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# Card Values for Three Notrump

In other articles in this series, we are basing our search of evaluators on efforts to determine very generally how many tricks we expect to take. But what happens if we instead want to zero in on a particular contract and level?

We represent our side's combined honor holdings as a vector `(A,K,Q,J,T)`, where `0<=A<=4`, `0<=K<=4`, `0<=Q<=4`, `0<=J<=4`, `0<=T<=4.` We let `P(A,K,Q,J,T)` be the probability we can make nine or more tricks in notrump if we hold `A` aces, `K` kings, `Q` queens, `J` jacks, and `T` tens.

Here are some rules that we know "intuitively:"

```(1) P(A+1,K,Q,J,T)>P(A,K,Q,J,T)   (A<4)
(2) P(A,K+1,Q,J,T)>P(A,K,Q,J,T)   (K<4)
(3) P(A,K,Q+1,J,T)>P(A,K,Q,J,T)   (Q<4)
(4) P(A,K,Q,J+1,T)>P(A,K,Q,J,T)   (J<4)
(5) P(A,K,Q,J,T+1)>P(A,K,Q,J,T)   (T<4)

(6) P(A+1,K,Q,J,T)>P(A,K+1,Q,J,T) (A,K<4)
(7) P(A,K+1,Q,J,T)>P(A,K,Q+1,J,T) (K,Q<4)
(8) P(A,K,Q+1,J,T)>P(A,K,Q,J+1,T) (Q,J<4)
(9) P(A,K,Q,J+1,T)>P(A,K,Q,J,T+1) (J,T<4)
```

(1)-(5) simply state that additional honors make it more likely we can make 3NT. (6)-(9) says that swapping a card with a card of higher rank makes it more likely we can make 3NT.

For the mathematicians in the house, these relations create a partial order on the vectors (A,K,Q,J,T).

We can come up with estimates for `P(A,K,Q,J,T)` with double-dummy simulations. Restricting to a limited range, we get a table which looks like:

```AKQJT  % freq     P
====================
...
13431   0.052  17.0%
13432   0.056  29.0%
13433   0.025  37.5%
13434   0.004  40.0%
13440   0.003  32.0%
13441   0.009  47.5%
13442   0.008  51.5%
13443   0.003  51.5%
13444   0.000  59.5%
...
```

The Work Point Count assigns the values 4, 3, 2, and 1 to the ace, king, queen, and jack, and traditionally, we say we have enough for game if we have 25 points. But is this an ideal set of values?

Let's say our goal is to bid all games which are better than 50% likely. Is it possible to come up with values for the honors card, and some limit, to get a "perfect" evaluator - one which seperates the rows into those with >50% and <50%?

The answer to that is "no." Look at these rows:

```(a) 24041  0.0136  60.0%
(b) 24121  0.3911  31.8%
```
If we have a perfect evaluator for honor cards, then we'd know from these two lines that the value of a queen is less than twice the value of a jack, because (b) adds a queen and subtracts two jacks from (a), and ends up not being a good game.

On the other hand:

```(c) 31332  0.5163  37.6%
(d) 31412  0.1744  50.6%
```
In this case, adding a queen and subtracting two jacks improves the odds.

So, we can only get close with this simple approach. So, how do we determine how "close" an evaluator is to ideal? By measuring the error, which we can do row by row and sum up.

If the evaluator says it is right to bid game, but the table says it is wrong, or visa versa, we define the error contributed by the row as the absolute value of the difference between the game percentage and 50%, multiplied by the frequency of the row.

For example, if we are using the Work Point Count, and setting a threshhold of 25 for game, then row (d) above would contribute an error of 0.6%*0.001744. This means that rows which are rare, and rows which are close to 50% games, are not weighted nearly as high.

Obviously, we can apply the same technique for any percentage - if we want to bid 40% games, for example, we'd adjust the above error estimate.

## The Sample

I've generated a considerable sample on the following conditions:
1. South is balanced.
2. There is no 8-card major fit
3. North is not "freakish" shape
4. Neither hand has a doubleton queen or jack without a higher honor (no Qx, Jx), and north doesn't have any singleton kings.

The last condition is slightly controversial, but I wanted to test for the "essential" values of honors. I will, at some later date, investigate the effect of less ideal conditions.

## The Results

So, without further ado, here are some data points for 50% 3NT contracts:
```        Card Values         Game    100*
A    K    Q    J    T   Goal    Error
4.00 3.00 2.00 1.00 0.00  25.00    38.99     Work point count
4.50 3.00 1.50 0.75 0.25  24.75    59.85     BUM-RAP
4.00 3.00 2.00 1.00 0.50  25.50    16.12     Work point plus 1/2 per T
4.25 3.00 2.00 1.00 0.50  26.25     9.70     " + 1/4 per ace
4.00 2.80 1.80 1.00 0.40  24.20     6.31     WPC- 1/5 per K&Q + 2/5 per T
```

The goals for each row are chosen to minimize the error. So if we are using the Work Point Count, we want to have 25 points or more. If we add 1/2 for a ten, we want 25.5 to have a 50% chance for game, and we get a significantly smaller error. We get a further gain by adding 1/4 point more for each ace.

If we are aiming for 40% games - say, at IMPs rather than MPs - we get different errors and goals:

```        Card Values         Game    100*
A    K    Q    J    T   Goal    Error
4.00 3.00 2.00 1.00 0.00  24.00    65.12     Work point count
4.50 3.00 1.50 0.75 0.25  24.50    67.06     BUM-RAP
4.00 3.00 2.00 1.00 0.50  25.00    12.15     Work point plus 1/2 per T
4.25 3.00 2.00 1.00 0.50  25.50     9.87     " + 1/4 per ace
4.00 2.80 1.80 1.00 0.40  23.80     9.18     WPC- 1/5 per K&Q + 2/5 per T
```
Note that the errors, when we are seeking 40% games, are different from the 50% games. In particular, all but one of the errors are increased, and BUM-RAP now does almost as well as Work Point Count.

As before, the fifth row (A=4, K=2.8, Q=1.8, J=1, T=0.4) is the best, but this time, only slightly better than the third. Also, in this case, WPC beats BUM-RAP by just a slight bit compared to the considerable difference in the 50% case.

So far, I've only tested card values by trial and error. But the more "fractional" the evaluator, the harder it is to use in bidding.

## Less Ideal Situations

If we remove the last condition on the above sample we get:

### Seeking 50% Games

```        Card Values         Game    100*
A    K    Q    J    T   Goal    Error
4.00 3.00 2.00 1.00 0.00  25.00    32.13     Work point count
4.50 3.00 1.50 0.75 0.25  25.00    61.76     BUM-RAP
4.00 3.00 2.00 1.00 0.50  25.50    16.86     Work point plus 1/2 per T
4.25 3.00 2.00 1.00 0.50  26.25     7.50     " + 1/4 per ace
4.00 2.80 1.80 1.00 0.40  24.40     6.71     WPC- 1/5 per K&Q + 2/5 per T
```

### Seeking 40% Games

```        Card Values         Game    100*
A    K    Q    J    T   Goal    Error
4.00 3.00 2.00 1.00 0.00  24.00    71.95     Work point count
4.50 3.00 1.50 0.75 0.25  24.25    61.21     BUM-RAP
4.00 3.00 2.00 1.00 0.50  25.00    20.40     Work point plus 1/2 per T
4.25 3.00 2.00 1.00 0.50  25.75    10.90     " + 1/4 per ace
4.00 2.80 1.80 1.00 0.40  23.80    13.17     WPC- 1/5 per K&Q + 2/5 per T
```
Note that BUM-RAP does slightly better than HCP for the 40% games, but the other three still significantly beat BUM-RAP.

## Conclusions

BUM-RAP is designed better for suit contracts than notrump, but another reason BUM-RAP might fail is that it is designed to work across all levels, and not geared towards the 3NT contract, specifically.

When we target the specific notrump contract level, we find that Work Point Count is as at least as good an evaluator as BUM-RAP, and that we can get a very good evaluator by tweaking the Work Point Count slightly.

The nice thing about the last evaluator, (A=4,K=2.8,Q=1.8,J=1,T=0.4) is that the total count in the deck is still 40, as with WPC and BUM-RAP. I call this evaluator the "Fifths evaluator" because you take WPC and subtract one fifth for each king and queen, and add two fifths for each ten.

## Caveats

Do not necessarily take these results to mean that you should use these new evaluators.

For example, if your partner opens 1 NT with a hand worth 14.6 to 17.4 in the Fifths valuator, then when should you bid 3 NT? Obviously, if you use the above rules, you would always bid 3 NT with a value of 9.8 in the same evaluator, because your side is always worth at least 24.4, which is our cut-off point for 50% games. You also won't want to bid 3NT with less than 7.0 points, because there is no chance you will have as many as 24.2 points. But do you want to bid with 7.2 points? 7.4? At what point do you invite? At what strength does partner accept?

The problem with the fractional values is that the borders become fuzzy. The actual "error" when you bid to 3NT is some combination of the errors for various ranges. For example, if you decide to always bid game with 9.0 points or more, and never for less, you get two sets of errors, one where this "underestimates" the total strength, and another where it over-estimates. Again, we'd need to come up with some definition of "error," here.