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Turner formula, or Equities Easily

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Stephen Turner

Jun 22, 1994, 4:42:51 AM6/22/94
Have problems remembering or working out those match equities? Then
use the new Turner formula!

The match equity is defined as the probability that a given player
will win the match, given the match score, and assuming perfect play
by both players. Kit Woolsey in his book "How to Play Tournament
Backgammon" has a table of these equities, for all scores where
neither player needs more than 15 points to win. They are useful in
making doubling decisions.

Rick Janowski invented a formula, known (unsurprisingly!) as the
Janowski formula, for working out these equities pretty accurately
without having to memorise them all. It is defined as follows.
Let L = number of points the leader still needs;
T = number of points the trailer needs;
D = the difference, i.e. (T - L).
Then the leader's equity in % is approximately

85 x 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 425/14, even to the nearest 1, in their heads. Hence
I have 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.

1) Neither formula works when L = 1, i.e. in the Crawford game. You'll
just have to learn those. (You don't need to know about post-Crawford;
there's no doubling decisions to be made that affect that).
2) Both formulae underestimate when L = 2, and 5 <= T <= 9. You might
like to add 2% to these equities with either formula if you're worried.
3) When T >= 13, neither formula 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 better at the beginning of the match
(or when one of you is still at the beginning!)
4) My formula does worse than Janowski'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.

Apart from these few cases, however, my formula is accurate to within
1% for all equities where T <= 12, and, I claim, is much easier to
calculate than Janowski's in all those cases.

Stephen Turner
Stochastic Networks Group, Statistical Laboratory,
University of Cambridge, CB2 1SB, England

PS Many thanks to Kit Woolsey for commenting on the above post. He
described to me another system called Neil's Numbers, invented by Neil
Kazaross, which is again basically the same, but you learn a short
table of multipliers instead of using my (24/T + 3) or Janowski's
85 / (T + 6). 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

Interpolate where no number is given. I think my method is easiest up
to T=12, but use one of the other two if you prefer!

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