11.5.4 Normalized equity

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 normalize the MWCs to some common scale. The most often used normalization is Normalized 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 usual 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 literature this term seems to be reserved for the cubeless money equity.