Monday, September 17, 2018

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
1110 110150
2110 110150
3110 -50400
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:

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
EstimatePsychVul Game Non-vul Slam

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.