The Stiff Queen, and Other Binky Points Oddities
If you check out the table for singleton values in the Single Hand Evaluator, the difference between a small stiff and a stiff queen in suit contracts is a mere 0.08 tricks. This means that, when considering just your own hand, the expected number of tricks that you will make in your best suit contract when holding a stiff queen is about 0.08 tricks higher than you'd expect holding a small singleton.
| Holding | Expected Tricks | Binky Points |
|---|---|---|
| Q | 8.88 | 0.08 |
| X | 8.80 | -0.17 |
| Difference | 0.08 | 0.25 |
On the other hand, the difference between a small singleton and singleton queen is 0.25 tricks in Binky Points.
Why the big difference?
The thing that causes most of the paradoxes with Binky Points is its additive nature - the aim of Binky Points is that the sum of your partner's hand and your hand should approximate the number of tricks you have.
Now, when you have a small singleton, partner, when evaluating her hand, knowing nothing about your singleton, will over-value high cards in that suit. So your Binky Points will need to adjust for this over-valuation by partner.
On the other hand, when you hold the stiff queen, partner's holding in the suit will have a lower chance of wastage - it is less likely that partner will be over-valuing cards in that suit. This accounts for the difference.
This actually might alter the way you bid. If partner knows you have a singleton in a suit, then the queen is worth about 1/4 a point. If partner is unaware of the singleton, then the queen is worth about 3/4 a point.
A similar effect occurs when you hold a very long suit. If you hold an 8-3-1-1 hand the expected tricks is 9.82 tricks, or 2.06 tricks more than expected for a 4-3-3-3 hand. In Binky Points, the difference is 1.8 tricks, and the reason is that partner is likely to over-evaluate the playing strength of shortness in your eight card suit. The difference is 3/4 a point. However, if you show your very long suit, partner won't over-value the stiff, so you gain your 3/4 point back.
There are, in fact, lots of cases like this.
| Holding | Expected Tricks | Binky Points |
|---|---|---|
| AXX | 8.69 | 0.77 |
| XXX | 7.62 | -0.84 |
| Difference | 1.07 | 1.61 |
When holding A-X-X, compared to X-X-X, Binky Points evaluates the ace as worth 1.61 tricks, while the single hand evaluator has the ace as worth a mere 1.07 tricks. Partner's valuation of her holding in the suit will be, on average, an under-estimation opposite A-X-X, but an over-estimation when you hold X-X-X. Binky Points corrects your holding's valuation to account for this. It turns out, that these two are about equal in this instance - partner will over-value her hand by 0.27 tricks, on average, when you hold X-X-X, and under-value her hand by 0.27 tricks when you have A-X-X.
Giving Partner the Right Information
One of the surprises in Binky Points is that it does not value honors in long suits as more valuable than honors in shorter suits. The hands ♠ A-K-8-6-2 ♥ K-J-5-4 ♦ 7-5-3 ♣6 and ♠ 8-7-6-5-2 ♥ A-K-5-4 ♦ K-J-5 ♣ 6 are, according to Binky Points, exactly the same value in suit contracts.
Yet expert practice is to think the first hand is stronger.
This is likely due to a variety of factors, but one is the "real world" consideration that, if you open one spade on the second hand, partner is likely to completely mis-evaluate her hand. With a singleton spade and a heart fit, she will devalue the stiff, and with Q-9-3 , she will grossly over-value her spade cards. In other words, opening marginal hands with bad suits is one of the best ways to end up too high or too low.
The first hand may not be "stronger" in the sense of expected tricks, but there is a stronger tactical value to opening one spade with that hand, because partner will re-evaluate her hand in the right direction more often.
Math Behind This
You might wonder how Binky can take into account partner's average hand while partner's usage of Binky takes into account your average hand.
If you've read the original Binky Points article, you might recall that we arrived at Binky Points by inverting two large matrices, both of the form I+A, where A was a probability matrix.
Much as you can write 1/(1+z) = 1-z+z^2-z^3... for suitable numbers z, so you can also write the inverse of the matrix I+A as:
(*) I - A + A^2 - A^3+...
This is multipled by the single hand evaluator to get our final table for Binky Points.
Interpret the "I" to mean that we take into account our own hand's expected number of tricks. The -A means "then subtract the average value of partner's expected number of tricks." The +A^2 then means, "but add back in what partner will, on average, think we think our expected number of tricks will be." The matrix inversion can be then seen as a kind of infinite recursion of "she expects that I expect that she expects that I expect..."
Math Note
The series (*) above does not converge in a standard meaning of convergence. That's not strictly relevant, because, as Tom Lehrer says, "The idea is the important thing." There are mathematical shenanigans to get around this, but what I really want to show is how the inversion of the matrix can be thought of as an infinite "she expects that I expect that she expects tha I expect..."