It appears that the XG engine on Backgammon Galaxy takes the match score into effect when assigning equity to specific positions (or equity losses/error values to specific moves).
For example, in the position below, I correctly dropped, reasoning that it was probably a close pass for money and therefore a big pass at 2a/4a due to the gammons. But I'm wondering specifically how the value -1.717 (the equity if it had gone double/take) is derived from the win probabilities/gammon probabilities/match score. I assume the calculation somehow takes into account a match equity table but I'm wondering if anyone can help me understand specifically how this works (and how these calculations result in the number -1.717). Can anyone point me in the right direction?
One problem with the calculation you are thinking about is that the percentage numbers that XG gives are cubeless. That is, they assume the rest of the game is going to be played out without the cube. In fact they assume the rest of the game is going to be played out a cubeless money game, which causes XG to err in matches sometimes, but that's sort of beside the point.
My main point is that equity numbers XG gives are cubeful, so there's a calculation based on the cubeless percentages and a match equity table and then the final equity numbers are adjusted for future cube value. The cube adjustment is all done under the surface and there's no way to replicate it.
At the specific match score in your position, I think you probably can do a calculation based on the percentages given and the XG match equity table to get the 1.717 number, but only because the cube is dead after double/take. Generally that's not true. I can guess how they do that calculation, but I'm not totally sure, so maybe someone more knowledgeable than me will respond.
So I read the article from Jeremy Bagai and it was quite dense but the concept actually isn’t too complicated. Here’s a simplified version (applied to the position above) if anyone is interested.
If I drop, it’s 2a/3a giving me match-winning chances (MWC) of 60%. We’ll call this MWC value (when the opponent cashes) “equivalent” to equity of -1.
On the other hand, if I cashed at this score, it’s 1a/4a Crawford giving me MWC of 81%. We’ll call this MWC value (when I cash) “equivalent” to equity of +1.
XG seems to think that taking in this position gives me about 51% match winning chances (unclear how this works under the hood, but roughly, the opponent wins 35% immediately from gammons/backgammons, then 50% of the 2a/2a games from his remaining 28% wins, a total of 49% match wins for him and 51% for me).
So if we’re calling 60% MWC equivalent to -1 equity, and 81% MWC equivalent to +1 equity, we can extrapolate to get our equity with 51% MWC:
-1 - (60%-51%)/(81%-60%)*(+1 - -1) = -1.85
This is a little off from -1.717 and it’s not clear where the difference comes from (perhaps my estimate of our double/take MWC is wrong or perhaps I made a mistake with the match equity table or perhaps I just rounded off too much in the intermediate calculations) but at least conceptually this is how these “equivalent to money game” equities work.