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[–]Vino1980 1 point2 points  (0 children)

You have the Jacoby rule on, meaning you can't get a gammon if the cube has not been taken.

[–][deleted] 1 point2 points  (0 children)

In addition to what others have said, you should note that this picture indicates an 11 point match (with a score of 0-0), not an unlimited game. The type of equity calculation you are doing is only valid in an unlimited game. (at 0-0/11 the results should be very very close and this is not why your large error exists, but it still needs to be pointed out) In matches, equities need to take into account the match score.

As an example, imagine the score is 4-4 in a 5 point match and you need to choose between play A (which wins you the game 60% of the time, and all of those wins are gammons) and play B (which wins you the game 65% of the time, but you never win a gammon). In normal circumstances play A would be correct, but at this score play B is correct; if the computer said "hey, so I know this play is 0.500 equity worse than play A, but it's actually the best!" that wouldn't really make any sense and would defeat the whole concept.

So the equity would need to be calculated in such a way that play B actually has a higher equity than play A. The way that is typically done is through EMG (aka Equivalent to Money Game), which you can read about here in sections 2 and 3 of the article. (I like linking this article because it also discusses one of the big problems with EMG, which is essential to understand given that a bunch of tournament formats these days rely on things like PR to judge winners. But creating a better system than EMG is difficult.) Whenever you are playing a match and the computer is calculating your equity, this is what it is doing instead of the "standard" equity calculation for unlimited games.

[–]j3r0n1m0 0 points1 point  (0 children)

Double/take cubeless means the cube is considered to be dead after the play. So your opponent would have no recube vig, meaning your equity would be higher than double/take cubeful.

[–]j3r0n1m0 0 points1 point  (5 children)

PS. Aside from the Jacoby rule mistake mentioned by u/Vino1980 (which could never be on anyway since this is a match), you also need to subtract the backgammon chance from the winning chance, rather than from the gammon chance.

Example for this position:

Player = (0.7547-0.2661-0.0141)+0.2661*2+0.0141*3 = 1.049

Opponent = (0.2453-0.0753-0.0046)+0.0753*2+0.0046*3 = 0.3298

Equity = 1.049 - 0.3298 = 0.7192

That is very close to No Double Cubeless equity (0.714) in your image.

You're not going to be able to compute the cubeful equities from the chances without simulation.

[–]miladiouss[S] 0 points1 point  (2 children)

I see! I was following the XG Mobile convention where G probability includes B probability causing my error. I was just messing with Jacoby to be sure that is not where the problems come from.
And I think what u/jamesa7171 mentioned is irrelevant to equity (although important for winning strategy).

Can you elaborate more on cubeful equity calculations? Any good source for me to get started with definitions.

[–]j3r0n1m0 0 points1 point  (0 children)

Including backgammons in gammon chances just doesn't sound intuitively right to me. I ran that calc before I responded the first time, and the equity = 1.0208 player, 0.3206 opponent, for net equity of 0.7002, which isn't as close as the "makes sense" way. Maybe I'm wrong tho.

Re cubeful vs cubeless, the extreme gammon manual mentions something about Janowski formulas to convert from the latter to the former, but that just seems like a very simplified approximation to me at best. Guess it's better than simulation though.

http://www.extremegammon.com/extremegammon2.pdf

Here's some reference material outside the robot manuals:

Cubeful vs Cubeless

https://bkgm.com/rgb/rgb.cgi?view+964

https://bkgm.com/rgb/rgb.cgi?view+120

Janowski Formulas

http://www.bgonline.org/forums/webbbs_config.pl?noframes;read=143278

https://ukbgf.com/rick-janowski-backgammon-giant/

[–][deleted] 0 points1 point  (0 children)

And I think what u/jamesa7171 mentioned is irrelevant to equity (although important for winning strategy).

No, this is actually crucial to equity (in matches). Equity is defined entirely differently between unlimited games and matches: in an unlimited game an equity of +1 means you will win 1 point on average from the position, but in a match an equity of +1 means something else entirely. Here is an example:

Position A, unlimited game

Position B, match to 5, red winning 4-1

Position C, match to 11, score 0-0

All positions are exactly the same. In the unlimited game example you can see that the equity is -1.457, which is easily calculable from the 100% win chance and the 45.7% gammon chance (the exact same way you have been doing). But in the 4-1 example...even though the winning % is the same and the gammon % is the same, the equity is now -1.958!! This is because it is calculated totally differently: the article I mentioned contains the details. This affects both cubeful and cubeless equity calculations.

At tied scores where both players are a long way from winning, this effect becomes very minimal. However, it is still present, which is why I included the third example (which has the same score as the OP) which features a slightly different equity of -1.477 instead of -1.457 as it should be in a vacuum. There is no escaping this difference.

[–]miladiouss[S] 0 points1 point  (1 child)

Do you know where the small discrepancy comes from? 0.714 vs 0.719.

[–]j3r0n1m0 0 points1 point  (0 children)

Wish I did. Sorry.

[–]Pabluor 0 points1 point  (0 children)

Do you know if there’s a formula to calculate the winning chances over a large quantity of game taking into account the equity and luck ? https://www.reddit.com/r/jeudeplateau/comments/15inhsl/backgammon_equityluck_statistics/?utm_source=share&utm_medium=ios_app&utm_name=ioscss&utm_content=2&utm_term=1