[ View Thread ] [ Post Response ] [ Return to Index ] [ Read Prev Msg ] [ Read Next Msg ]

BGonline.org Forums

Undefined equities

Posted By: ah_clem
Date: Saturday, 17 October 2026, at 2:42 p.m.

In Response To: Undefined equities (Timothy Chow)

Interesting article. Some quick takeaways:

1) The undefined equity only occurs if the cube is allowed to be turned arbitrarily many times. So, for match play the equity is always well defined (even though we tend to look at EMG instead of the MWC equity) And if also doesn't happen for any club with house rules that limit the cube value.

2) The equity value for when the cube is turned n times would be the limit as n -> ∞. Since the growth is exponential (order is 2ⁿ) the limit must be infinite . But each turn there is some finite probability that the on roll player will roll a six and the cube. Is this probability low enough that the exponential growth out-races it? My math skills are pretty rusty, and probability was never my field, but I'm not seeing it.

3) Since the Zare position is symmetric, the equity would seem to grow simultaneously to ∞ and -∞. Which is a sure sign of "undefined".

4) I popped the position into XG and its verdict is a clear R/T. Playing it out resulted in two dances followed by red entering with a 63. So, while the redoubling could go on forever it would seem like it doesn't, or at least it doesn't often enough for the exponential growth to dominate. Maybe I'm missing something here.

5) Here it is, FWIW:





White is Player 2

score: 0
pip: 115
Unlimited Game
Jacoby Beaver
pip: 115
score: 0

Blue is Player 1
XGID=aBBBBBa---aaAaAA---AbbbbbA:1:1:1:00:0:0:3:0:10
Blue on roll, cube action?

Analyzed in Rollout No redouble Redouble/Take
Player Winning Chances: 57.49% (G:46.32% B:9.51%) 57.61% (G:46.34% B:9.40%)
Opponent Winning Chances: 42.51% (G:32.69% B:6.49%) 42.39% (G:32.61% B:6.64%)
Cubeless Equities +0.316 +0.634
Cubeful Equities
No redouble:+0.354 (-0.216)±0.007 (+0.347..+0.361)
Redouble/Take:+0.570±0.015 (+0.554..+0.585)
Redouble/Pass:+1.000 (+0.430)
Best Cube action: Redouble / Take
Rollout details
2592 Games rolled with Variance Reduction.
Moves: 3-ply, cube decisions: XG Roller
Double Decision confidence:100.0%
Take Decision confidence:100.0%
Duration: 43.0 seconds

eXtreme Gammon Version: 2.10

Messages In This Thread

 

Post Response
Your Name:
Your E-Mail Address:
Subject:
Message:

If necessary, enter your password below:

Password:
 

 

[ View Thread ] [ Post Response ] [ Return to Index ] [ Read Prev Msg ] [ Read Next Msg ]

BGonline.org Forums is maintained by Stick with WebBBS 5.12.