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The Price of Information
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This will be a rather long article. What's worse, it is not supposed to be funny, and it is doubtful whether reading it will improve your bridge game. Just so you know.

Stayman or Blast?

Sitting north in a team match, vulnerable, you hold AJxx Qx Kxxx xxx and hear partner open a vulnerable 1NT, showing 15-17 HCP. 3NT may be the best game even when partner holds four spades, but it isn't so likely. In the long run 4 wins about 2.4 IMPs over 3NT, which is quite a lot, so you bid Stayman. Partner responds 2, and you bid 3NT, ending the auction. That is too bad, as you could have blasted 3NT immediately, but it was worth looking for the 4-4 spade fit, and anyway, we are in the same contract so nothing is lost. Right?

Wrong! By bidding Stayman we have told the opponents that we hold four spades and fewer than 4 hearts, while partner holds 4 or 5 hearts and fewer than 4 spades. Both pieces of information may help them on opening lead, and the information about partner's hand may help them during later defense.

While most will agree that our Stayman auction will cost a few IMPs in the long run, it is hard to quantify how much the leaked information will cost on average. It would be useful to add a price tag to bidding Stayman, as we could then determine whether it was actually worth bidding Stayman in the first place, or if we should have blasted 3NT right away, or if it would perhaps be better to play entirely different methods.

There are several ways in which we can try to quantify the price of information. Perhaps the best way is to look at a large database of hands played in real life, preferably from high-level play. For example, we could look at the average number of tricks that are dropped on defense after a Stayman auction, and compare it to the average number of tricks dropped after a quick 1NT : 3NT. I think that that would be quite interesting, but I haven't done it, and I will not discuss it here. Perhaps some readers will be able to do such an investigation more quickly than I can.

What I will discuss here is how the price of information can be quantified using only double-dummy simulations.

It seems somewhat contradictory to use double-dummy simulations to investigate how much the leaked information is worth. After all, the auction is completely irrelevant to double-dummy analysis.

Well, double-dummy analysis can be used to quantify the price of information, but it takes a bit of effort. The analysis that I will present here looks only at how much the leaked information helps the opponents on opening lead. It would be nice to also find out how much the defense can benefit later in the hand, but while it seems theoretically possible to do this using double-dummy analysis, I am not sure that it can be done in practice. At the moment I am happy to only consider opening leads.

Blind lead simulations.

Imagine the following 4-step experiment.

Step 1. We randomly deal a deal consistent with the Stayman auction to 3NT. North gets our exact hand, South gets a (14?) 15-17 notrump opening, with semi-balanced hands according to taste, and for East and West we try to exclude hands that would have acted, perhaps by overcalling or by doubling Stayman.

Step 2. From this deal, we first only take the West hand, that is, the hand on opening lead against 3NT. We then deal the other three hands 200 times consistent with one of two auctions: 1NT : 3NT, or 1NT : 2 :: 2 : 3NT. In either case we do not force North to hold our hand, because West does not know our hand when choosing an opening lead. For each set of 200 hands, and for each of those hands, we look at how well each lead fares in comparison with the optimal lead. Since we are playing a team game, we use IMP scoring to determine the difference with the optimal lead. Then, for each of the two auctions, we select the lead that on average gives away the fewest IMPs. We call these two leads the blind double-dummy leads. They may be identical, but they need not be, because the 200 deals are selected using different specifications.

Step 3. Now consider the original deal found in step 1, and see how many tricks can be taken after each of the blind double-dummy leads. Again we compute the difference in IMPs. We would expect the blind double-dummy lead found using all the available information to do better in the long run, but it is certainly possible that on any given deal the uninformed lead scores better. Sometimes inferior plays score better.

Of course when the two leads are identical, as will often be the case, we do not have to compute anything. In that case the IMP difference will be 0.

Step 4. We repeat the above three steps for a large number of hands, say 400. Yes, I do know that 400 is not so large a number, I will get back to that on the next page.

We compute the average number of IMPs that the informed double dummy leads score better than the uninformed double dummy leads. This average number we regard as the Price of Information. Since the available information will also help the defending side during the rest of the play, I would think that the actual price is actually a bit higher.

I should note now that double-dummy leads are different from leads in practice. Contrary to at the table leads, they are not signals. They assume that after the opening lead everybody plays double-dummy. This makes the lead of an ace much more attractive, since you will always find the optimal switch. The lead of two or three small cards becomes more attractive as well. Leading from Q10x may cost a trick when partner has xxx, but leading from xxx when partner has Q10x will at most cost a tempo, because if the suit can be picked up, the double-dummy declarer will find it. To me that is the weakest part of this study, and I would be very happy if somebody would do the real data study I mentioned on the previous page.

Significance and Optimization.

(Feel free to skip this page, it definitely won't improve your bridge results, but is included to address statistical concerns.)

Using only 400 deals is a rather small number to get accurate results, but as we need to perform 400 (twice 200) double-dummy simulations to determine the blind double-dummy leads for each of these west hands, the computer is actually running over 160,000 double-dummy simulations. That takes a considerable amount of time, at least on my computer, so increasing the number of hands is not appealing. Can we increase the accuracy of our experiment without dramatically increasing the time it takes to run the experiment?

There is a trick that we can use to increase the significance greatly, without increasing the amount of time needed to run the simulation.

Suppose that in Step 2, when we take the West hands from the original deal, we really throw away the rest of these deals. That's right, we forget the South and East hands. After we have determined the two double-dummy leads in Step 2, we can deal new East and South hands for the comparison in Step 3. Since there was nothing special about the original deal and we are using the same specifications, we are not doing anything wrong by dealing new hands. But instead of dealing only 1 new hand, we could do the comparison for, say, 100 deals for each West hand. After all, since we invested so much energy in finding the two double-dummy leads, it would be wasteful to then use those leads for only 1 data point!

If we use 100 deals for each West hand, we get 40,000 data-points instead of 400 without significantly increasing the computer time. Since these 40,000 data-points are dependent, it is not as good as obtaining 40,000 independent data-points, but it is much better than having only 400 independent data-points. Exactly how much better this is cannot be computed without knowing more about the probability distributions, but once the work is done we can use our data to determine the significance of the simulation.

I found that using the numbers given above provide quite accurate results (usually a standard deviation of less than 0.1 IMPs).

 

The price of bidding Stayman

So what is the price of bidding Stayman? Here are the prices of the two different auctions ending in 3NT:

1NT : 2 :: 2 : 3NT: 0.49 IMPs. 

1NT : 2 :: 2 : 3NT: 0.06 IMPs. 

What is going on here? How can the price of the second auction be so much lower than the first? Isn't the second auction at least as revealing as the first? Did something go wrong?

No, this is not a measurement error. In fact, you may be able to deduce why the difference between these two auctions is so large.

Keep in mind that we are only considering the benefit to the opponents on opening lead. Moreover, we are using optimal double-dummy leads.

If you ever took a quick look at the book called Winning Notrump Leads, by Taf Anthias and David Bird, then you probably remember that after the auction 1NT - 3NT you should almost always lead a major suit. And what do you think that you should lead after 1NT - 2  - 2 - 3NT? Right, a major suit, perhaps even more so than after 1NT - 3NT. Even though this Stayman auction gives a lot of information, which will be useful during the defense later in the hand, it makes little difference on opening lead.

On the 400 West hands for each of the two auctions, after 1NT : 2 :: 2 : 3NT the double-dummy leads were identical 89 times, but after 1NT : 2 :: 2 : 3NT they were identical 247 times.

The auction 1NT : 2 :: 2 : 3NT is quite different. Knowing that opener has 4-5 hearts and responder 4 spades makes it much more appealing to lead a minor suit. Thus, the information leaked by this auction is quite useful on opening lead.

Should you bid Stayman?

So, should you bid Stayman on this hand? In order to determine this we also need to determine how often you have a 4-4 spade fit, and how much you lose by bidding 3NT when you have a 4-4 spade fit. On the previous page I stated that 4S is about 2.4 IMPs better, an enormous amount. But that is a double-dummy number, and does not take into account that the opponents are also more likely to mislead against 3NT when we have a 4-4 spade fit. In order to get more realistic numbers we should redo the whole study, but now for the auctions 1NT : 2 :: 2 : 3NT :: 4 and 1NT : 2 :: 2 : 4.

Here are the results:

Opener has no 4-card major:

Probability: 0.34

Blasting gains: 0.06

Opener has 4+ hearts:

Probability: 0.38

Blasting gains: 0.49

Opener has 4+ spades:

Probability: 0.22

Blasting gains: -1.26

Opener has both majors:

Probability: 0.07

Blasting gains: -0.39

 

The conclusion is that blasting 3NT on average loses about 0.1 IMPs, assuming the opponents make the optimal blind double-dummy leads, and all play is perfect afterwards. What does this means for real bridge, where people lead differently, and do not defend perfectly afterwards? I don't know, but it does seem to me that there is a lot more to be said for blasting 3NT than I expected beforehand.   

Is Puppet Stayman perhaps better?

In recent years it has become more popular to search for a 4-4 major suit fit using Puppet Stayman. Instead of 1NT : 2 :: 2 : 3NT, the auction would go:

1NT - 3

3 (no 5-card major) - 3 (4 spades)

3NT

 

or

 

1NT - 3

3 (5 hearts) - 3NT

 

The major difference with standard Stayman is that in the first auction, it is not known whether opener has 4 hearts. So perhaps you would expect the price of Puppet Stayman to be less than that of standard Stayman? Well, I suspect that this is indeed the case when one considers the whole play of the hand. But I found that when only considering opening leads, Puppet Stayman actually does slightly worse than standard Stayman. The cause is of course the very small relevant information leakage of the auction 1NT : 2 :: 2 : 3NT.

The price of Forcing Stayman

If you are also a regular reader of Kit's Corner, then you may recognize the hand AQ8 K84 Q8765 A2 from He Will Continue. Partner opens a mini notrump, showing 10-12 HCP, what should you do?

As the mini notrump could easily contain a 5-card major, the player holding this hand set out to find out if that was the case. Unfortunately, he had to do so using Forcing Stayman, and they had the following auction:

W
N
E
S
1NT
P
2
P
2
P
2NT
P
3
P
3NT

It should be obvious to the reader that revealing so much about opener's shape (a balanced hand with 4 hearts and 4-5 diamonds) comes at a tremendous cost. Perhaps it is better to blast 1NT - 3NT? Kit wrote in his article:

Is the concealment factor worth risking getting to the wrong game? This is a matter of philosophy more than anything else.

Usually when people say that something is a matter of style, or philosophy, what they mean is that bridge is too difficult a game to determine the optimal strategy. And usually they are right. But here I believe that the optimal strategy is actually quite clear.

If you do have a 5-3 major fit, then 4M will usually be better than 3NT. Classical double-dummy simulations (not using blind leads) suggest that playing in 3NT loses about 1.2 IMPs on average (a bit more if partner has spades, a bit less if partner has hearts). So how much do we lose by bidding Forcing Stayman and not finding a 5-3 major fit? The procedure described above indicates the information leakage of the given auction costs about 1 IMP! And note this is only the price of bidding Stayman when considering the opening lead. Clearly telling so much about opener's hand will also hurt us during the play of the hand.

Considering how rarely opener will have a 5-card major, it is clear to ignore the possibility of having a 5-3 fit, and to bid 3NT right away.

Further applications of blind lead simulations

I think that the idea of blind lead simulations can be useful to more kinds of questions. Here is a hand that I recently posted on Bridge Winners:

South
xx
10x
10xx
Q9xxxx
W
N
E
S
2NT
P
?
 

Vulnerable at IMPs, partner opens 2NT (balanced 20-21). Should you bid 3NT?

Double-dummy simulations revealed that you should barely bid 3NT holding the club king instead of the queen, and certainly not when holding the queen. I don't find those simulations very convincing. Despite the fact that I like to play with double-dummy simulations, I think that it is very dangerous to draw conclusions about real-world bridge. It seems to me that on this hand, the play will often be straightforward, but the opening lead can be vital. Thus, the double-dummy declarer should on average make 3NT considerably less often than one would in practice.

Using blind lead simulations we can correct for this fact. Again we deal 400 deals consistent with the auction, throw away everything but the West hands, for each of the West hands deal two sets of 200 new deals consistent with 2NT-pass and with 2NT-3NT, and determine the blind double-dummy leads after both of these auctions. (These leads will often be the same, so maybe we could just work with the double dummy leads against 3NT, but while we are at it we might as well do things properly.)

Then, we deal 100 new deals for each of the West hands and determine the double-dummy results in 2NT and in 3NT after the earlier found double-dummy leads. Which contract is better?

I found that passing 2NT is still better, but now the difference is only 0.3 IMPs per hand, while a classical double-dummy analysis gave a much larger difference of 1.7 IMPs. My hesitant conclusion is that passing 2NT is still the right call, but not by that much.

Acknowledgement

All simulations were done using the program deal, written by Thomas Andrews. Thanks to Meike Wortel and Simon de Wijs for programming help and stimulating discussions.

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