## Calculating and Using Match Equities

In match play, particularly when making doubling decisions, it is useful to know one's match equity. This is the probability that one will win the match from a given score, assuming perfect play by both players.

## Kit Woolsey's match equity table

This table is reproduced with permission from Kit Woolsey's book How to Play Tournament Backgammon.
```
1   2   3   4   5   6   7   8   9  10  11  12  13  14  15

1       70  75  83  85  90  91  94  95  97  97  98  98  99  99
2   30  50  60  68  75  81  85  88  91  93  94  95  96  97  98
3   25  40  50  59  66  71  76  80  84  87  90  92  94  95  96
4   17  32  41  50  58  64  70  75  79  83  86  88  90  92  93

5   15  25  34  42  50  57  63  68  73  77  81  84  87  89  90
6   10  19  29  36  43  50  56  62  67  72  76  79  82  85  87
7    9  15  24  30  37  44  50  56  61  66  70  74  78  81  84
8    6  12  20  25  32  38  44  50  55  60  65  69  73  77  80
9    5   9  16  21  27  33  39  45  50  55  60  64  68  72  76

10    3   7  13  17  23  28  34  40  45  50  55  60  64  68  71
11    3   6  10  14  19  24  30  35  40  45  50  55  59  63  67
12    2   5   8  12  16  21  26  31  36  40  45  50  54  58  62
13    2   4   6  10  13  18  22  27  32  36  41  46  50  54  58
14    1   3   5   8  11  15  19  23  28  32  37  42  46  50  54

15    1   2   4   7  10  13  16  20  24  29  33  38  42  46  50
```

The numbers in the top row and the left-hand column represent the distance the two players are away from the winning score. The figure in the table then represents the probability that the left-hand player will win the match, in percent. This takes into account the probability of gammons occurring at various match scores. For example, in a 15-point match, if I am leading 9:7, I am 6-away (from the winning score) and my opponent is 8-away. Consulting the 6th row and 8th column of the table, I see that my winning probability is then 62%.

It should be noted that if either player is 1-away, it is taken to be the Crawford game. The trailer's equity at 1-away T-away post-Crawford can be calculated as twice the equity for the 1-away S-away Crawford, where S is the next odd number greater than T. For example, the equity at 1-away 3-away or 1-away 4-away post-Crawford is twice the equity for 1-away 5-away Crawford, that is twice 15%, or 30%. Note, however, that there are no doubling decisions to be made that involve post-Crawford equities.

## Formulae for approximating the table

Obviously, one does not want to learn the whole of the above table! Therefore some people have devised formulae which approximate the table to a high degree of accuracy. In this section I discuss the three methods I know.

The formula described in Kit Woolsey's book How to Play Tournament Backgammon is Janowski's formula, invented by Rick Janowski. Suppose the trailer is T-away, and that the difference between the two scores is D. Then the leader's equity in percent is approximately

```      (85 D)
50 + ------- .
(T + 6)

```
For example, the equity at 3-away, 8-away is
```     50 + (5 x 85) / (8 + 6)
=  50 + 425 / 14
=  80%
```
to the nearest %, which agrees with Kit's table.

The only problem with Janowski's formula is that many people find it hard to work out (for example) 425 / 14, even to the nearest 1, in their heads. Hence I invented the following formula which is much easier to calculate. The Turner formula is

```     50 + (24/T + 3) x D
```
To take the above example again, we divide 24 by T ( = 8) to get 3, add 3 to get 6, multiply by D ( = 5) to get 30 and add 50 to get 80, the same result as before.

You will notice that the above example comes out in whole numbers. This is not just because it was a well chosen example. Because 24 is such a nice number, my formula comes out in whole numbers when the trailer is 3-away, 4-away, 6-away, 8-away or 12-away. The other numbers up to 12 are not too bad either:

```24 / 5 = 4.8;
24 / 7 = nearly 3 1/2;
24 / 9 = 2 2/3;
24 /10 = 2.4; and
24 /11 = 2 2/11.
```
For example, at 3-away, 7-away, the Turner formula has (24/7 + 3) = nearly 6.5, multiply by 4 to get nearly 26, so the equity is nearly 76. Janowski's formula gives 50 + 340/13 = about 76.2, Kit's table gives 76.

Mathematical readers will notice that the two formulae are essentially the same, but whereas Janowski uses the multiplier 85 / (T + 6), I use (24 / T + 3). Neil Kazaross has invented another system, which he calls Neil's Numbers, which relies on learning the following short table of numbers instead of calculating a multiplier. The table is

```         T    3  4  5  6  7  8  9  10  11  12  13  14  15
multiplier   10  9  8  7     6          5               4
```
One should interpolate where no number is given. For example, at 3-away, 8-away, the difference is 5 and the multiplier is 6, so we get the equity (5 x 6) + 50 = 80 again. At 3-away, 7-away, the multiplier is half way between 6 and 7, so we take 6.5 and get the equity
(4 x 6.5) + 50 = 76 again.

Notes:

1. None of these formulae works when L = 1, i.e. in the Crawford game. You'll just have to learn those.
2. Stick to one formula or another within one calculation: don't mix them up.
3. All the formulae underestimate slightly when L = 2, and T is between 5 and 9 inclusive. You might like to add 2% to these equities with any of the formulae if you're worried.
4. When T is 13 or more, none of the formulae is great but mine is bad. Also the arithmetic in mine isn't any easier by then. If you're playing long matches, you might find Janowski's formula or Neil's numbers better at the beginning of the match (or when one of you is still at the beginning!).
5. My formula does worse than Janowski's and Neil's when T = 11 and L = 2, 3; also when T = 12 and L = 2, 3, 4. All these equities are in the 90-95% range anyway.

I (of course!) think my formula is best when T is at most 12. It is accurate to within 1%, except as noted above, and, I claim, is much easier to calculate than Janowski's formula in all those cases, and doesn't require learning Neil's numbers. But you should choose the one that you find most useful.

## Using the match equity to influence doubling decisions

The problem of making doubling decisions in matches is a complicated one, even if we know the match equities and the probability of winning this game. For simplicity, therefore, let us consider situations in which the game effectively ends after this roll. Typically these are situations in which the player doubling can either hit a shot or finish bearing off this roll, thus winning the game, or miss and lose the game (by the opponent finishing bearing off immediately, or by being redoubled and having to concede); or situations in which the cube is dead because the player taking the double will then have enough points to win the match on winning the game, and thus will not redouble.

For such special situations, the correct doubling strategy can be worked out exactly. For example, suppose I am losing 13-11 in a 15 point match. I have two pieces left on my one point. My opponent has one piece left on his two point and one on his three point. He doubles to 2. What should I do?

These calculations are most easily worked out by using the following formula for my take point (the probability of winning above which I can take the cube). If I drop the cube, I will have a certain match equity. From this base, the amount I gain by taking and winning will be called my gain. The amount I lose by taking and losing will be called my risk. Then my takepoint is

```     risk
----------- .
risk + gain
```

In a money game, I would be risking 1 point (the difference between -1 and -2) to gain 3 (the difference between -1 and +2), so my take point would be at 25%. In the above example, I carry out a similar calculation as follows. If I drop, the score would be 14-11, and my equity, consulting the equity table, would be 17%; if I take and win the score would 13-13, so my equity would be 50%; if I take and lose, I would lose the match, and my equity would be zero. So I am risking 17% to gain 33%, and my take point is therefore

```    17
-------   =   34% .
17 + 33
```
Now a quick count reveals that my opponent bears off if and only if he doesn't roll a 1, so I win 11 out of every 36 games, or 31% of the time. So I should take the cube in a money game, but drop at the match score indicated.

Where does the above formula come from? Suppose my equity is A if I drop, B if I take and win, and C if I lose. Suppose furthermore that I have a probability p of winning. If I take my equity is then

``` p B + (1 - p) C .
```
So I should take if and only if
```                 p B + (1 - p) C  >  A

if and only if   p (B - A + A - C)  >  (A - C)

A - C              risk
if and only if   p  >  -----------------  =  -----------
(B - A) + (A - C)     gain + risk
```

Other calculations we can do precisely are when it is not the last roll of the game, but the person being doubled can, because of the match score, immediately redouble. For example, suppose I am winning 11-7 in a 15 point match. I hold the cube at 2. When should I double?

If my opponent takes, he can immediately redouble so we would be playing for the match, that is for an equity of 0% or 100%. If I don't double and win, the score is 13-7, with an equity of 88%. If I don't double and lose, the score is 11-9, with an equity of 64%. This time I am risking 64% (the difference between the two losing equities) to gain 12% (the difference between the two winning equities), so my doubling point is

```    64
-------   =   84% .
64 + 12
```
(We can do a calculation similar to the above one to show that this is the correct doubling point). Similarly, if my opponent drops he has an equity (at 13-7 down) of 12%. So he will take if his probability of winning this game (with the cube dead) is greater than 12%.

In general we have to take into account the probability of gammons, and of redoubles, and how many and how big our market losers are, and it is not clear how to do this. For example, in the final example above, I would probably want to try and play for a gammon if there was still a reasonable chance of that, and only double if that became unlikely.

## Other Relevant Documents

The standard text on match play strategy (cube and chequer play) is Kit Woolsey's book How to Play Tournament Backgammon. As for resources on the Web, Tom Keith has written an article on how to compute a match equity table, and one on match play doubling strategy. Claes Thornberg has written a program to calculate match equities given different gammon rates and skill levels.