gnubg works with many different kinds of equities. The equity is defined as the expected value of the position. However, this value can be expressed in several different metrics and may be calculated with or without taking the effect of the cube into consideration. In the following section we will describe the equities used and calculated by gnubg.
This is the value of the position in money game, e.g., if you equity is +0.4 an you are playing money game with a $1 stake, you will win $0.40 on average. The money equity can be calculated with or without taking the effect of the doubling cube into consideration, or cubeful or cubeless. The cubeless equity can be calculated from the basic formulae: 2*p(w)-1+2(p(wg)-p(lg))+3(p(wbg)-p(lbg)). Evaluating the cubeful equity is much more difficult; it can either be estimated from the cubeless equitity by using transformations as outlined by Rick Janowski or by constructing a neural net that directly outputs cubeful equities. gnubg uses the former approach (see chapter Cubeful equities).
In match play we're generally not particular interested in the outcome of the individual games as much as the outcome of the entire match, so the interesting quantity for match play is match winning chance (MWC). As for the money equity the MWC can be calculated with and without the effect of the doubling cube. The MWCs are generally calculated with the use of a match equity table, which contains the chance of winning the match before a game starts, e.g., if the score is 0-0 in a 1pt match each player has 50% chance of winning the match before the game starts assuming they're of equal skill.
The cubeless MWC is calculated as: MWC(cubeless) = p(w) * MWC(w) + p(l) * MWC(l) + p(wg) * MWC(wg) + p(lg) * MWC(lg) + p(wbg) * MWC(wbg) * p(lbg) * MWC(lbg).
For example, if the w/g/bg distribution is 0 30 60 - 40 10 0 and the match score is 1-3 to 5 with the cube on 2 the cubeless MWC is:
MWC(cubeless)= 30% * 50% + 30% * 0% + 30% * 100% + 10% * 0% + 0% * 100% + 0% * 0% = 45%,
so the cubeless MWC is 45%.
Evaluating the cubeful MWC is more difficult, and as for the cubeful money equity it's possible to estimate cubeful MWCs from transformation on the w/g/bg distribution or directly calculate it from neural nets. gnubg uses the former approach, but the formulae are currently not published.
It's generally very difficult to compare MWCs. For example, it's hardly worth mentioning a 0.5% MWC error at DMP where as it's a huge error at 0-0 to 7. It is therefore of interesting to normalise the MWCs to some common scale. The most often used normalisation is Normalised Money Game Equity (NEMG) where the MWC for any game is transformed into the same interval as money game, i.e., -3 to +3 (due to anomalies at certain match scores the NEMG can go below -3 and above +3). The transformation is linear:
NEMG = 2 * (MWC-MWC(l))/(MWC(w)-MWC(l)) - 1
In other words, extrapolation with the following two extrapolation points: (MWC(w),+1) and (MWC(l),-1).
For example, suppose the score is 3-1 to 5 with the cube on 2: MWC(l)=0% and MWC(w)=50%:
| MWC | NEMG
|
| 0% | -1
|
| 25% | 0
|
| 50% | +1
|
| 75% | +2
|
| 100% | +3
|
Note that a w/g/bg distribution of 0 100 100 - 0 0 0 gives a NEMG of +3 whereas the corresponding money equity is only +2. This is because the gammon price is high for that particular score. When both players are far from winning the match, e.g., 0-0 to 17 or 1-0 to 17, NEMG is very close to the ususal money equity.
NEMG can be calculated from both cubeless and cubeful MWCs.
A word of caution: A cubeless NEMG calculated from a cubeless MWC could be named "cubeless equity", but in most backgammon litterature this term seems to be reserved for the cubeless money equity.