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The live cube equity is the equity assuming that the equity changes continuously, so that doubles and takes occurs exactly at the double point and take point. For gammon-free play this is the well-known take point of 20%. Janowski derives the more general formula
TP = (L-0.5)/(W+L+0.5)
where W is the average cubeless value of games ultimately won, and L is the average cubeless value of games ultimately lost. For example, for the following position
Cubeful example 1
[[cubefuleq-ex1.png]]
GNU Backgammon evaluates
| Win | W(g) | W(bg) | L(g) | L(bg)
| |
| static: | 0.454 | 0.103 | 0.001 | 0.106 | 0.003
|
and hence W=(0.454 + 0.103 + 0.001)/0.454=1.229 and L=(0.556+0.106+0.003)/0.556) = 1.196. For gammon-free positions, e.g., a race, W=1 and L=1.
The live cube equity is now based on piecewise linear interpolation between the points (0%,-L), (TP,-1), (CP,+1), and (100%,+W): if my winning chance is 0 I lose L points, at my take point I lose 1 point, at my cash point I cash 1 point, and when I have a certain win I win W points:
mgtp
Equity
1.5 ++------------+-------------+-------------+-------------+------------++
+ + + + cubeless ****** +
| live cube####*****
1 ++ ###cubeful*****$$++
| ####$$$$ ***** |
| ###$$ ***** |
0.5 ++ ###$ **** ++
| ###***** |
| ##**** |
| ***** |
0 ++ *****# ++
| ******## |
| *****$### |
-0.5 ++ *****$$### ++
| ***** $$### |
| **** $$$$$### |
-1 ++ *****########## ++
***##### |
+ + + + + +
-1.5 ++------------+-------------+-------------+-------------+------------++
0 0.2 0.4 0.6 0.8 1
Wins
For match play there is no simple formula, since redoubles can only occur a limited number of times.
The live cube take point is generally calculated as
TP(live, n Cube)=TP(dead, n cube) * (1 - TP(live, 2n cube)
So to calculate the live cube take points for a 1-cube at 3-0 to 7 we need the live cube take points for the 4-cube and the 2-cube. For the position above and using Woolsey's match equity table the live cube take point are:
| Cube value | TP for Black | TP for White
|
| 4 | 0% | 41%
|
| 2 | 15% | 38.5%
|
| 1 | 24.5% | 27.3%
|
The calculation of these are left as an exercise to the reader.
Ignoring backgammons, the gammon rates for White and Black are 0.106/54.6=19% and 0.103/0.454=22%, respectively. If White wins the game his MWC will be
81% * MWC(-3,-7) + 19% * MWC(-2,-7) = 78%
and if Black wins his MWC will be
78% * MWC(-4,-6) + 22% * MWC(-4,-5) = 41%.
If White cashes 1 point he has MWC(-3,-7)=76% and if Black cashes he has MWC(-4,-6)=36%. Analogous to money game the live cube MWC is calculated as piecewise linear interpolation between (0%,22%), (24.5%,24%), (72.7%,36%), and (100%,41%) (from black's point of view):
mptp
MWC
0.42 ++------------+-------------+-------------+-------------+------------++
+ + + + cubeless ********
0.4 ++ live cube #****#++
0.38 ++ cubeful**$$$$$++
| #***** |
0.36 ++ #**** ++
| ***** |
0.34 ++ ****# ++
| *****## |
0.32 ++ *****### ++
0.3 ++ *****#### ++
| ****$#### |
0.28 ++ ****$$$### ++
| ***** $$#### |
0.26 ++ ***** $$$### ++
| **** $$$$$#### |
0.24 ++ ****########## ++
0.22 ****#### ++
+ + + + + +
0.2 ++------------+-------------+-------------+-------------+------------++
0 0.2 0.4 0.6 0.8 1
Wins