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Old 10-12-2021, 09:28 AM   #1
sdfsgf
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Why isn't no-double equity proportional to win probability?

In other words, why is the red curve in the image below not a straight line between the points (0%, -1) and (100%, +1)?

I would have thought our no-double equity would be E = p - (1-p) or (equivalently) E = 2p - 1. This red curve starts and ends at the same points, but is a bit steeper in the middle and less steep near the ends. Any insight on why this happens? Am I missing something obvious?



(The image is from this article but I couldn't find an answer there).

EDIT: It also seems unlikely that it's related to gammons, since if gammons were included (e.g., 15% of our wins are gammons) I would expect our ND equity to be greater than 1 when we reach 100% win probability.

Last edited by sdfsgf; 10-12-2021 at 09:41 AM.
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Old 10-12-2021, 01:20 PM   #2
_Z_
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Re: Why isn't no-double equity proportional to win probability?

The no double line assumes you don't double for a roll, then use the cube correctly after that. Same as what XG reports as the no double equity for a cube decision.

So if you are 95% cubeless, your "no double" equity is pretty much 1.0 because it will almost always go double/pass next roll since it's unlikely your win % will drop from 95% all the way down to something where your opponent can take. I mean it depends on the exact position, but this graph appears to assume something like average volatility. I haven't read the article though.

Last edited by _Z_; 10-12-2021 at 01:31 PM.
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Old 10-12-2021, 02:31 PM   #3
sdfsgf
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Re: Why isn't no-double equity proportional to win probability?

That makes perfect sense, thanks for pointing that out. It makes sense why the red curve has a less-steep slope near the ends. And I guess the reason it’s steeper in the middle (above the 2p-1 line) must be because of the possibility that, although we don’t double this roll, we could improve to having a profitable double on our next turn? Subtle point that I didn’t really appreciate before.
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