BGonline.org Forums

Cubeful equity differences between HyperGammon and GambleGammon

Posted By: MK
Date: Sunday, 25 October 2026, at 5:09 a.m.

What prompted me to post this is what Chow said about "some of XG rollouts being off by a lot" while talking about his paper. See:

https://www.bgonline.org/forums/webbbs_config.pl?read=222004

For decades I have argued that the bots' cubeful equity calculations are increasingly inaccurate going backwards from simple end-of-game positions towards more complex middle-of-game and then beginning-of-game positions.

I always had to say "inaccurate by an unknown amount" since there is no way to compare them to equities calculated differently but I believed that they were inaccurate enough for me to feel justified in calling cubeful equities for near-beginning-of-game positions "totally bogus".

Today it just occurred to me that we can actually do such comparisons at least for some positions using GnuBG, since it offers playing HyperGammon against claimedly "perfect" opponents and because HyperGammon is not a true variant of GambleGammon but just a specific "play from" position that can be played using the regular GnuBG database or against the claimedly "solved" hyperX databases.

Below are three cube action analyses of a 2-checker HyperGammon's starting, middle and end positions using GnuBG to play HyperGammon and then using it to play regular GambleGammon.

As you can see that the two bots wildly disagree on the cubeful equities of and the recommended cube actions for what are nothing more than simple late game GambleGammon positions and that they only start to agree towards the final positions.

I guess this vindicates my decades-long argument that the so-called "cube skill theory" is mostly bullshit except in simple late game positions.

Considering that all HyperGammon positions are basically near-end-of-game GambleGammon positions, I'll let you all draw your own conclusions about "by how much more cubeful equities for near-beginning-of-game GambleGammon positions can be off" by comparing as many more positions as you need to convince yourselves.

I don't think there is a way to quantify those inaccuracies but I have demonstrated with my bot-vs-bot experiments that early errors in a game don't effect the outcome as much as predicted by the bots, not even anywhere close.

This means, as I had also demonstrated with my earlier human-vs-bot experiments that you can ignore/defy what bots think and try deploying early alternative strategies to beat the bots, from which errors you can recover later in the game if your strategy goes in the wrong direction.

BTW, being able to use it to do things like this shows how much superior GnuBG is compared to all other existing bots out there.

MK

============== Position 1 ============================

Position ID: AABAAQAACgAAAA Match ID: cAkXAAAAAAAE

Evaluator: Hypergammon-2

Player Opponent Position 346 346

Owned cube : +0.3278 Centered cube : +0.2092 Centered cube (Jacoby rule) : +0.2046 Opponent owns cube : +0.0109

Win W(g) W(bg) L(g) L(bg) Equity Cubeful static: 53.5 41.8 4.2 34.9 3.0 +0.152 +0.209 1 ply: 53.5 41.8 4.2 34.9 3.0 +0.152 +0.209 2 ply: 53.5 41.8 4.2 34.9 3.0 +0.152 +0.209 3 ply: 53.5 41.8 4.2 34.9 3.0 +0.152 +0.209 4 ply: 53.5 41.8 4.2 34.9 3.0 +0.152 +0.209

Cube analysis 4-ply cubeless equity +0.152

53.5 41.8 4.2 - 46.5 34.9 3.0 Cubeful equities: 1. No double +0.209 2. Double, pass +1.000 (+0.791) 3. Double, take +0.022 (-0.187) Proper cube action: No double, take (19.2%)

==========================================

Position ID: AABAAQAACgAAAA Match ID: cAkXAAAAAAAE

Evaluator: Crashed

Win W(g) W(bg) L(g) L(bg) Equity Cubeful static: 61.4 0.0 0.0 0.0 0.0 +0.228 +0.344 1 ply: 54.4 0.0 0.0 0.0 0.0 +0.089 +0.134 2 ply: 56.2 0.0 0.0 0.0 0.0 +0.123 +0.183 3 ply: 54.7 0.0 0.0 0.0 0.0 +0.095 +0.138 4 ply: 53.5 0.0 0.0 0.0 0.0 +0.070 +0.104

Cube analysis 4-ply cubeless equity +0.070

53.5 0.0 0.0 - 46.5 0.0 0.0 Cubeful equities: 1. No double +0.104 2. Double, pass +1.000 (+0.896) 3. Double, take -0.177 (-0.282) Proper cube action: No double, beaver (23.9%)

==========================================

4-ply rolled out equities after 1,000,000 trials are:

HyperGammon +0.163 +0.233 GambleGammon +0.070 +0.105

============== Position 2 ============================

Position ID: AAACAoAIAAAAAA Match ID: cAkAAAAAAAAE

Evaluator: Hypergammon-2

Player Opponent Position 263 322

Owned cube : +1.0458 Centered cube : +1.0006 Centered cube (Jacoby rule) : +0.8428 Opponent owns cube : +0.8156

Win W(g) W(bg) L(g) L(bg) Equity Cubeful static: 70.5 61.2 10.5 21.0 1.6 +0.899 +1.001 1 ply: 70.5 61.2 10.5 21.0 1.6 +0.899 +1.001 2 ply: 70.5 61.2 10.5 21.0 1.6 +0.899 +1.001 3 ply: 70.5 61.2 10.5 21.0 1.6 +0.899 +1.001 4 ply: 70.5 61.2 10.5 21.0 1.6 +0.899 +1.001

Cube analysis 4-ply cubeless equity +0.899

70.5 61.2 10.5 - 29.5 21.0 1.6 Cubeful equities: 1. No double +1.001 2. Double, take +1.631 (+0.631) 3. Double, pass +1.000 (-0.001) Proper cube action: Too good to double, pass (0.1%)

==========================================

Position ID: AAACAoAIAAAAAA Match ID: cAkAAAAAAAAE

Evaluator: Crashed

Win W(g) W(bg) L(g) L(bg) Equity Cubeful static: 72.6 0.0 0.0 0.0 0.0 +0.452 +0.680 1 ply: 74.8 0.0 0.0 0.0 0.0 +0.495 +0.709 2 ply: 73.9 0.0 0.0 0.0 0.0 +0.477 +0.722 3 ply: 73.9 0.0 0.0 0.0 0.0 +0.478 +0.743 4 ply: 72.0 0.0 0.0 0.0 0.0 +0.440 +0.693

Cube analysis 4-ply cubeless equity +0.440

72.0 0.0 0.0 - 28.0 0.0 0.0 Cubeful equities: 1. No double +0.693 2. Double, pass +1.000 (+0.307) 3. Double, take +0.692 (-0.001) Proper cube action: No double, take (0.3%)

==========================================

4-ply rolled out equities after 1,000,000 trials are:

HyperGammon +0.892 +1.000 GambleGammon +0.407 +0.649

============== Position 3 ============================

Position ID: BAAARAAAAAAAAA Match ID: cAkAAAAAAAAE

Evaluator: Hypergammon-2

Player Opponent Position 29 3

Owned cube : +0.6111 Centered cube : +0.6111 Centered cube (Jacoby rule) : +0.6111 Opponent owns cube : +0.6111

Win W(g) W(bg) L(g) L(bg) Equity Cubeful static: 80.6 0.0 0.0 0.0 0.0 +0.611 +0.611 1 ply: 80.6 0.0 0.0 0.0 0.0 +0.611 +0.611 2 ply: 80.6 0.0 0.0 0.0 0.0 +0.611 +0.611 3 ply: 80.6 0.0 0.0 0.0 0.0 +0.611 +0.611 4 ply: 80.6 0.0 0.0 0.0 0.0 +0.611 +0.611

Cube analysis 4-ply cubeless equity +0.611

80.6 0.0 0.0 - 19.4 0.0 0.0 Cubeful equities: 1. Double, pass +1.000 2. Double, take +1.222 (+0.222) 3. No double +0.611 (-0.389) Proper cube action: Double, pass

==========================================

Position ID: BAAARAAAAAAAAA Match ID: cAkAAAAAAAAE

Evaluator: Bearoff2

Player Opponent Position 10 3

Bearing off Bearing at least one chequer off Rolls Player Opponent Player Opponent

0 0.000 0.000 100.000 100.000

1 80.555 100.000 0.000 0.000

2 19.445 0.000 Average rolls Bearing off Saving gammon

Player Opponent Player Opponent Mean 1.194 1.000 0.000 0.000 Std dev 0.396 0.000 0.000 0.000

Effective pip count:

Player Opponent EPC 9.755 8.167 Wastage 4.755 5.167

EPC = 8.167 * Average rolls Wastage = EPC - pips

Win W(g) W(bg) L(g) L(bg) Equity Cubeful static: 80.6 0.0 0.0 0.0 0.0 +0.611 +0.611 1 ply: 80.6 0.0 0.0 0.0 0.0 +0.611 +0.611 2 ply: 80.6 0.0 0.0 0.0 0.0 +0.611 +0.611 3 ply: 80.6 0.0 0.0 0.0 0.0 +0.611 +0.611 4 ply: 80.6 0.0 0.0 0.0 0.0 +0.611 +0.611

Cube analysis 4-ply cubeless equity +0.611

80.6 0.0 0.0 - 19.4 0.0 0.0 Cubeful equities: 1. Double, pass +1.000 2. Double, take +1.222 (+0.222) 3. No double +0.611 (-0.389) Proper cube action: Double, pass

==========================================

4-ply rolled out equities after 1,000,000 trials are:

HyperGammon +0.618 +0.618 GambleGammon +0.618 +0.618

==========================================