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## Committing a match equity table to memory

The following is the most recognized backgammon match equity table, namely "Rockwell-Kazaross", generated by David Rockwell and Neil Kazaross by extensive rollouts on eXtremeGammon. It represents the probability for a player to win the match, in function of the number of points each player needs to complete the match. For instance, the entry E(-3,-5) is the probability for a player to win the match when he is three points away from the match length, and his opponent is five points away. The entries where one player is only one point away are the "pre-Crawford" probabilities, that is, given that the next game is played under the Crawford rule (this makes sense because the post-Crawford probabilities are almost never used for calculating take points).

 Me/Opp -1 -2 -3 -4 -5 -6 -7 -8 -9 -10 -11 -1 50 67.69 75.12 81.38 84.19 88.69 90.72 93.23 94.4 95.93 96.64 -2 32.31 50 59.9 66.86 74.34 79.9 84.21 87.52 90.17 92.3 93.93 -3 24.88 40.1 50 57.14 64.77 71.16 76.25 80.49 84.02 87.06 89.44 -4 18.62 33.14 42.86 50 57.74 64.31 69.97 74.62 78.83 82.41 85.4 -5 15.81 25.66 35.23 42.26 50 56.66 62.67 67.82 72.55 76.71 80.27 -6 11.31 20.1 28.84 35.69 43.34 50 56.28 61.66 66.79 71.31 75.34 -7 9.28 15.79 23.75 30.03 37.33 43.72 50 55.49 60.86 65.63 70.02 -8 6.77 12.48 19.52 25.38 32.18 38.34 44.51 50 55.44 60.37 64.99 -9 5.6 9.83 15.98 21.17 27.45 33.21 39.14 44.56 50 55.02 59.79 -10 4.07 7.7 12.94 17.59 23.29 28.69 34.37 39.63 44.98 50 54.85 -11 3.36 6.07 10.56 14.6 19.73 24.66 29.98 35.01 40.21 45.15 50

Great info, but some shortcuts are necessary if one would like to memorize such a table. Indeed, with some effort it is possible to retrieve all those values with a precision less than 0.5%. There are two separate methods, a very simple one for the Crawford equities, and a more difficult one for all the other equities in the table.

A trivial first observation is that only the values of the coloured squares have to be memorized. The values on the diagonal (even scores) are obviously 50%, and E(-m,-n) = 100% - E(-n,-m) (switching opponents).

### 1. Crawford equities

Here we are going to work on the trailer's equities, that is the E(-n,-1). The leader's equities will of course follow from

`E(-1,-n) = 100% - E(-n,-1)`
. The three first values (starting with E(-2,-1) since E(-1,-1)=50%) should be memorized rounded to the nearest integer, that is :
```• E(-2,-1) = 32
• E(-3,-1) = 25
• E(-4,-1) = 19```
Those are in fact only two values to memorize if you notice that the fact that E(-3,-1) is almost exactly 25% follows from the fact that two consecutive wins are needed to win from -3,-1 Crawford down, except in the very rare case of a backgammon win. Remain only 32 and 19 to memorize until now.

How about the higher values of n ? They follow from the odd rule and the even rule :

```• Odd rule : E(-n,-1) = 1/2 * E(-(n-3),-1), if n is odd ("three points less mean twice less chances").
• Even rule : E(-n,-1) = 60% * E(-(n-2),-1), if n is even ("two points less mean 60% of the chances").```

The method is now as follows for a number over 4: start with E(-4,-1)=18, apply the even rule as many times needed, and apply the odd rule at the end if n is odd, rounding to the nearest half integer at each step. This leads to a correct approximation ! Here is how it would work for the last entry, E(-11,-1) :

```• Start with E(-4,-1) = 19
• We must go up to E(-8,-1) with the even rule before applying the odd rule ; the first step is E(-6,-1) = 60% * 19 = 11.5% (rounded).
• E(-8,-1) = 60% * 10.5 = 7% (rounded)
• E(-11,-1) = 1/2 * 7% = 3.5%
```
The actual value was not 3.5%, but 3.36%. Not too bad a mistake. It would have been even better if we hadn't rounded the values : we would have ended with 3.42%.

### 2. Non-Crawford equities

How were people retrieving match equities before Neil Kazaross invented the "Neil's numbers" method ? I guess they had better be good at remembering tons of numbers. Fortunately for us, Neil's way is much simpler though a bit less precise, and I will propose a way of making it more precise with a minimum of memory effort. But let's proceed in order; first of all, what are Neil's numbers ? They are given by the following table :

 Trailer's score (in n-away) 3 4 5 6 7 8 9 10 11 Leader's "point value" 10 9 8 7 6 5

The leader's point value is the extra equity that one extra point gives to the leader, given the trailer's score. Cool thing, this value is not far from constant for a fixed trailer's score, which means that any entry in the match equity table can be calculated as 50% plus the score difference times the appropriate Neil's number. In formula :

`E(-m,-n) ≃ 50% + (m-n) * Neil(n), for 2 <= m <= n`
Note that this time we are computing the leader's equities.

Oh, but aren't there some numbers missing in Neil's table ? Those are meant to be interpolated, in other words Neil(7) = 6.5, Neil(9) = 5 2/3, Neil(10) = 5 1/3, and the table can even be continued, inserting one more gap between each integer value (so e.g. Neil(15) = 4, with intermediate values at 3 3/4, 3 1/2 and 3 1/4). Actually this makes Neil's numbers pretty easy to remember : start with Neil(n) = 13-n for n <= 6, and then insert one gap, two gaps, three gaps, etc.

All known stuff until there, but as it is also known, not all values thus obtained are correct, especially when the leader is two points away. I suggest to learn the necessary corrections in a visual way with the help of the following table :

 Me/Opp -1 -2 -3 -4 -5 -6 -7 -8 -9 -10 -11 -1 50 67.69 75.12 81.38 84.19 88.69 90.72 93.23 94.4 95.93 96.64 -2 32.31 50 59.9 66.86 74.34 79.9 84.21 87.52 90.17 92.3 93.93 -3 24.88 40.1 50 57.14 64.77 71.16 76.25 80.49 84.02 87.06 89.44 -4 18.62 33.14 42.86 50 57.74 64.31 69.97 74.62 78.83 82.41 85.4 -5 15.81 25.66 35.23 42.26 50 56.66 62.67 67.82 72.55 76.71 80.27 -6 11.31 20.1 28.84 35.69 43.34 50 56.28 61.66 66.79 71.31 75.34 -7 9.28 15.79 23.75 30.03 37.33 43.72 50 55.49 60.86 65.63 70.02 -8 6.77 12.48 19.52 25.38 32.18 38.34 44.51 50 55.44 60.37 64.99 -9 5.6 9.83 15.98 21.17 27.45 33.21 39.14 44.56 50 55.02 59.79 -10 4.07 7.7 12.94 17.59 23.29 28.69 34.37 39.63 44.98 50 54.85 -11 3.36 6.07 10.56 14.6 19.73 24.66 29.98 35.01 40.21 45.15 50

The values in this table are the same than in the previous one, but it is the colours which are interesting. Every coloured cell means that there is a correction to make after calculating the equity with Neil's numbers. Here is the colour code :