Fuzzy Squeeze Theory
The Value of Worthless Cards
Suppose you have 432 of spades. In any traditional squeeze analysis, you can pitch these cards. They cannot win a trick. But suppose the position is this:
We can see that declarer always has three tricks by taking the spade finesse. But what if declarer can make 3NT without taking the spade finesse, and if declarer takes the spade finesse and it loses, declarer goes down? Then declarer will not take the spade finesse.
But if declarer plays the K of spades and you show out, then declarer can safely take the spade finesse.
So you needed at least one spade. Pitching spades also makes it look like you do not have the queen. If the situation is this
Then pitching two spades probably solves declarer's problem of which way to finesse spades, if declare needs to finesse for the ninth trick.
And here is another deadly one
If declarer needs another trick, declarer will probably take the spade finesse -- unless you first pitch 4 spades.
In a trump contract, declarer typically retains trumps in hand so that he can safely take the spade finesse. This means there is less pressure on you to hold small cards in the suit to disguise the distribution. There is also less pressure on you anyway, because declarer is not playing out all of this winners.
So the problem here usually arises in the no trump contract, which is the natural home of the fuzzy squeeze.
Squeezing out Winners
Of course, if you have the ace and declarer has the king, you have to save your ace so that declarer's king isn't good. But what I am talking about are pure winners. For example, you have the ace and declarer does not have any cards in the suit.
If declarer's loser count is one, then you are trying to win a trick. Winners are irrelevant and they can be pitched.
If it is a trump contract and declarer has retained trumps, then your winners usually are not relevant and can be pitched.
The remainder of the time, winners are always useful. First, and most obviously, if and when you get in, you can cash them.
But secondly, they are a threat. Suppose you have enough trump in your hand to set the contract. Then, if declarer has enough tricks for contract, declarer cannot take any risks for an overtrick.
Note that for this purpose, what counts is appearance. Suppose it will take 3 tricks to set the contract (not counting the one you win before cashing your winners). If you have two and your partner has one, but declarer does not know this, declarer cannot afford to let you win a trick. However, declarer can finesse into your partner (assuming declarer knows you have some winners), because if partner has an entry to your winners, then you have only two. But if you and your partner have a total of 4 cards in the winner suit, then it is not safe for declarer to lose a trick to either of you.
So everything is an asset. Your partner has set up a suit, you might think that you need to save only one entry to partner's suit. But pitching one card from that suit might make it safe for declarer to make an attempt for the overtrick, such as taking a finesse that might lose.
Note that if you want declarer to finesse, you might try to throw winners to make it safe for him to do so.
Squeezing out Low Cards
Suppose your partner is guarding hearts and you have to pitch a heart from Qx. You have to pitch what might be a useful high card, so you will probably throw your small hearts. But when you do that, it is easier for declarer to throw you in in hearts. Or you will be thrown-in in another suit, but the stiff Q will not be a good exit card.
Squeezing out irrelevant cards
Suppose dummy has a four card spade and you have 5 spades. You can guess that your fifth spade is never going to be a winner. Therefore, it is perfectly safe to pitch a spade. You will never need it.
However, when you pitch a spade, looking at 4 in the dummy, that suggests you have 5 spades. This could be valuable information to declarer.
The point is that even the most innocuous of pitches can hurt you. Again, in a trump contract, you can have a long suit that will always be worthless because declarer has trumps. So pitches from this suit are safe. So once again, the fuzzy squeeze tends to occur at no trump.
Home of the Fuzzy Squeeze
To review, declarer usually does not cash all of his trumps in a trump contract. So you tend not to be squeezed. Your winners are much less value, and given declarer a picture of the distribution is less of a problem. So you have more free cards to pitch when declarer is retaining cards.
For all of these reasons, the natural home of the fuzzy squeeze is a no trump contract.
In 6 no trump, the issue is usually a traditional squeeze or endplay. So the fuzzy squeeze usually isn't found here.
The 3 no trump contract is very conducive to the fuzzy squeeze. The squeeze then occurs when declarer runs a long suit. Technically, just having to pitch one card can be a problem. But it is really a problem when you feel uncomfortable pitching a card and you still have 3 more pitches to find. Find a 3 no trump contract where declarer is running a 6-card suit and you will probably find an interesting fuzzy squeeze.
1 no trump is probably less so the home of a fuzzy squeeze. In this contract, there is usually room to pitch cards. But this is the second most likely place to find it. And I really don't know, maybe it is just as common here.
Hand 3 presents a situation where, if declarer plays double-dummy, he already has two of the remaining tricks, and if the defender discards a heart, declarer can take all three of the remaining tricks. So the defend shed a spade.
The problem was, declarer was unlikely to play the hand double-dummy. Declarer was probably, though not necessarily, going to just cash out winners. The discard of a spade then gave away a trick, while the discard of a heart would not.
So, how to put this, even though it was not a real squeeze from traditional analysis, the defender was still forced to give up an important asset. In this case, if the defender knew how the declarer was going to play the hand, he could have discarded in a way so as not to lose a trick.
But the reality is that defender's don't know how declarer will play the hand. So they have to give up asseets that might be of value for one line of play, not knowing if that line of play will be taken.
This is not ignoring the fact that declarer may choose a line of play based on what you discard. It is just acknowledging the fact that declarer might not choose the optimal double-dummy line of play and ideally you would like to retain cards that will help in one possible line of play, just in case declarer chooses that line of play.
Fuzzy Squeeze Theory
What is fuzzy squeeze theory? There is little to no concern with a double-dummy analysis if the players cannot place the cards.
There is a concern with appearances -- if it appears like you could have enough winners to set the contract, that is usually enough to deter declarer, whether or not you actually do have enough winners.
Because the analysis is not double-dummy, there is an acknowledgement of assets that merely serve to hid the distribution and high-cards from declarer. As Marvin Levine notes, the fuzzy squeeze squeezes out information.
There is an acknowledgement that in your state of ignorance, it could be that any of the cards in your hand is potentially valuable. When you feel squeezed, you probably are being fuzzy squeezed. There is less concern with pseudo-squeezes versus real squeezes.
Certainly, there is no clear structure for the squeeze as there is for the traditional squeeze analysis.
The "Winner Squeeze", described elsewhere on this website, is somewhat transitional between traditional squeeze theory and fuzzy squeeze theory. It has a formal structure, so it is analyzed with traditional squeeze analysis. But it notes the importance of winners, which is usually a key part of the fuzzy squeeze. Also, the "pseudo" winner squeeze is particularly effective -- essentially, the better choice is usually to discard winners. This blurs the distinction between real squeeze and pseudo squeeze.