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Equity Disparities

Posted By: Nack Ballard
Date: Wednesday, 20 November 2024, at 10:57 p.m.

I think this issue may have previously come up on this site, but either not so early in the game or not to this degree.

Below are two positions. (For further reference, scroll down to see diagrams.) In one, Blue has opened with 21$ (i.e., 13/11 6/5) and in the other 21S (i.e., 24/23 13/11). In both cases, White has replied with 51S (i.e., 24/23 13/8), and Blue now has double 1s to play.

On the left is a list of candidate plays for 21$-51S-11, each followed by its equity (and the same equity repeated in parentheses but with four decimals).

On the right is a list of "matching" plays, in that each one reaches the identical position to the one reached on its left.

The issue is that the left-side equities differ from the right-side equities, in every case by the fourth decimal place, and in most cases (all but one) even in the third decimal place.

21$-51S-11 .........................21S-51S-11
----------------------------------------------------------------------
24/21 6/5 .........= .250 (.2500) ..23/21 6/5(2) ......= .255 (.2550)
24/22 11/10 6/5 ...= .235 (.2352).. 23/22 11/10 6/5(2) = .235 (.2347)
24/23(2) 11/10 6/5 = .211 (.2110) ..24/23 11/10 6/5(2) = .214 (.2144)
24/22 6/5(2) ......= .204 (.2038).. 23/22 6/5(3) ......= .206 (.2064)
24/23 24/22 6/5 ...= .201 (.2015).. 24/22 6/5(2)...... = .203 (.2033)
24/23 11/9 6/5 ....= .192 (.1919) ..11/9 6/5(2) .......= .194 (.1943)
24/23(2) 6/5(2) ...= .183 (.1834) ..24/23 6/5(3) ..... = .180 (.1799)
24/23 11/10 6/5(2) = .180 (.1800) ..11/10 6/5(3).......= .179 (.1787)
24/23 8/5 .........= .180 (.1797) ..8/5 6/5 ...........= .184 (.1842)
----------------------------------------------------------------------

When you punch up the position(s), do you get the same play equities that I do? If not, please report any differences you get, either in the three-decimal or four-decimal numbers (as you like). To toggle from one to the other, look under the Options tab and click on "Use 4th decimal for equity." I'm especially interested in the larger differences (in red).

Diagrams are below (side by side if your screen is wide), for additional reference.

Thanks in advance for any feedback.

Nack





White is Player 2

score: 0
pip: 161
Unlimited Game
Jacoby Beaver
pip: 164
score: 0

Blue is Player 1
XGID=-aa--AD-C--AdD---d-e----B-:0:0:1:11:0:0:3:0:10
Blue to play 11

1.XG Roller++24/21 6/5eq: +0.2500
Player:
Opponent:		56.15% (G:15.05% B:0.62%)
43.85% (G:11.13% B:0.42%)
2.XG Roller++24/22 11/10 6/5eq: +0.2352 (-0.0149)
Player:
Opponent:		55.88% (G:15.17% B:0.64%)
44.12% (G:11.88% B:0.46%)
3.XG Roller++24/23(2) 11/10 6/5eq: +0.2110 (-0.0390)
Player:
Opponent:		54.70% (G:15.69% B:0.62%)
45.30% (G:11.41% B:0.49%)
4.XG Roller++24/22 6/5(2)eq: +0.2038 (-0.0462)
Player:
Opponent:		55.09% (G:14.95% B:0.60%)
44.91% (G:11.93% B:0.47%)
5.XG Roller++24/23 24/22 6/5eq: +0.2015 (-0.0486)
Player:
Opponent:		55.14% (G:14.42% B:0.61%)
44.86% (G:11.70% B:0.45%)
6.XG Roller++24/23 11/9 6/5eq: +0.1919 (-0.0581)
Player:
Opponent:		54.63% (G:15.70% B:0.65%)
45.37% (G:12.22% B:0.54%)
7.XG Roller++24/23(2) 6/5(2)eq: +0.1834 (-0.0666)
Player:
Opponent:		54.04% (G:15.69% B:0.61%)
45.96% (G:11.71% B:0.49%)
8.XG Roller++24/23 11/10 6/5(2)eq: +0.1800 (-0.0700)
Player:
Opponent:		54.31% (G:15.52% B:0.63%)
45.69% (G:12.52% B:0.54%)
9.XG Roller++24/23 8/5eq: +0.1797 (-0.0703)
Player:
Opponent:		54.21% (G:15.66% B:0.62%)
45.79% (G:12.04% B:0.50%)

eXtreme Gammon Version: 2.10




White is Player 2

score: 0
pip: 161
Unlimited Game
Jacoby Beaver
pip: 164
score: 0

Blue is Player 1
XGID=-aa---E-C--AdD---d-e---AA-:0:0:1:11:0:0:3:0:10
Blue to play 11

1.XG Roller++23/21 6/5(2)eq: +0.2550
Player:
Opponent:		56.25% (G:15.06% B:0.65%)
43.75% (G:11.19% B:0.43%)
2.XG Roller++23/22 11/10 6/5(2)eq: +0.2347 (-0.0203)
Player:
Opponent:		55.83% (G:15.47% B:0.67%)
44.17% (G:12.00% B:0.48%)
3.XG Roller++24/23 11/10 6/5(2)eq: +0.2144 (-0.0407)
Player:
Opponent:		54.72% (G:15.97% B:0.65%)
45.28% (G:11.36% B:0.49%)
4.XG Roller++23/22 6/5(3)eq: +0.2064 (-0.0486)
Player:
Opponent:		55.10% (G:15.00% B:0.61%)
44.90% (G:11.84% B:0.45%)
5.XG Roller++24/22 6/5(2)eq: +0.2033 (-0.0517)
Player:
Opponent:		55.02% (G:14.53% B:0.60%)
44.98% (G:11.51% B:0.44%)
6.XG Roller++11/9 6/5(2)eq: +0.1943 (-0.0607)
Player:
Opponent:		54.74% (G:15.58% B:0.66%)
45.26% (G:12.34% B:0.53%)
7.XG Roller++8/5 6/5eq: +0.1842 (-0.0708)
Player:
Opponent:		54.20% (G:15.84% B:0.62%)
45.80% (G:11.92% B:0.48%)
8.XG Roller++24/23 6/5(3)eq: +0.1799 (-0.0751)
Player:
Opponent:		53.96% (G:15.96% B:0.61%)
46.04% (G:11.94% B:0.52%)
9.XG Roller++11/10 6/5(3)eq: +0.1787 (-0.0764)
Player:
Opponent:		54.36% (G:15.22% B:0.60%)
45.64% (G:12.22% B:0.51%)

eXtreme Gammon Version: 2.10

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