Aces and Spaces: Checking Binky Points

Binky points treat aces as very strong cards. For example, in suit contracts, the value of A-x-x is 1.6 tricks better than x-x-x, while in notrump, the difference between these two is 2.3 tricks.

Why is that? Obviously, the ace is usually worth its one trick. Where do the other tricks come from?

Part of it is that it improves the value of partner's holding. If partner holds K-x-x in the suit, the A-x-x holding makes sure the king is worth a full trick.

Another advantage is in controlling the tempo - you have extra time to establish your tricks.

But this tempo addition becomes less useful when you know that you have no source of tricks. If you hold the hand:

♠	A432
♥	A32
♦	A32
♣	A32

you have no source of tricks, so the tempo advantage obviously gives you no advantage unless partner has a source of tricks.

This is an example of where we expect the additive nature of the Binky evaluators to over-estimate the value of the hand.

Suit Contracts

I dealt out 5000 hands for each balanced pattern (4333, 4432, and 5332,) holding all four aces, and no spots higher than an eight. I then used GIB to determine the number of tricks this hand can take in a suit contract. The results were:

                Hand          Tricks  Tricks-binky(*)    Binky
    Axxxx Axx Axx Ax   9.91 +/- 1.62    6.62 +/- 0.90     6.68
    Axxx Axxx Axx Ax   9.73 +/- 1.62    6.48 +/- 0.87     6.59
    Axxx Axx Axx Axx   9.36 +/- 1.70    6.15 +/- 0.90     6.33
         Average(**)   9.56             6.33

	     (*) This is the difference between the number of tricks
	     and the Binky Points evaluation of partner's hand. If
	     Binky Points work, then this difference should be
	     close to the value of the last column.
                 
             (**) The average is a weighted average because each of
             these hand types are not equally likely.

So, in each row, the Binky evaluator over-estimates its strength slightly, with the 5332 pattern off by only 0.06, the 4432 off by 0.11, and the 4333 off by 0.18. So Binky Points do overestimate aces-and-spaces hands slightly, with the flattest hands off by the most. 0.18 Binky points is about the difference between K-J-x and K-J-T.

Notrump Contracts

One assumption about Binky Points is that they estimated the number of tricks available when the contract is played from the best side. In reality, of course, we don't always play contracts from the right side. For instance, in the "aces and spaces" hands, we are often opening or rebidding notrump, even though it is much more likely that double-dummy, the partner is the better hand to play.

                Hand          Tricks     Tricks-binky    Binky
    Axxxx Axx Axx Ax   8.42 +/- 1.97    6.67 +/- 1.20     6.85
    Axxx Axx Axx Axx   8.33 +/- 1.91    6.55 +/- 1.20     6.83
    Axxx Axxx Axx Ax   8.23 +/- 1.93    6.46 +/- 1.18     6.78
             Average   8.32             6.55

So in notrump, assuming the aces-and-spaces hand is declarer, Binky Points over-estimate the value of these hands by 0.18 tricks for 5332s, 0.28 for 4333s, and 0.32 for 4432s. In notrump, 0.32 Binky points is the difference between KT9 and KJT.

Note the more extreme standard deviations than in suit contracts. Even with all suits stopped, our notrump deviations are high.

Compare the above table to this table:

                Hand          Tricks     Tricks-binky    Binky
    KJxx KJx KJx KJx   8.47 +/- 2.17    6.18 +/- 1.02     5.99
    KJxx KJxx KJx KJ   8.04 +/- 2.32    5.77 +/- 1.14     5.75
    KJxxx KJx KJx KJ   8.02 +/- 2.37    5.76 +/- 1.14     5.72
             Average   8.30             6.01

I'm not sure what conclusion to draw from this table. In these hands, Binky Points under-value the hand, even when we force the hand as declarer. But being declarer is likely preferred in this case - you have tenaces in every suit. It's interesting that the deviation for "Tricks" is higher than aces-and-spaces, but when we take partner's strength into account, the deviation is less than the aces-and-spaces deviation. That means that these hands are riskier, but that partner is better able to gauge our trick-taking ability based in her hand.

Just for fun, I thought I'd check king-queen-and-spaces hands:

                Hand          Tricks     Tricks-binky    Binky
    KQxx KQx KQx KQx   9.14 +/- 1.93    7.42 +/- 0.92     7.31
    KQxxx KQx KQx KQ   8.65 +/- 2.18    6.93 +/- 1.00     7.02
    KQxx KQxx KQx KQ   8.60 +/- 2.15    6.92 +/- 1.02     6.97
             Average   8.93             7.22

These hands are significantly stronger than the aces-and-spaces hands, despite being the same strengths in BUM-RAP and other modern evaluators.