Strategy and Information

Strategy and Information

Let's consider a common single-dummy problem:

In this layout:

A J 6 5

we want to take all four tricks

W E

K 4 3 2

Should we play top honors or should we finesse ? – assuming that the suit is breaking 3-2:

W E		W E	All other breaks make our goal impossible to achieve. Thus, for the purpose of our analysis, those breaks can be eliminated.
Qxx–xx		xxx–Qx

We are facing a dilemma, since:

– if the missing cards are divided: Qxx–xx – only finessing works

– if the missing cards are divided: xxx–Qx – only playing for the drop works

and we don't know what the actual layout is.

Let's forget the present deal and let's concentrate on working out a rule for the future, say 100 deals like this one. What gives us a higher ratio of success:

– always finessing ? or

– always playing from the top ?

On that future 100 deals the missing cards will be dealt, on average, like this:

xxx–Qx	1098–Q7	10 times	40 times	Why were all the elementary breaks assigned the same (10%) frequency ? But why not ? Is the 9, at shuffle time, anything different than the 7 or the Queen ?
	1097–Q8	10 times
	1087–Q9	10 times
	987–Q10	10 times
Qxx–xx	Q109–87	10 times	60 times
	Q108–97	10 times
	Q107–98	10 times
	Q98–107	10 times
	Q97–108	10 times
	Q87–109	10 times

The table above clearly indicates that:

§ always finessing we we'll succeed 60 times

§ always playing for the drop we we'll succeed 40 times

which means that we should always finesse.

Let’s check what happens if we deviate from the advised line and play for the drop from time to time, say one time in ten:

1098–Q7	10 times	1 success	4	As you can see we'll succeed only 58 times in 100 deals (58%) while invariably finessing assures 60%.
1097–Q8	10 times	1 success
1087–Q9	10 times	1 success
987–Q10	10 times	1 success
Q109–87	10 times	9 successes	54
Q108–97	10 times	9 successes
Q107–98	10 times	9 successes
Q98–107	10 times	9 successes
Q97–108	10 times	9 successes
Q87–109	10 times	9 successes

There are, however, bridge players who complain against the calculations presented above arguing that:

Agreed, it's really true but only before we start playing the suit !

However, if we play the King (and both opponents follow) and then lead a small card from South, awaiting a card from West – the result of the calculations will be different.

At that point there will be only two remaining cards which can be distributed in two, equally probable, ways: Q–x or x–Q.

Thus, finessing is as good as playing for the drop (50% for each line).

Our calculation is significantly better as it takes into account the extra information given by the fall of three spot cards.

That reasoning, although superficially correct, is, however, a fallacy and can be countered in several ways:

If those two breaks (Q-x, x-Q) were equally probable that would mean that at this point of the play the defenders could detach those two cards (the Queen and the small card), shuffle them and redistribute again between the two hands.

But it was the entire deck that was shuffled !

The information we receive watching the small cards played by the defenders doesn't increase our knowledge about the actual distribution.

Provided that (and both sides of the argument agree with that) the defenders follow suit completely randomly, never displaying any habits (eg always in ascending order) or regular patterns.

It was clear beforehand (ie before the play started) that at the decisive point of the play some three small cards will have been revealed. We could equally well ask West to show us his two small cards and East – to show us his small card, and our chances of success wouldn't improve.

If the point we have just made seems vague to the reader, we recommend its more emphatic version:

There are thirteen hearts lying on the table (face down), divided into two groups:

in the smaller one – 3 cards, in the bigger one – 10 cards.

Where is the Ace ?

We bet on the bigger part, of course, and are willing to accept the odds of 10:3 on.

Then, our opponent shows us 9 cards selected from the bigger stack (of course he doesn’t select the Ace to show, if it is there).

Are we going to bet on the smaller part (three cards face down) now and accept the odds of 3:1 on ?

This kind of reasoning doesn't nullify our adversaries' argument about making use of surplus information, though. It's indeed true that :

The calculation of the chances should take into account

not only shuffling but also the defenders' plays !

Let's calculate the probabilities again, this time taking into account the way defenders play their spot cards and assuming that they do it randomly.

Let's make it: 89–7 ( West followed with the 8, and then the 9 – East followed with the 7 )

If so, the result of the shuffling was either D98–107 or 1098–D7, and both of them are equally probable.

Let's analyze 480 deals (that number taken to avoid fractions) featuring those two distributions (240 for each), along with all possible ways of playing the spot cards by defenders:

distribution	discards
Q98–107 240 times	98–10	60 times	and we'll see that on those 100 deals where spot cards were played 98–7 (see · in the table): Q98–107 distribution will occur 60 times 1098–Q7 distribution will occur 40 times Thus, the probabilities remain the same as the original ones: finessing – 60%, from the top – 40%, which means that randomly revealing three spot cards by defenders doesn't change anything.
	89–10	60 times
	98–7	60 times
	89–7 ·	60 times
1098–Q7 240 times	109–7	40 times
	910–7	40 times
	108–7	40 times
	810–7	40 times
	98–7	40 times
	89–7 ·	40 times

Let's check now what happens if defenders do not adhere to randomness of playing their spot cards ?

Let’s assume that they always play their spot cards in ascending order:

distribution	discards	success
Q109–87	910–7	1	We can notice here an interesting phenomenon: § In four cases (marked 1) the distribution is fully disclosed. § In all other cases (marked 1/ 2) we can as well finesse as play for the drop (both chances are equal). eg 89–7 played points towards one of the two equally probable distributions marked with the letter x.
Q108–97	810–7	1
Q107–98	710–8	1
Q98–107	89–7	1/ 2 x
Q97–108	79–8	1/ 2 y
Q87–109	78–9	1/ 2 z
1098–Q7	89–7	1/ 2 x
1097–Q8	79–8	1/ 2 y
1087–Q9	78–9	1/ 2 z
987–Q10	78–10	1

As we can see, there is no constant and uniform strategy in that case. The way we play varies with the way defenders play they spot cards.

Summing up the number of successes (1) and partial successes (1/2) we obtain 7 successes in 10 possible distributions which means that owing to nonrandomness of defenders' plays the chance of catching the Queen rose to 70%.

Readers are invited to examine how other “cunning” ways of following suit affect declarer's chances of success. For example:

1) West, holding xxx, conceals the smallest card – East plays the higher from xx.

2) West, holding xxx, conceals the middle card – East always plays the lower from xx.

3) East plays the higher from xx (to “suggest” Qx) – West follows suit in ascending order.

Strategy and Information (2)

Let’s start with the solutions of the exercises we suggested to the readers ending the part one:

variation 1 = 70 %
variation 2 = 80 % !
variation 3 = 90 % !!

Let’s analyze, for example, the most impressive variation 3:

	A J 6 5
?		?
	K 4 3 2

West follows suit in ascending order;

East plays the higher from xx (to suggest Qx).

Distribution W – E			Played W–E			Probability of success	Therefore, declarer’s chance of success is indeed not less than 90% ! As we can see, „cunning” plays by defenders don’t achieve the desired objective, quite on the contrary – declarer’s chance of success significantly rises (!), provided he can determine (which is often possible) the kind of trickiness employed by defenders.
Distribution W – E			Played W–E			Probability of success
Q109	–	87	910	–	8	1
Q108	–	97	810	–	9	1
Q107	–	98	710	–	9	1
Q98	–	107	89	–	10	1
Q97	–	108	79	–	10	1
Q87	–	109	78	–	10	1/ 2
1098	–	Q7	89	–	7	1
1097	–	Q8	79	–	8	1
1087	–	Q9	78	–	9	1
987	–	Q10	78	–	10	1/ 2

The best defenders’ strategy is, therefore, to play randomly insignificant cards, and any „deceptive” plays should be consider naivety, especially after the publication of this article.

It may happen, however, that even completely random plays by defenders alter declarer’s probabilities to the extent that he has to change his line of play. Although it doesn’t apply to the problem of “catching the Queen missing five cards”, it applies to the following:

	7 6 5 4
?		?
	A K 10 9 8

S played the Ace: W followed with the Jack, E – with the three.

S crossed to the table and led a spot – E followed with the two.

What now – to finesse or to play the King ?

Declarer looked up the relevant tables and found there the following probabilities:

distribution QJ–32 = 6,78 %

distribution J–Q32 = 6,22 % , which would mean that playing from the top is better.

However, the tabular probabilities apply only to shuffling the entire deck !

To obtain correct results we have to take into account defenders’ plays. Let’s do it, then, assuming their plays are random:

distribution

plays

probability

QJ–32

Q–32

Q–23

J–32 ·

J–23

1,695 %

1,685 %

1,695 %

6,78 %

One can see that:

If J–32 appeared

the probabilities are as follows:

QJ–32 = 1,695 %

J–Q32 = 3,110 %

J–Q32

J–32 ·

J–23

3,110 %

6,22 %

Transforming this, so as to the sum of the probabilities adds up to 100%, we obtain that at the crucial point of the play:

distribution QJ = 35,28 %

distribution J–Q32= 64,72 % !!!

Since defenders played their cards randomly (thus, in optimal fashion) we could expect that nothing would change. It turned out to be the opposite: the probabilities were altered so significantly that the optimal strategy changed. Finessing on the second round of the suit is best (64.72%), winning by a huge margin over playing from the top which originally (based on shuffling alone) might be considered best (52%).

We could arrive at the conclusion that „something will change” in a different way, reasoning like follows:

If West had QJ, then – assuming he plays his cards randomly – he could equally well drop the Queen as the Jack. When he drops the Jack (as he did) the probability of his holding QJ decreases (and at that – exactly by half).

The reasoning outlined above is known in bridge literature as the

Principle of Restricted Choice (PRC):

Any play chosen by a player makes it more likely that his choice was restricted,

ie he had not an equally good play as an alternative.

The wording of the PRC isn’t precise enough, though, and it can lead us astray if we face a more subtle situation. When in doubt, it’s always better to conduct a precise tabular analysis.

The problem of correcting the probabilities is too complex to be enclosed in one universal rule. Let’s content ourselves with the following general advice:

The calculation of probabilities should take into account not only the fact that the deck was shuffled but also all calls and plays made by the opponents and, generally – EVERYTHING THAT HAS HAPPENED AT THE TABLE !

Yes – everything. For example, the facts established on occasion of the following deal:

AJxx

AQxx

S plays 7NT on a diamond lead, preceded by

a detailed inquiry about the bidding and a… long huddle by W.

so W „had” to lead one of these suit for safety reasons.

K10xx

KJxx

Who has the Heart Queen ?

E, for sure, if we assume that W had reasons to think before making the opening lead !

The reasoning is simple:

If W had the Heart Queen he wouldn’t be thinking that long before leading.

If we don’t take the huddle into account, the chance of E having the Queen is 2 / 3.

Check it yourself !

If it turns out that it’s W who has the Queen after all, don’t make a fool of yourself by complaining about the huddle (“What were you thinking about?”, “Why were you inquiring about the bidding”)!!! But that is another issue.

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