Another suit-combination

[Stephen Tu]

And yet East must as a matter of fact have K,K10 or Kx and you propose to lose to four of those 5 holdings

 

Yes, because you aren't looking at the bigger picture. As declarer, the holdings we can pick up for 4 tricks where our choice of plays matters are the following:

K(2.82), KT(3.39), Kx (10.17), Kxx(10.17)

 

Say the Q is not covered. What should we do next? It doesn't matter if RHO never covers from Kx, but if he ever covers, clearly Kxx will be more likely. So we are choosing to pick up Kxx. A defender knows this, so he should not be covering from Kx, at least not very much, see below.

 

Say the Q is covered. What now? KT is 3.39%. Finesse is better IFF (prob stiff K) + (prob Kx) * (prob opp covers) > 3.39. So if 2.82 + (10.17) * (n) > 3.39 you should hook. But since we are playing against optimal defenders, who know they are supposed to duck close to 100%, (since they know we then will try to pin T), n is going to be very small, so the odds favor playing them for KT & therefore drop.

 

If you propose to finesse on the way back, essentially you are losing the 3.39-2.82 to defenders who never cover, and a smaller fraction if they cover a small fraction. But if they cover too much, then finessing on the way back is better.

 

Note this is all talking about the 4-4 break and needing all 4 tricks, usually an IMP situation. If 5-3 and can no longer pick up say stiff K, or if MP, numbers & considerations change. At MP, it's actually usually best to cater to Kx & stiff K, because although this makes taking 4 tricks less likely (by a little over 0.5%), you avoid losing 2 tricks to KTxx offside (8.5%).

[dburn]

And yet East must as a matter of fact have K,K10 or Kx and you propose to lose to four of those 5 holdings

Let us examine this in the context of the more familiar "restricted choice" position:

 

North (dummy)

A10876

 

South (declarer)

K954

 

I, South with a few hundred thousand dollars on the line, play the six from dummy. East plays the queen, I win with the king on which West plays low. Then I lead a low card, on which West plays low...

 

Halo will argue that East "must as a matter of fact" have started with either QJ or Q. Since QJ is a priori more likely than Q, why should I (who will finesse against the jack every time, and win almost two times out of three) "propose to lose" to an original holding of QJ with East? Even if QJ were not a priori more likely than Q but exactly as likely as Q, isn't it "just a guess"?

 

No, it isn't. I'm sure Halo knows this as well as I do, and that he will now know what was wrong with drawing the conclusions he drew from "must as a matter of fact" in his earlier statement. If not... well, this conditional probability is perhaps counter-intuitive stuff.

[Halo]

I am aware of the logic around genuine 'restricted choice', and entirely bought into it. As you know the principle doesn't contradict a priori odds - indeed how could it possibly.

 

The position in this post simply is not restricted choice in the classic sense of your H versus HH holding example.

 

My point is very simple. Once an optimum 'perfect knowledge of history' strategy is established for this holding, it is possible for me to choose to exploit this information, unpredictably, by adopting a different strategy that poses a different problem to you.

 

I can't do that in a genuine restricted choice situation.

 

Your advice in this post is excellent, but it is based on assumptions about the unrestricted behaviour of players. Sound advice generally speaking, but if a lot was at stake against a known expert, I would personally just play the a priori odds, as a rational alternative to 'he knows that I know that he knows' games'.

[Stephen Tu]

I am aware of the logic around genuine 'restricted choice', and entirely bought into it. As you know the principle doesn't contradict a priori odds - indeed how could it possibly.

 

The position in this post simply is not restricted choice in the classic sense of your H versus HH holding example.

 

You have to make assumptions on how the defenders will behave. If you posit a "perfect defender", then you must assume they are pursuing a strategy that minimizes your expectation. It's been clearly shown that the best strategy for a defender here is not to cover with Kx, at least not more than a few %. Then the best declarer can do is pick up KT & Kxx. Any other strategy for declarer will take 5 tricks less often.

 

Even in the classic restricted choice situation you are making these assumptions. In the Q/J stiff vs. QJ tight case, you are assuming that defender is randomizing sufficiently from QJ such that the stiff honor case is more likely. If he was known to favor one or the other honor in too great a margin (> 92%), then you could exploit that by favoring the drop if that honor appeared. Defender has a wide latitude to randomize here, from QJ as long as he plays differently 1 time in 10 the declarer can do no better than to simply follow the restricted choice line of playing for stiff honor.

 

For the discussed suit combo, the situation is different, if defender played from Kx randomly (~50%) he would be exploitable, declarer switches to hooking on the way back. Declarer would then pick up Kxx, K, and the portion of Kx that covered. But for the purposes of the problem we are assuming optimal defense, which means the defender will minimize the frequency of covering from Kx, so that Kx + K frequency is less than KT. This causes the declarer the greatest difficulty. Also note that once declarer chooses to pick up KT + Kxx, he cannot be exploited, that is the best he can do against best defense.

 

You can never properly compare plays based solely on a priori odds if it is a combination where defender has a choice of plays. You must make a reasonable assumption of the defenders strategy. On some combos, it is reasonable to assume the defender's strategy is random, that they would play each of two cards 50/50. But on other combos, like this one, if a good defender that is a bad assumption. If a good defender is not covering from Kx, it's a poor choice to play him for Kx if he covers, even though Kx is a likely holding.

 

Say you play against this defender 1000 times with this holding.

You can expect him to hold Kx about 102 times. Say he plays 99 times x, 3 times K. In that same 1000 he will hold stiff K about 28 times, KT about 34 times. Doesn't that make it clear that you should play him for KT if he covers? 34 > 28 + 3.

 

My point is very simple. Once an optimum 'perfect knowledge of history' strategy is established for this holding, it is possible for me to choose to exploit this information, unpredictably, by adopting a different strategy that poses a different problem to you.

 

No, if either party decides to adopts the game theory optimal line for his side, the other party can not exploit him. You can always guarantee yourself the game theoretical result, it just fails to maximize if your opponent is doing something exploitable. It is only if you think your opponent is deviating from the game theory optimal strategy that you can exploit him. But if you try to exploit this, you yourself will be the one exploited if your assumption was wrong.

 

So the end result is that with known experts battling, Kx doesn't cover, declarer picks up Kxx & KT, defense gets two tricks with K & Kx. Neither side can do any better if the other side doesn't err.

[dburn]

Of course, if I were playing for a billion dollars in the final of the World Pairs Championship and not the Bermuda Bowl, I would play low from dummy on the second round if the queen were not covered, and finesse against West's ten if the queen were covered (the strategy I have described as drop-finesse).

 

Bridge is not an easy game. But then, the only person who said that it is was a Conservative politician.

[Halo]

But for the purposes of the problem we are assuming optimal defense, which means the defender will minimize the frequency of covering from Kx, so that Kx + K frequency is less than KT. This causes the declarer the greatest difficulty.

Thanks Stephen. Your exposition is very lucid.

 

In a practical sense, though, we have to be sure people understand that if I deviate from the optimal roughly 7 times in a thousand rather than 5 or 6, you can start to exploit me by adapting your strategy. If you cannot tell the difference between 5 times in a thousand and 7 times in a thousand I start to make a small gain.

 

The extent of my deviation from 'optimal' should in fact be the biggest number greater than 7 per thousand that I think I can get away with. This is a game of incomplete information.

 

I suppose it is equivalent to the frequency of bidding on tram tickets before the opponents loosen up on penalty doubles etc.

[Stephen Tu]

In a practical sense, though, we have to be sure people understand that if I deviate from the optimal roughly 7 times in a thousand rather than 5 or 6, you can start to exploit me by adapting your strategy. If you cannot tell the difference between 5 times in a thousand and 7 times in a thousand I start to make a small gain.

 

No, you aren't quite getting it. If your opponent is playing the unexploitable line, if you deviate, you are not making any gains, small or otherwise. You have absolute zero effect. Even if you cover all the time from Kx, if your opponent doesn't change, you've still gained nothing, he is picking up the same KT, Kxx he decided to in the first place. He's only *failed to exploit you*. That is different from you gaining.

[dburn]

I suppose that after all, the well-known combination we are discussing here may have something in common with Fred Gitleman's new combination that inspired the original post in this thread: the capacity to astound.

 

Fred's offering was:

 

North (dummy)

J108

 

South (declarer)

K65432

 

Declarer runs the jack if East plays the seven, but plays the king if East plays the nine. Fred and others were surprised that declarer's strategy should vary with the low card that East plays.

 

Here, with

 

North (dummy)

QJ98

 

South (declarer)

A432

 

an optimal defender sitting East against an optimal declarer will never make a trick with Kx at matchpoints, but will always make one at IMPs. Now is not this ridiculous, and is not this preposterous? - a thorough-paced absurdity; explain it if you can.

[BillHiggin]

an optimal defender sitting East against an optimal declarer will never make a trick with Kx at matchpoints, but will always make one at IMPs. Now is not this ridiculous, and is not this preposterous? - a thorough-paced absurdity; explain it if you can.

How are we supposed to explain a false statement.

 

Are you claiming that declarer will for some reason adopt an inferior line of play at MP scoring. Perhaps he does not recognize that the defender is "optimal" and hopes for typical MP foolishness?

 

Declarer only deviates from drop-pin when he is seeking to exploit foolishness on his right. This does not vary by scoring!

[Halo]

No, you aren't quite getting it. If your opponent is playing the unexploitable line, if you deviate, you are not making any gains, small or otherwise. You have absolute zero effect. Even if you cover all the time from Kx, if your opponent doesn't change, you've still gained nothing, he is picking up the same KT, Kxx he decided to in the first place. He's only *failed to exploit you*. That is different from you gaining.

Stephen

 

I do understand.

 

The reason in practice that it is so important to know how little the optimal strategy (play for Kxx and K10) gains is because the declarer/defender or both may adapt for minimal reasons.

 

I gave you one reason in the spirit of the post.

 

Consider the real world. If you had a proof or even an inference about the opponents distribution. For example, you could draw trumps and they were 23 or 32. For example, opponents lead suggested a side suit distributed 24. Your optimal line is so marginal that both experts could already be adapting their strategy if the frequency of Kxx v Kx shifted by a very small amount - if they were genuine experts. How easy do you think this game is. Not just maths and stamina.

[Stephen Tu]

Are you claiming that declarer will for some reason adopt an inferior line of play at MP scoring. Perhaps he does not recognize that the defender is "optimal" and hopes for typical MP foolishness?

 

David is absolutely right at MP scoring. The idea is that at MP scoring, you cannot afford to play for Kxx onside, because a good LHO will be ducking frequently from KTxx (i.e. some good chunk of 8.47%), and if you play the J next round you are toast. By playing for Kx/K/x instead, you lose a small % in picking up all 4 tricks, but you gain substantially when LHO has KTxx & ducks by not losing 2 tricks to him.

 

I think it's actually the same for IMP scoring, since the advantage in the "optimal for 4 tricks" line is so small, .5% extra of game bonus is not worth 6% extra undertricks. Best line for IMP scoring / Best for MP scoring / best average max tricks / best for taking maximum possible tricks are not always the same. State of match considerations could come into play.

[dburn]

an optimal defender sitting East against an optimal declarer will never make a trick with Kx at matchpoints, but will always make one at IMPs. Now is not this ridiculous, and is not this preposterous? - a thorough-paced absurdity; explain it if you can.

How are we supposed to explain a false statement.

 

Are you claiming that declarer will for some reason adopt an inferior line of play at MP scoring. Perhaps he does not recognize that the defender is "optimal" and hopes for typical MP foolishness?

 

Declarer only deviates from drop-pin when he is seeking to exploit foolishness on his right. This does not vary by scoring!

What do you mean by "an inferior line of play at MP scoring"?

 

It is well known that in a great number of positions at MP scoring, declarer should not adopt his best line of play for the contract; instead, he should seek to combine a good chance of making the contract with a good chance to minimise the number of tricks he goes down, if go down he must.

 

Indeed, there are some easily exhibitable positions in which declarer should not even try to make the contract at all. But for the most part, what declarer seeks is a line of play that makes the maximum number of tricks per deal.

 

In this case, the line we have described as pin-drop makes four tricks (against optimal defence) roughly 16.4% of the time. But it makes two tricks roughly 8.5% of the time (when an optimal West ducks the queen from K10xx, as he always will). It makes three tricks the rest of the time, and it makes 3.04 tricks per deal.

 

The line we have described as drop-finesse makes four tricks about 15.8% of the time. But it never makes only two tricks, and it makes 3.13 tricks per deal.

 

Assume that in the 11-table 1,000-board final of the World Open Pairs Championship, I sit South at one table and Bill Higgins sits South at the other ten tables. By a statistical miracle, every board is the same - all Souths arrive in a 27-point 3NT that needs four tricks from QJ98 facing A432.

 

I will play drop-finesse throughout (if the queen is not covered, lead the eight from dummy and run it unless the king appears; if the queen is covered, win with the ace and play to the eight).

 

The Higginses will play pin-drop throughout (if the queen is not covered, run the jack; if the queen is covered, play to the jack).

 

East will almost never cover from Kx, and West will never win the first round from K10xx.

 

I am English, so I will assume English matchpointing - two for a win, one for a tie, none for a loss. The East holdings are small enough in number to tabulate here. When East has:

 

K10xxx it does not matter; we will all go down one.

 

K10xx it does not matter; we will all go down one (for different reasons, though).

 

Kxxx it does not matter; we will all make three.

 

K10x it does not matter; we will all go down one.

 

Kxx it does matter; I will get 0 match points for down one and the Higginses 11 match points apiece for making three. This will happen on 102 of the boards, so Bill is ahead of me by 1,122 matchpoints at this stage.

 

K10 it does matter; I will get 0 match points for down one and the Higginses 11 match points apiece for making three. This will happen on 34 of the boards, so Bill is ahead of me by 1,496 matchpoints at this stage.

 

Kx it does matter; I will get 20 match points for making three and the Higginses 9 match points apiece for down one. This will happen on 102 of the boards, so Bill is ahead of me by 374 matchpoints at this stage.

 

K it does matter; I will get 20 match points for making three and the Higginses 9 match points apiece for down one. This will happen on 23 of the boards, so Bill is ahead of me by 108 matchpoints at this stage.

 

So much for the cases where East has the king. There is only one relevant case where West has the king (we have not explicitly discussed singleton king, but both Bill and I will cash the jack next and go down one because of what Halo does not yet think of as "classical" restricted choice, even though it really is). However:

 

When West has K10xx, I will get 20 matchpoints for down one and the Higginses 9 matchpoints each for down two. This will happen on 84 of the boards, so I am ahead of Bill by 906 matchpoints at this stage.

 

And there aren't any more stages. Really, Bill, there aren't. An inferior line of play at IMP or rubber bridge or total-point scoring may nevertheless be the correct line of play at MP scoring. You knew that all the time, though. Maybe you just didn't know why.

[dburn]

I observe that I have just subtracted 108 from 924 and arrived at an answer of 906.

 

This might foster in some minds the notion that I am a mathematician, because it is well known that mathematicians cannot do simple arithmetic to save their lives.

 

I assure you that I am no mathematician of any kind, but I am typing this post (and the one that preceded it) after a friend's birthday party.

 

I see, however, that Halo has invited Stephen (who clearly is a mathematician, and an excellent one at that) to "consider the real world". Mathematicians should not be invited to do that. Nor, at times, should I.

[dburn]

It should also be said (before I go to bed, which I should have done some hours ago) that:

 

If you are in 7NT in a pairs tournament with this combination, you should follow the Higgins pin-drop strategy, for it will not matter whether you go down one or down two - the field will not be in your contract.

 

If you are in a 27-point 3NT in a pairs tournament with this combination, you should (for the reasons I have given above) follow the drop-finesse strategy.

 

If you are in the real world, you can do what you like, for it won't make all that much difference in the grand scheme of things. Well it was said by the bard:

 

To some people a squirrel's a squirrel,

To others a squirrel's a squrl.

Since freedom of speech is the birthright of each,

I can only this fable unfurl:

A virile young squirrel named Cyril,

In an argument over a girl,

Was lambasted from here to the Tyrol

By a churl of a squrl named Earl.

 

Ogden Nash

[Stephen Tu]

when an optimal West ducks the queen from K10xx, as he always will

Maybe not?

 

There is only one relevant case where West has the king (we have not explicitly discussed singleton king, but both Bill and I will cash the jack next and go down one

 

Huh? I thought if West has singleton K and you cash the J you make it. That shows that actually from KTxx West should take it 1/3 of the time, that way if West takes you don't get to freely cash the J.

 

"Optimal" defense is complicated!

[Halo]

If your aim is to apply mathematics to Bridge for practical reasons, then approaches that cannot be applied at the table are interesting but eventually unfruitful.

 

If this was always a 'purist' discussion independent of reality, I apologise for intruding.

[Stephen Tu]

If your aim is to apply mathematics to Bridge for practical reasons, then approaches that cannot be applied at the table are interesting but eventually unfruitful.

 

Exactly what about the mathematical reasoning do you think cannot be applied practically at the table? You can always assign your own estimate of probabilities to an opponent's choice of plays if you feel he is deviating from correct defense & wish to try & exploit him. Just be aware that when you try to exploit someone, usually that leaves yourself open to be exploited if you are wrong about them. Ignoring the opponent, just playing for the theoretical optimal combinations, avoids you being on the losing end of any mind games. It also avoids you having the winning end also, as it fails to punish opponents who deviate from the proper frequencies of actions. But against a good player just guaranteeing not losing is probably right.

 

 

Math will always tell you exactly what to do. If opp does A between x% & y%, choose line 1, else if if < x% line 2, else if > y% line 3 etc. You just have to make an accurate determination of what the opp is doing if you wish to maximize vs. them. Or you give up on maximizing, assume they are playing properly, and guarantee yourself the normal level of success.

 

If you do not want to use math, how else do you propose to solve the problem accurately?

[han]

I observe that I have just subtracted 108 from 924 and arrived at an answer of 906.

 

This might foster in some minds the notion that I am a mathematician, because it is well known that mathematicians cannot do simple arithmetic to save their lives.

 

I assure you that I am no mathematician of any kind, but I am typing this post (and the one that preceded it) after a friend's birthday party.

 

I see, however, that Halo has invited Stephen (who clearly is a mathematician, and an excellent one at that) to "consider the real world". Mathematicians should not be invited to do that. Nor, at times, should I.

Clearly you are no mathematician because you are talking crap. Mathematicians tend to be very very good at doing simple arithmetics. Stick to suit combinations and poetry I say. B)

[Halo]

Math will always tell you exactly what to do. If opp does A between x% & y%, choose line 1, else if if < x% line 2, else if > y% line 3 etc. You just have to make an accurate determination of what the opp is doing if you wish to maximize vs. them. Or you give up on maximizing, assume they are playing properly, and guarantee yourself the normal level of success.

Stephen

 

I give in.

 

But I promise you next time you meet me and ask my name, I'll be disguised of course, and say 'dburn'. Maybe you have already modelled it - for every personating 'dburn' there is a 'halo', and the best of good fortune if that is what you conclude.

[dburn]

Mathematicians tend to be very very good at doing simple arithmetics.

Ernst Eduard Kummer (1810-1893), a German algebraist, was rather poor at arithmetic. Whenever he had occasion to do simple arithmetic in class, he would get his students to help him. Once he had to find 7 x 9. "Seven times nine" he began. "Seven times nine is er... ah... ah... seven times nine is..."

 

"Sixty-one", a student suggested. Kummer wrote 61 on the board.

 

"Sir," said another student, "it should be sixty-nine."

 

"Come, come, gentlemen, it can't be both!" Kummer exclaimed. "It must be one or the other."

 

Another great mathematician, Paul Erdos, suggested that Kummer in fact solved the problem himself as follows: 7 x 9 can't be 61 because 61 is prime; it can't be 65 because that is a multiple of 5; 67 is prime; 69 is clearly too large; so that leaves 63.

 

But if it's poetry you want, here is an indication of how to pronounce the Hungarian name of Erdos:

 

A conjecture both deep and profound

Is whether a circle is round.

In a paper by Erdos

That's written in Kurdish,

A counter-example is found.

[han]

I wasn't alive yet when Kummer was so I'll have to believe you on your word. And it logically follows that all mathematicians are bad at simple arithmatics, doesn't it? How wrong I was!

 

I like the poem, who's Erdos?

[ceeb]

who's Erdos?

If you're not kidding, Google "most prolific mathematician".

 

Charles

[han]

I was kidding, or perhaps hustling is the word.

[skjaeran]

I was kidding, or perhaps hustling is the word.

Hustling is indeed the word Han. :P