[Halo]

I agree with Fred. Traditionally restricted choice refers to cases that can also be solved with probability/vacant spaces.

If your argument is based on what a class of players will mostly do, in solving a Bridge problem, the results may be similar to restricted choice, but only incidentally.

[cherdano]

djehuti, on Oct 20 2007, 12:12 PM, said:

Wow grosvenor: someone to play the 9 with Q9x

I am really glad to learn I am not the only one to be so sick of thinking about that...

[kenrexford]

cherdano, on Oct 21 2007, 10:48 PM, said:

djehuti, on Oct 20 2007, 12:12 PM, said:

Wow grosvenor: someone to play the 9 with Q9x

I am really glad to learn I am not the only one to be so sick of thinking about that...

Three of us. Probably more.

[ralph23]

kenrexford, on Oct 21 2007, 04:22 PM, said:

So, this was a Restricted Choice problem.

The really interesting "rest of the story" is that this situation illustrates a very strange type of restricted choice analysis. I wonder if there are others that we have not thought of, where someone has one restrained situation, a completely different and unrelated constrained situation, and an option that somehow attaches itself to each.

http://www.acbl-dist...rg/artic003.htm

Article using the "Monty Hall problem" as a teaching tool to explain RC in bridge.

http://bridge.thomas...ds/triple2.html

Setting out a "subtle form of RC" similar in principle to the example created by Ken.

[jdeegan]

fred, on Oct 18 2007, 04:34 PM, said:

Summary: if East follows with the 7, play small. If East follows with the 9, play the King.

To me it is really strange that the correct play depends on whether East follows with the 7 or the 9. Who would have thought this could possibly matter?


Fred Gitelman
Bridge Base Inc.
www.bridgebase.com

I dunno. If I'm playing against Fred, I can correctly guess the opponents' holding in the diamond suit, and I hold the Q9; all I have to do is play the 9, and I have a sure two tricks since Fred says he will always play the king. For me, much of the fascination of bridge is the battle of wits between participants.

Against lesser players, the play of the 9 would virtually deny the queen. It just 'feels right' to play the queen in order to have a chance to promote the 9. Or maybe it is just following the rule: 'always cover an honor with an honor'.

Change the honors around a little bit and make the length of the diamond suit ambiguous, and you might have a real 'head scratcher' from RHO's point of view. I guess the lesson here is to always try to anticipate the opponent's problem(s) and try to learn as much from his/her 'tells' as possible.

[Stephen Tu]

Quote

I dunno. If I'm playing against Fred, I can correctly guess the opponents' holding in the diamond suit, and I hold the Q9; all I have to do is play the 9, and I have a sure two tricks

It doesn't matter what you do, since if you cover, he's finessing on the way back. The game theory unexploitable optimal declarer line simply gives up on the opponent holding Q9, so you can do whatever you want.

However, against opponents who don't know it's right to go up K on the 9, in principle you want to cover, because that maximizes the magnitude of their mistake. This is exploitative play. The more you cover, the bigger their mistake is as it reduces the number of times their play is correct.

However, if you cover too often (> 91.67%), you open yourself up to a counter-exploit, that is declarer not hooking on the way back if you cover. He will pick up the Q9s you cover instead of the stiff Qs. But this declarer (who goes up with K on the 9, but plays you for Q9 if you cover) in turn can be exploited, but in this case by not covering w/ Q9. So which way you go depends on which of the 3 types of declarer you are facing if you want to maximize your expectation.

The complicating matter, which I tried to point out earlier, is that in real life it isn't double dummy for you, and if declarer has only 5 not 6 you definitely want to cover to have any chance at all (want declarer to play partner for A9x).

But in any case, if 9 played, K > ducking as long as defender EVER covers. Whether to play for stiff Q or Q9 w/ Q covering is supposed to be a separate decision. (Playing to pick up Q9 ONLY is clearly the worst of all worlds).

[joshs]

Ok I will speak up as a math guy.

We all agree that if RHO plays the 7 we should run the J. This play works just as well on the 2-2's and picks up 1 3-1 (stiff 9 on your left). Playing the K picks up no 3-1's (you still have 2 losers if there is a stiff Q on your left).

If RHO plays the 9, assuming no stupidity, the only relevent holdings are Q9 and A9 and A97 (playing the 9 from AQ9 is stupidity).

Let us first assume that the defender never plays the 9 from A97, and plays the Q with probability p where 1>p>0 with Q9 and always plays the 9 from A9.


Here the conditional probability of the defender having the A9 given the play of the 9 is greater than the probability of him having the Q9 given the play of the 9 (how much greater depends on the value of p). Hence the K is the better play.

If you include the A97 as a possible holding you again are comparing the probability that the defender has A9 OR A97 given the play of the 9, with Q9 given the play of the 9.
Since P(A9)>P(Q9) we conclude that
P(A9 OR A97) >P(Q9) since the OR only makes the probablity greater.
so the K is still the right play.

Finally, since on optimal play the play of the 9 from A97 never gains and sometimes loses (declarer would have gone wrong here, but instead went right if you played the 9), we also conclude that playing the 9 from A97 is stupid, although my partners have been known to do that play in the desire to give suit preference...).

If you want the exact odds here, given a value for p, we apply Bayes rule to the relevent cases:

Q9 and A9 are equally likely
The 9 is played 100% of the time from A9
The 9 is played 1-p% of the time from Q9
Let X=A9 holding
Let Y=the 9 is played
P(X Given Y)=P(Y Given X) * P(X)/P(Y)

P(Y Given X)=1
P(X)=0.5
P(Y)=1/2 *1 +1/2*(1-p)=1-p/2

Hence P(X Given Y)=0.5/(1-p/2)>0.5 since P>0
If P=1/2 (Q and 9 plays are equally likely) then we get 0.5/0.75=2/3 that most people assume in restricted choice situations.

But the inequallity doesn't change no matter what p is as long as neither the Q nor the 9 play occurs 100% of the time.

Similarly in standard restricted choice where you can play the Q or J from QJ doubleton, playing the player for the singleton is more liekly then the QJ doubleton no matter what frequency you play each card as long as its not 0 or 100%.

Josh

[jdeegan]

Question 1. I lead the J and Mrs. G, who always covers an honor with an honor, plays the Q. You are saying eschew the finesse on the way back. I understand and agree. So far so good.

Question 2. I lead the J and Mr. X, 8000 masterpoints, but whose emotions are written on his sleeve, plays the Q after emitting a definite 'tell'. My best guess is that even given that he is not above coffee housing with a stiff Q, the odds of his having started with Q alone are half the a priori odds (first he has to have started with the stiff Q, then he has to decide to coffee house). I cover with the K. It goes A. Later, I lead small toward the dummy, and LHO plays the 7. How now brown cow?

[dburn]

Speaking of Grosvenors, what should you do if RHO plays the ace on the jack, then the nine when you next lead the ten from dummy?

[keylime]

For all the mathematical attempts at reasoning, I've yet to besides Fred's statement see anyone really assess the more valid point, which is the auction and table feel element. Bayes' theorem (which I am versed in) works only under controlled conditions - but when is bridge really an experiment with control variables and set criteria in order to derive a conclusion?

If you're at the club game, you are far more likely to get this problem correct than wrong, because of the ingrained "cover an honor with an honor" mentality. Additionally, you should be able as the superior player that you are, sense if there's certain honor combinations out there (i.e. if LHO holds the AQ9, they should lead differently than holding A-9 versus Q-7).

However if you are let's say at a National, then you can not necessarily follow the Zia rule (if they don't cover, they don't have it). Players of our peer group know that tip already; thusly they are more prone to duck smoothly and put you to a guess. Now, some deduction enters mathematically (especially if they have bid in the auction, now you get an inferential count of sorts on the first trump play - that and opening leads of course).

[Stephen Tu]

Quote

For all the mathematical attempts at reasoning, I've yet to besides Fred's statement see anyone really assess the more valid point, which is the auction and table feel element

Huh???? I already laid out all the reasoning one needs. The question is *DOES YOUR RHO KNOW YOU HAVE 6 CARDS IN THE SUIT*. If he knows this, and he knows you are capable of the "right" play when the 9 appears, then he should be capable of ducking from Q9 enough that if he does cover you should hook on the way back to make sure you pick up stiff Q. This minimizes your losses if you are wrong about your opponent's tendencies, at most you give up about .5%, where as if he ducks a lot & you go up with the T after the Q covers you'll be wrong by several percentage points.

But if he thinks you can have 5, really he must cover, ducking can only lose then, as you no longer have reason to put up the K. The cover is absolutely necessary in this case to put you to a guess.

And to answer the earlier question, if you have 6, and RHO coffeehouses before covering, play the T on the way back and call the director for your trick back if it turned out he did it with stiff Q, you'll get your trick back. It's only if he covers smoothly that you have a problem.

[hrothgar]

keylime, on Oct 25 2007, 05:40 PM, said:

For all the mathematical attempts at reasoning, I've yet to besides Fred's statement see anyone really assess the more valid point, which is the auction and table feel element. Bayes' theorem (which I am versed in) works only under controlled conditions - but when is bridge really an experiment with control variables and set criteria in order to derive a conclusion?

Any discussion about using "Table Feel" involves deviating from a mathematically optimal line based on a subjective phenomena.

Comment 1: If you don't understand what constitutes the optimal line, you can't make an informed decision about whether or not to deviate away from it.

Comment 2: Discussion about table feel aren't particularly fruitful because you are (essentially) discussing some kind of gut feeling.

The initial posting might have some interesting content "I knew that the percentage line was to play for the drop, but I chose to hook the jack". However, its hard to extend the conversation much from there...

[ArtK78]

Stephen Tu, on Oct 25 2007, 10:17 AM, said:

And to answer the earlier question, if you have 6, and RHO coffeehouses before covering, play the T on the way back and call the director for your trick back if it turned out he did it with stiff Q, you'll get your trick back. It's only if he covers smoothly that you have a problem.

You can scream for the director all you like, but you will never get your trick back.

You take advantage of your opponents' mannerisms at your own risk. Your opponent may have to appear before a conduct and ethics committee if his coffeehousing is deemed to be premeditated, but the trick won will stand.

[Stephen Tu]

Are you a director Art? Pretty damn sure I'd get an adjustment if the guy hitched noticeably with stiff Q:

Quote

Law 72:

D. Variations in Tempo or Manner

    1. Inadvertent Variations

        It is desirable, though not always required, for players to maintain steady tempo and unvarying manner. However, players should be particularly careful in positions in which variations may work to the benefit of their side. Otherwise, inadvertently to vary the tempo or manner in which a call or play is made does not in itself constitute a violation of propriety, but inferences from such variation may appropriately be drawn only by an opponent, and at his own risk.
    2.  Intentional Variations

        A player may not attempt to mislead an opponent by means of remark or gesture, through the haste or hesitancy of a call or play (as in hesitating before playing a singleton), or by the manner in which the call or play is made.

F.
2. Player Injured by Illegal Deception

    if the Director determines that an innocent player has drawn a false inference from a remark, manner, tempo, or the like, of an opponent who has no demonstrable bridge reason for the action, and who could have known, at the time of the action, that the action could work to his benefit, the Director shall award an adjusted score

Inference from an opponent's inadvertent variations is at your own risk. But if they are intentionally trying to deceive, as with hitching with a singleton, you can & will get an adjustment in my experience.

[jdonn]

Stephen Tu, on Oct 25 2007, 01:26 PM, said:

Are you a director Art? Pretty damn sure I'd get an adjustment if the guy hitched noticeably with stiff Q:

You are right about the quoted rule, but in practice it's (almost) impossible to get this ruling in your favor. The player simply says either "I didn't do it", "I played at the same speed as always", or "That tempo is normal for me" and there it has to die. If you have been seeing people get this ruling then point me to your directors, they are one of a kind.

[Stephen Tu]

I agree it has to be a pretty blatant hitch, I have seen the ruling, but probably not more than twice ever. I wouldn't call unless it was an egregious tempo break.

[Stephen Tu]

If the hitch is blatant, you have redress coming, so you get it right either at the table or via director call by not playing for stiff Q, it's a 100% play. Naive has nothing to do with it. If this is an informal game with no director in a place where coffeehousing is tolerated, then the factors behind your choice are changed.

[joshs]

Well you certainly can include table feel in your analysis:

Suppose you feel that with A9 defender will hesitate q% of the time before playing the 9, and with Q9 the defender will hesitate z% of the time before playing which ever card he plays, which again we are assuming he plays the Q with probability p..

Now your information is:
Case 1: Played the 9 and hitched
Case 2: Played the 9 and didn't hitch

Again we apply Bayes rule.

Lets say we want the optimal play in case 1:
Probability of him having the A9 given he played the 9 and hitched is =
(Prob of playing the 9 and hitching given the holding of A9)*(Prob having the A9)/
(Prob playing the 9 and hitching)

Probability of playing the 9 and hitching given the holding of A9=q
Prob of having the A9=1/2
Prob of playing the 9 and hitching=1/2*q+1/2*(1-p)*z

So our answer is:
(q/2)/(q/2+z/2-pz/2)

Let try this with some real numbers:

a. p=1/2 q=1/10 z=1/5
P(A9 Given the hitch and the play of the 9)=(1/20)/(1/20+1/10-1/20)=1/2

So its a even money guess.

b. p=3/4 q=1/10 z=1/2
P(a9 Given the hitch and the play of the 9)=(1/20)/(1/20+1/4- 3/16)=4/9
So you should run the J

and so on.

I will leave it as an excercise to the reader that if you make the symplifying assumption that p=1/2 then you should run it, given a hestiation, whenever z>2q.

In general you should run the J whenever z-pz>q

(For the record, I do not beleive p=1/2 here, if the expert defender is sure you have at least 6 cards, but might have 7, p<1/2. If the expert defender is sure you have at most 6 cards, but might have 5 p>1/2)

Finally note that you as declarer should feel free to hitch while doing this calculation