[FrancesHinden]

There's a reason I started down this route.

The QJ example, the AJ9 puzzle, and the when-to-cover-from-Kx puzzle all have the same issue: the 'optimal' strategy in terms of trick expectation is usually to play alternative cards with anything from a range of probabilities. Playing each of the Q and J with 50% probability is 'a' rather than 'the' optimal strategy (apologies to Hannie, by the way, who also pointed this out earlier but I missed it).

I think there's a mildly interesting bridge point here, rather than a probability theory point.

If you have discussed this with your partner, or played together a lot, you may have an agreement about how often you play each card, and as such it should be disclosed to the opponents. I want to take the Kx question as a good example: dummy has QJ98x, declarer is known to have Axx, when should you cover from Kx? Suppose in addition declarer cannot pick up a 4-1 break for some reason (needs to take a ruff in the short hand later, say). If I remember rightly, the answer is to cover anything between 0 and 1/3 of the time.

Now, if you play an infinite number of boards it doesn't matter which of the optimal strategies you play. But if you never cover from Kx, and reveal this to declarer, declarer will always get it right when you have K10 doubleton, because that's now the only relevant holding on which you cover. He'll go wrong more often when you have Kx, which compensates you in expectation calculations.

But if I'm playing a match against a team I expect to beat, I want to minimise random swings and rely on my skill. So I don't want to lose a game swing because I happen to have K10 doubleton and oppo get that right for certain.

I'm not going to use game theory terminology, because I'll probably get it wrong (I was a fluid dynamicist), but I think the optimal strategy against a worse team should satisfy two criteria:

i) You cannot improve your trick expectation by changing it (this I think is the 'nash equilibrium' point)
ii) You minimise the difference in trick expectation between your alternative plays

So in fact, I think playing the Q or the J from QJ with 50% probability _is_ the optimal strategy after all.

Unless I'm playing against a much better team than my team. Then I want to make things swingier. So I should tell them that I always play the Q from QJ doubleton (and pray for singleton queens), that I never cover from Kx (and pray for no K10 doubletons) and so forth.

I may have drivelled on for too long here, but am I making any sense at all?

[whereagles]

FrancesHinden, on Apr 25 2006, 04:34 PM, said:

I may have drivelled on for too long here, but am I making any sense at all?

Not much. It's kinda pointless because it might decide 1 match out of 5000 or so.

[hrothgar]

FrancesHinden, on Apr 25 2006, 07:34 PM, said:

Now, if you play an infinite number of boards it doesn't matter which of the optimal strategies you play. But if you never cover from Kx, and reveal this to declarer, declarer will always get it right when you have K10 doubleton, because that's now the only relevant holding on which you cover. He'll go wrong more often when you have Kx, which compensates you in expectation calculations.

...

I may have drivelled on for too long here, but am I making any sense at all?

The academics who have studied game theory and cards have typically focused on poker. With this said and done, there are a number of examples from poker which are directly applicable to some of the questions that we're discussing.

For example, there's been some very interesting analysis done surrounding 5 card draw that focused on the optimal number of cards to draw if you get dealt three of a kind. I've seen a couple good proofs that demonstrate that the equlibirum solution is to draw one card. (If you draw two cards, your chance to improve your hand. However, the math works out such that its more advantageous to conceal whether you have three of kind, two pair, or are drawing to complete a four flush)

The reason that I brought this up is that the same equilibirum solution also included a number of so-called mixed strategies. It turns out that optimal play requires deliberately randomizing your behaviour when you have a four-flush. You want to draw to the four-flush X pecent of the time and fold (1-X) percent of the time. The authors of the paper suggested that the "best" way to randomize is to use the entropy inherent in the your hands. (If you hold the following card combinations draw, if you hold anything else fold)

The same principles could easily be applied to the game of bridge. For example, suppose that you need to decide whether or not to rise with the King. Create the following rule:

Think back to the previous hand. If the length of your Spade suit was equal to the length of your Heart suit, cover. Otherwise duck.

As I recall, this rule will lead you to cover roughly 17% precent of the time.

For what its worth, this types of practices create some interesting disclosure issues. From my perspective, player who use these types of mixed strategies can be required to describe the existence of a mixed strategy to the opponents and register their "keys" with the Tournament Directors. However, their partner can't have any information regarding the particular key that is being used on any given day.

[han]

No problem at all Frances, you explained things much better than I did. (and I was way to lazy to compute the 91.5%)

Matt, it would be nice if you could explain your calculations without using sophisticated notation. I think that this is something that many bridge players are able to understand, but not if they are not familiar with the notation.

[cherdano]

FrancesHinden, on Apr 25 2006, 06:34 PM, said:

But if I'm playing a match against a team I expect to beat, I want to minimise random swings and rely on my skill. So I don't want to lose a game swing because I happen to have K10 doubleton and oppo get that right for certain.

I'm not going to use game theory terminology, because I'll probably get it wrong (I was a fluid dynamicist), but I think the optimal strategy against a worse team should satisfy two criteria:

i) You cannot improve your trick expectation by changing it (this I think is the 'nash equilibrium' point)
ii) You minimise the difference in trick expectation between your alternative plays

So in fact, I think playing the Q or the J from QJ with 50% probability _is_ the optimal strategy after all.

Unless I'm playing against a much better team than my team. Then I want to make things swingier. So I should tell them that I always play the Q from QJ doubleton (and pray for singleton queens), that I never cover from Kx (and pray for no K10 doubletons) and so forth.

I may have drivelled on for too long here, but am I making any sense at all?

It does make sense to me. However, I would formulate it more simply: you want to maximize the winning probability. A game swing in against a weak team may increase your winning chances by 5% and your chances to lose by 10%, so you can just use these numbers instead of expected IMPs.

Then you can again define an optimal strategy as a Nash equilibrium by saying a strategy from your side consists of both your plan as defender and your team mates' plan as declarer, and similarly for the opposing side.
(For those who don't know the term: Nash equilibrium is a situation where either side would lose if it changed its strategy.)

For the QJ example, I would expect that there is still a range of probabilities that are all optimal strategy, just that it is getting smaller.

This sounds like enough fun to analyze completely that I might do so on a rainy afternoon...

Arend

[Cascade]

hrothgar, on Apr 26 2006, 05:03 AM, said:

For what its worth, this types of practices create some interesting disclosure issues. From my perspective, player who use these types of mixed strategies can be required to describe the existence of a mixed strategy to the opponents and register their "keys" with the Tournament Directors. However, their partner can't have any information regarding the particular key that is being used on any given day.

Why?

If I use a mixed strategy and sometimes I do why does the director need to know. Not only that I am free to deviate from my own mixed strategy on any particular hand.

I don't see any advantage in telling the director my "key" and it might be a disadvantage if on another day that director becomes my opponent.

[hrothgar]

Cascade, on Apr 25 2006, 11:06 PM, said:

hrothgar, on Apr 26 2006, 05:03 AM, said:

For what its worth, this types of practices create some interesting disclosure issues.  From my perspective, player who use these types of mixed strategies can be required to describe the existence of a mixed strategy to the opponents and register their "keys" with the Tournament Directors.  However, their partner can't have any information regarding the particular key that is being used on any given day.

Why?

If I use a mixed strategy and sometimes I do why does the director need to know. Not only that I am free to deviate from my own mixed strategy on any particular hand.

I don't see any advantage in telling the director my "key" and it might be a disadvantage if on another day that director becomes my opponent.

The primary problem with mixed strategies is one of disclosure: How can players accurately describe bidding agreements that are inherently random? In an ideal world, players would have enough experience with their opponent's methods that they could make accurate judgements regarding the frequencies that they randomize. In practice, its necessary to trust the opponents when they describe their methods.

Unfortunately, this creates some very clear incentive problems. Why should I as a player accurately describe the Probability Density Function that I use when I randomize? Equally significant, assume that I did provide an inaccurate PDF... How would anyone ever know? (Please note: I am not claiming that any one pair would necessarily cheat. Rather, that the rules set should not be designed in such a way that people have easy ways to cheat). You may WANT to be able to based your randomization on inherently subjective considerations. However, I think that its best for everyone if you aren't allowed to.

The most logical way to eliminate this possibility is to force players to tie their hands. Players have the right to randomize/adopt mixed strategies. However, if they do so the TDs need some mechanism to determine whether they are accurately disclosing their methods. In turn, this requires that players describe the "keys" when they randomize their methods.

Please note: There is no reason that keys need to remain static between sessions. Indeed, I'd argue that players should probably be required to vary their keys for fear that their partner's might "key in" on whats going on... In all honesty, its not particularly difficult to devise a set of different keys that could be used for any given mixed strategy.

(As usual, this posting is only half serious. I severely doubt that anyone would ever bother going to this much trouble. Equally significant, I doubt that the existing Zonal authorities would ever want to deal with all the issues associated with opening this can of worms. With this said and done, if your sole concern is coming up with a fair way to allow so-called "random" bids I suspect that you'll need to implement something similar to this.

For whgat its worth, I storngly beleive that anyone who claims that they use 1S as a "random" overcall of a strong club opening should be forced to adopt these types of restrictions. If you can't accurately describe your methods you don't get to use your methods.

[Echognome]

Hannie, on Apr 25 2006, 05:22 PM, said:

No problem at all Frances, you explained things much better than I did. (and I was way to lazy to compute the 91.5%)

Matt, it would be nice if you could explain your calculations without using sophisticated notation. I think that this is something that many bridge players are able to understand, but not if they are not familiar with the notation.

A fair point Han. I am also certain that many people will NOT be interested in it. So for those that are not, just ignore the rest of what I'm about to say. For those that are, read on.

The beginning of my thread was discussing simply the line that declarer would take. In effect, declarer has the option of playing for the drop or playing a two-way finesse. However, in playing for the finesse one way, he always succeeds, whereas playing for it the other way, he has a guess. In essence, if he guesses correctly the first time, he doesn't need to guess the second time.

Now, let us take the example given by Gerben and assume that declarer has:

AT98

K5432

If declarer guesses to play small to the A, he wins all the tricks. It doesn't matter what card East plays.

If declarer guesses to play that 8 (or 9 or 10) from dummy and sees the J while winning the K, he must guess whether to finesse West for the missing Q or to play for the drop. He leads another small card from dummy and sees the last small card from West. He now has to make the decision. So there is only one card outstanding and he is guessing whether the suit is laid out as:

Q76 opposite J or
76 opposite QJ

Now we get into the discussion of the often cited "restricted choice". The idea is simple enough in saying that IF East has J singleton, then East will have to play the J on the first round. If East has QJ doubleton, then East can play the J or the Q on the first round. Now before we get into East making a choice (and thus altering the odds), let us consider the odds ex ante. That is to say, what are the odds of the various holdings of the EW cards before we observe E and W's plays to the first couple tricks. Here I used suitplay to speed up the calculations. If you are interested in those combinatorics, that can be discussed separately.

West-----East----Prob
Void-----QJxx----4.78%
x---------QJx-----12.43%
xx--------QJ------6.78%
J----------Qxx----6.22%
Jx--------Qx------13.57%
Jxx-------Q-------6.22%
Q---------Jxx-----6.22%
Qx-------Jx-------13.57%
Qxx------J--------6.22%
QJ-------xx-------6.78%
QJx------x--------12.43%
QJxx----Void-----4.78%

So these are the odds of the various holdings of East and West. However, we can rule out several of the possibilities given that we have seen three cards. Namely, we now know that East holds the J and West holds the two x's. So we know that it is one of the following two possibilities:

West-----East----Prob
xx--------QJ-----6.78%
Qxx-------J------6.22%

Note that it is more likely ex ante that East holds QJ doubleton than J singleton. On that premise it may seem better to play for the drop than the finesse. And actually that premise would be true IF East always played the J from QJ doubleton. As an exercise, we can consider the ex ante conditional probabilities that East holds J or QJ. That is to say, what if we asked the following question: "Now that we know East holds either QJ or J, what are the odds he holds QJ compared to J?" We "condition" on the argument that it must be one of QJ or J and then look at the relative probabilities on these. Now we note that East holds QJ or J some 13% of the time. So the "conditional" probability that East holds QJ doubleton is 6.78/13 = 52.2% and the "conditional" probability that East holds J singleton is 6.22/13 = 47.8%. So if this were the only consideration, then we would be better off playing for the drop rather than the finesse.

However, we now add the premise that when East holds QJ doubleton, East might sometimes play the J and sometimes play the Q. So we now need to define a variable that I called p to be the probability (or if you like proportion of time) that East plays the J when he holds QJ doubleton. Another way of saying this is that when East holds QJ doubleton, he is only playing the J a certain percentage of the time and we are calling that percentage p. If you want, think of p as the percentage of time you "true card" and 1 - p as the percentage of time you "false card". However, when East holds J singleton, East has no choice but to play the Jack. Thus when East holds J singleton, the J is always a true card. So we can adjust our table to now think of the ex ante probabilities (given that East falsecards with percentage p) as:

West------East-----Prob
xx---------QJ-------6.78p%
Qxx--------J--------6.22%

We now make the exact same calculations as before, with the added wrinkle that the total percentage is no longer 13% but is 6.78p% + 6.22%.

So the odds that East holds QJ doubleton are now:

Conditional Probability of QJ = 6.78p/(6.78p+6.22)
Conditional Probability of J = 6.22/(6.78p + 6.22)

Which means that whether we play for the finesse or the drop depends entirely on the value of p! Let us consider the two extreme cases.

Suppose that p = 1. This means that East always plays the J from a holding of QJ doubleton. In this case, we can use the odds calculated above and play for the drop.

Now suppose that p = 0. This means that East never plays the J from a holding of QJ doubleton. The consequence of this is remarkable. We can now RULE OUT QJ doubleton as a possibility and play for the finesse.

My main point here is that p matters. If East takes a strategy of ALWAYS playing the J or ALWAYS playing the Q, then it should affect our decision and we can use that knowledge to our advantage. But if East sometimes plays the J and sometimes the Q, we can think about how often East plays one card or another in making our decision.

The last step is to associate the probabilities on the holdings above with the relevant line. When East holds QJ doubleton we want to play for the drop. When East holds J singleton, we want to play for the finesse. So we compare them via an inequality. Playing for the finesse must be better than playing for the drop when:

Conditional Probability of J > Conditional Probability of QJ

or

6.22/(6.78p + 6.22) > 6.78p/(6.78p+6.22)

Since the denominators are the same, we can rewrite this as:

6.22 > 6.78p or

p < 6.22/6.78 or

p < 0.915 = 91.5%

And here is where we get our rule. When East is looking at QJ doubleton, he wants declarer to finesse. Thus he should play the J from QJ doubleton less than 91.5% of the time.

Finally, we can go through the entire exercise again when East instead plays the Q on the first round rather than the J. Luckily, our new problem is completely equivalent to our old one and we find that East should play the Q from QJ less than 91.5% of the time. Now we have our solution for East. When you hold QJ doubleton, it is best to randomize between playing the Q and the J. However, your randomization should not be too extreme (in this case).

[Echognome]

By the way, is it the case that anywhere up to 91%, but not 92% will find my prior post uninteresting?

[pclayton]

Why do I have visuals of Richard and Matt standing on the 2nd floor of a dormitory in New. Engalnd in January scribbling these formulas on a window pane in crayon and talking to imaginary people?

[han]

Quote

Why do I have visuals of Richard and Matt standing on the 2nd floor of a dormitory in New. Engalnd in January scribbling these formulas on a window pane in crayon and talking to imaginary people? tongue.gif tongue.gif

Hard to say without knowing you better Phil. Perhaps it is because you don't understand them and prefer to think of people you don't understand as being lunatics who have no idea what life is about? Perhaps because in most of the Western world intellectuals are often seen this way? I'm just guessing here, as I said, I can't read your mind from such a distance.

[hrothgar]

I suspect that you've been watching too many Russell Crowe movies...
(BTW, New Jersey is a mid Atlantic state)

[han]

I have seen the movie and enjoyed it. I understood the joke (and even thought it was funny, but don't tell that to Phil), but I made a serious reply.

[adhoc3]

Game Theory Professor said: The best strategy is opponent's strategy --- but make sure you will do it better.

So for weaker team, don't try to perform such analysis because your opps are better than you, just do it randomly. For stronger team, better play the game/cards in normal way because you're the better one. -------In either case: no one need the intellegent % calculation.

May be this is useful for two teams who are the exactly same competitive...

Good lucky and have fun!

Jack

[jdeegan]

When it comes to playing the Q from Q10x, isn't the correct answer always? Otherwise declarer will always go right since the nine is by far his percentage play if you dont 'split'. In short, you convert a sure loss into a possible win. Educate me, show me where I am wrong.

jdeegan, MA, PhD, PDQ, ASAP

[han]

jdeegan, on Apr 27 2006, 12:11 PM, said:

When it comes to playing the Q from Q10x, isn't the correct answer always? Otherwise declarer will always go right since the nine is by far his percentage play if you dont 'split'. In short, you convert a sure loss into a possible win. Educate me, show me where I am wrong.

jdeegan, MA, PhD, PDQ, ASAP

You are not exactly wrong, but I think that you are missing something. If you always play high then the percentage play for declarer is to come back to hand and play low to the nine. So you also always lose with this hand.

[kfgauss]

Hannie, on Apr 27 2006, 08:32 PM, said:

jdeegan, on Apr 27 2006, 12:11 PM, said:

When it comes to playing the Q from Q10x, isn't the correct answer always?  Otherwise declarer will always go right since the nine is by far his percentage play if you dont 'split'.  In short, you convert a sure loss into a possible win.  Educate me, show me where I am wrong.

jdeegan, MA, PhD, PDQ, ASAP

You are not exactly wrong, but I think that you are missing something. If you always play high then the percentage play for declarer is to come back to hand and play low to the nine. So you also always lose with this hand.

There's something to be said for this practical approach. We presume everyone will put in the 9 if you play low (assuming they don't know something about your habits), but if you play high, there will be some who won't expect that from H10x(x..) and will play for KQx(x..).

Andy