrong.  It's therefore a toss-up.  If the game is likely to go down on a wrong finesse or a bad break, then you shouldn't bid it.  What you've been taught as odds of 5:3 are now no better than evens.  That's because the VP scale isn't linear.

20 point VP Scale:

IMPs	VPs
0	10-10
1-2	11-9
3-4	12-8
5-7	13-7
8-10	14-6
11-13	15-5
14-16	16-4
17-19	17-3
20-23	18-2
24-27	19-1
28-	20-0

In general, if you're already well ahead (or behind) in a Swiss match, the decisions that you make will be less significant than otherwise because the slope of the VP scale is lower than it is at the start of the game or if there have been no big swings.  However, when you're up, the upside of a good decision is always less than "normal". Conversely, when you're down, the downside of a bad decision is less than normal. Let's look at another example: to bid or not to bid a non-vulnerable slam.  At the start of the match, you need at least a 50% chance of making the slam for bidding it to be right: you risk 11 to gain 11 (non-vulnerable).  But suppose that the slam arises later in the set and you estimate that you are down by 10 imps because you missed bidding an easy vulnerable game.  What odds do you need for the slam now?  If you make the right decision and win 11 imps that is worth 5 VPs.  If you make the wrong decision and lose 11 imps that's 4 VPs away. In other words, you should be bidding any slam that has at least a 44% chance of making.

Odds summary: expressed as reward:risk

Estimate	Psych	Vul Game	Non-vul Slam
-30	3:0	2:0	3:0
-20	4:2	4:2	4:2
-10	5:4	4:2	5:4
0	5:6	4:3	5:5
10	4:7	4:2	4:5
20	2:5	2:2	2:4
30	0:4	0:1	0:3

In the table above, we assume that the pysch (or other swingy action) stands to gain 12 IMPs if it succeeds but will lose 15 IMPs if it crashes and burns.

As an aside, in a recent flight A Swiss, we were perhaps slightly ahead after five boards and bid 6♦, going down. On the last board, I decided it was therefore right to push to an iffy 6♣. It went down too. Chances of winning that match were close to nil. But, we had done better on the first five than I thought, the other team also bid the first slam going down, and we still came out comfortably ahead!

Now, here (finally) is the important point.  Notice that it's not so much a question of bidding the slam to make 5 versus losing 4 when you're down by 10.  It's more that you should be contrary (also known as "swingy") when you are losing and, conversely, follow the herd when winning.  If you're behind and you think that your opponents will be in this slam, then you might consider not bidding it.  If there are twelve easy tricks, you will be another 4 VPs in the red. But suppose that it goes down at the other table while you make a conservative 450, then you will gain 5 VPs.  If you think they won't be in it, then bid it.  Now, of course, we need to have an idea of who our opponents are.

But, if we estimate that we are ahead in a Swiss (or KO), then we should play down the middle. We should bid all normal games, normal slams, etc. The other team will (or should be) swinging a bit. Let them. I was once in a KO (many years ago now) where my team was up by 24 at the half. I took my foot of the gas pedal a little and didn't bid a game that was a reasonable vulnerable game, thinking that I should be conservative. We ended up losing the match when the opponents won 10 on that board and some others. Being conservative doesn't mean not bidding games. It means bidding all games that you expect to have a decent play, but not stretching to thin games.

I will conclude with a horror-story which happened just yesterday (we are now in 2018). A certain team had had some considerable successes in a Swiss and was in fact 16 IMPs ahead at that point. Building on that success, with the same feeling gamblers get: "I can do no wrong," our hero psyched a preempt. You guessed it: lose 16 for a tie. That cost 6 VPs! Was there much of an upside? Hard to say. The best it could likely achieve would be that opponents talked out of game, or stampeded into bidding too much. Perhaps a 10 Imp gain? Or they might brush it off and the result would be a push. Even in the best case, the pysch would gain only 3 VPs. So, when you're ahead, stay ahead by bidding and playing according to the book!

* I wrote this article back in 2014 before the new Victory Point Scale which uses fractions instead of integers.

Anna Karenina

Ordinary bridge hands are all alike. Every extraordinary bridge hand is extraordinary in its own way. The Anna Karenina principle--with apologies to Tolstoy.

That's not to say that you can relax on the ordinary hands. Far from it, especially at matchpoints. A defensive slip in a routine 4H contract, for example can easily give you an absolute bottom. Yet, the extraordinary hands are, typically, where most of the IMPs and matchpoints flow. Sometimes, we have to be on our guard right from the moment we pick the hand up. But how do we recognize such hands?

Here are the clues you might notice when you pick up the hand:

• extreme distribution ("Goulash" hands, for instance);
• non-purity (short suits with honors, long suits without);

And here are further red flags that pop up as the auction progresses:

• high-level preempts (or interference);
• partner bids your short (singleton or void) suit;
• somebody puts down the red card or, especially, the blue card;
• dummy has a long suit.

Of course, there are many other danger signs that arise as we declare or defend a hand, but by that time, everyone at the table already knows a lot about the hand. This article is about early indications of trouble.

When we recognize such a hand, we need to sit up straight, and gather our concentration. The two early indicators are suggestions that the hand we are about to play will not conform to the "law of total tricks". There are likely to be more total tricks (in the first type) and fewer (in the second type). We must therefore be on our guard.

Here's a hand that came up recently in a BBO Speedball, that's to say matchpoints, where you are the dealer and at favorable vulnerability:  ♠A2 ♥ K ♦A765 ♣J87652.

The "non-purity" bell should be ringing loudly in your head! Are you going to open this hand? Hard to pass a hand with two aces along with two other face cards. But what are you going to do when partner responds one of a major? Rebid that moth-eaten club suit? You certainly can't reverse into diamonds. What about opening one diamond? Now, you will not be embarrassed by having to make a 2C rebid. But it does distort the hand. So, you recognize immediately that this hand looks like trouble. Nevertheless, you forge ahead into the unknown with one club.

It gets worse. LHO overcalls 1NT and partner doubles. This is always a tense situation when partner doubles 1NT after we have opened a minimum hand. Do we actually have sufficient firepower to defeat the contract? What if RHO passes? Would we dare rebid 2C when LHO probably has much better clubs?

We breathe a sigh of relief when RHO bids 2H. This is not alerted but after our pass, LHO bids 2S and partner now bids 3H and RHO passes so, pretty clearly, RHO has a weak hand with spades.

Are your alarm bells still ringing? They should be. Could anything worse have happened? Yes, partner might have doubled again. So, what does partner have? He has good hearts and they were probably his main reason for doubling last round. What about strength? Well, 3H isn't forcing and isn't game so he probably doesn't have opening count. He has a good chance to make this contract. Let's leave well alone and pass.

But, is that really the right call? We know (and presumably partner doesn't) that our clubs are absolute trash. And we can be pretty sure partner won't want to lead hearts if LHO decides to bid 3S. What will partner lead? A trump? Yes, maybe. A diamond? That would be nice but will he find that lead? How about bidding 4H?

Insane, you say? I don't think so. Clearly, there must be some play for 4H. We have the King of partner's good suit. We've got two aces on the side. And, if partner is short in clubs, there will be no wastage there. 4H could actually be a good advance save against 3S, especially if it's not doubled. This is matchpoints, after all and -50 or -100 beats -140.

So, you pass and, as expected, LHO bids 3S. Partner doubles and your worst nightmare has been realized. If you bid 4H now, you are definitely getting doubled and this could be -300 on a bad day (it is a bad day!). But, if you pass, partner will probably lead a club since your bidding--1C followed by three passes--strongly suggests a weak hand with long clubs. Away will go dummy's losing hearts and -730 will be the result.

Are we happy with our original opening bid now? Are we happy that we passed over 3H? No, we are not but there are no undos in bridge.

The result? -930 (0%). Even worse than we feared. Partner (I was that partner) could have saved the day by cashing the HA. Or leading spades or diamonds and overtaking our HK return. But he woodenly led a club and that was that.

-300 would have been worth 6%, -100 was worth 26%. +500 (somehow, steering partner away from a club lead) would have been worth 100%.

Breaking BADD

What's "BADD," I hear you ask. It stands for Bridge-related Attention Deficit Disorder. However hard I try, I just can't break it. I try to concentrate, I stay hydrated, I skip lunch. I get the occasional coffee. It all helps, but somehow I just can't stay 100% focussed.

This last week I played four days of bridge (and evening side games) at the Granite State Getaway tournament in Nashua, NH. During that time, I revoked, exposed three penalty cards forgot my notrump range, and various other lapses of concentration too numerous to mention.

But today's was such a classic, and involved such an unlikely parlay that I just have to tell you about it. It's like one of those terrible plays that a goalkeeper makes when he doesn't properly clear the ball. A perfect example of that occurred just this week in the World Cup matchup of Argentina vs. Croatia. Caballero, the Argentine goalie, failed to properly clear the ball and Rebić scored a brilliant goal to help Croatia to a 3-0 win.

So back to my play. Here was the board:

As you can see, our bidding was reasonably sophisticated. 2S was the "impossible spade," showing a good club fit and a game invitation. 3H suggested that spades may not be a problem, but was checking on the heart situation. 3NT is a good contract, making 9 tricks on a spade lead and 10 on any other lead. I got a heart lead and soon set about making overtricks.

The H3 was led, I played low from dummy and took the queen with the ace. I ducked a diamond to the ten and East's jack. A heart came back to West's king and after cashing the spade ace, he returned a heart to my jack. On this trick, I had to find a discard. At this point, I had nice rock-solid tricks and was interested in making the rest. In fact, I have the rest of the tricks just by cashing one spade, two diamonds and five clubs in addition to the two hearts already in the bag. At this point, they had three tricks so that's all there is.

But somehow, my counting was off and I thought I should at least try to generate an extra trick. In order that I wouldn't have to decide now whether to pitch the small diamond or the small spade, I pitched the totally unnecessary ace of clubs. Now, I set about cashing the top spade and then the clubs. I cashed the king and queen, then overtook the ten with the jack. At this point I realized (duh!) that there were no more losers so I faced the C8 and claimed the rest. "Not so fast," says West and produced the nine. Since I had no longer a stopper in spades, the defense took the next two tricks for down 2.

Pretty embarrassing! This "coup" should really have a name. Here's what was necessary to pull it off:

1. miscount the tricks so that I was in effect looking for a tenth trick with a squeeze that was extremely unlikely to succeed and, so, "unblock" one of the high honors in my club suit;
2. unblock the high spade (for the phantom squeeze, I could equally have chosen to keep the SK and unblock the DA);
3. not notice that East showed out of clubs at the first trick and then overtake the ten with the jack (when I could just as easily have used the DK as the reentry to my hand);
4. not notice even then that I had a total of seven tricks facing my way and didn't need a club for my contract: all I had to do was to cash the top two diamonds.

Four separate errors in one hand. Of course, errors 2 thru 4 all arose as a consequence of error #1. But, at any point, I could have quite easily recovered.

The his perhaps the most egregious, hilarious, and simply bizarre display of incompetence I've ever encountered, let alone, perpetrated!

So, how does an otherwise reasonably intelligent brain come up with this sort of thing? I don't know. I just call it BADD and do what I can to avoid it.

My teammates were remarkably understanding. This 10-IMP error (the contract at the other table was 5C down 1) cost us 3.45 victory points which normally would have meant a drop of several places in the standings. But, remarkably, it made no difference at all! We were sixth overall and second in X for a little over 11 masterpoints.

And, to add icing to the cake, England beat Panama 6-1 this morning in the World Cup!

Sometimes a bad split can be your friend

We all know, when we are declaring a contract, that bad splits are the enemy. It's nearly always harder to make our contract when suits don't split well. Nearly always!

Let's take the following contract as an example:

Clearly, this is going to be no problem if diamonds split 2-2. So, we draw two rounds of trumps and duck a diamond. Another heart comes back and you ruff. What do you pitch from dummy, by the way?

Now you play to the DA but unfortunately, RHO pitches a heart. Down one.

Not so fast! You aren't down yet. Cross back to your hand with the SK (RHO plays the Q) and, using vacant places and restricted choice you confidently run the nine. It wins and you are able to pitch your losing diamond on the fourth round of spades. You didn't pitch that little spade earlier, did you?

I wish I could tell you that I played like this. I could have. I should have! But I didn't. I just lamely conceded a diamond for down one and got on with the next board. As Eddie Kantar writes in his wonderful book Take All Your Chances, you should never just give up when there's even a glimmer of a hope. Notice that if spades had split a more normal 3-3 or 4-2, there would have been no hope at all. Well, it would make sense to run all your trumps and hope for a bad discard. But against good defenders there'd be no hope.

But the moral of the story is: sometimes a "bad" split is better than an even split. And, when there's a massive preempt at the table, this becomes even more likely.

The full layout:

If you're interested in the odds of the S9 winning the trick at T8 (as described above), then we know that East started with at least 8 hearts (based on the bidding and the opening lead), a diamond, two clubs and the spade Q. He has one vacant place for either the SJ or a ninth heart. West has shown 3H, 2C, 3D and so has five vacant places for the SJ. Taking restricted choice into account, the odds are 10 to 1 that running the nine will be successful.