[Bug-gnubg] Match math mysteries

[Petr Kadlec]

Hi all!

When recently discussing a match with a friend (with help of gnubg's 
analysis), I figured something which seems strange to me:
Situation: we are playing a match and I am trailing at 3-away, 2-away. 
During the game, the opponent doubles and I take. BUT NOW: What should I 
do? After some (painful...) thinking, I believe I have a compulsory double, 
since when I lose the game, I lose the match, but if I win with 2-cube, I 
will still be 1 point away, but with a 4-cube whoever wins the game, wins 
the match; so that I have nothing to lose.
More exactly (at least I will try, I am _really_ not good at bg theory), I 
see it this way:

At 2-cube, my MWC2 = Ps * C + Pg + Pbg, where Ps,Pg,Pbg are probabilities 
of single/gammon/backgammon win, C is MWC when leading 1-away, 2-away 
(which is, I believe, 0.7 according to Woolsey MET).

At 4-cube, my MWC4 = Ps + Pg + Pbg

As C < 1, always MWC4 > MWC2, so that I should double, no matter of my 
winning chances (except the situation Ps=Pg=Pb=0, but that is obviously 
uninteresting).

Now you may ask, where I see any problem? The problem is that according to 
gnubg analysis, the situation is "Optional double, take". OPTIONAL? HOW SO?
I have tried more positions, and even when gnubg says the position is 
"double, take", the difference between this and "no double" is very small. 
I would expect the difference to be equal to (1 - C) * Ps, for C = 0.7 it 
is 0.3 * Ps. For example, in position 3N0BAIB77wEAAA, gnubg reports 
P(win)=0.227, with double/take MWC (correctly) equal to that, but no-double 
MWC is reported to be also equal to 22.67%, although I would expect it to 
be 0.7 times smaller (ca 16%). But a rollout result is the same -- no 
difference...

I tried to look to the source to find some explanation, but when something 
I believed to be a MWC was sent to eq2mwc, I gave up. :)

Where am I wrong? [I am unable to believe it could be gnubg who is wrong 
here...] Could someone please help me understand that?

Thanks,
Petr Kadlec

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