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What is the lowest winning % for Too good?
Posted By: Rick Janowski
Date: Sunday, 7 July 2013, at 8:18 p.m.
In Response To: What is the lowest winning % for Too good? (rambiz)
For redoubles, there are three theoretical bounds, one where the cube is dead for both players, TG (0,0), another where the cube is live for both players, TG (1,1) and the third one, your case where the cube is live for the cubeowner, but always dead for the opponent even if he takes at some future point. For initial cubes, assuming no Jacoby, there is a fourth case, TG (0,1) again where the cube is live for one player but dead for the other, but switching players. In redoubles, TG (0,1) becomes the same as the dead case TG(0,0) because the player who would otherwise have a live cube has no access to it. The three cases may be expressed by the following equations (two being repeated from my earlier post):
TG (0,0) = (L + 1) / (W +L)
TG (1,1) = (L + 1) / (W +L +0.5)
TG (1,0) = (L + 1) / (W +L) * (L + 0.5) /(L + 1)
When W =3 and L = 1, then the TG (1,0) is 3/8 as Bob analysed. I also fully agree with Bob that live cube too good points correspond directly with live cube cashpoints. It is almost a paradox that where the favourite wins gammons or backgammons and where the cube is live, doubling is only possibly correct at precisely the cashpoint. At any lower probability it is not good enough, whilst at any higher probability it is too good. Even at the cashpoint doubling is optional rather than mandatory.
TG (1,0), the onesided live/dead cube situation is relevant to last roll positions or other situations where the cube will be capped after the next double. My detailed investigations of cube efficiency in races and bearoffs (using Sconyer’s database) indicated that outside these situations, TG (1,0) is far less useful as a limit than TG (1,1), where the cube is live for both sides.
Of course, in practice the 100% (or very high) backgammon win scenario cannot lead to positions with win probabilities, because to yield losses it is necessary to be hit in the bear off. When hit, not only will games be lost but a large proportion of games will still be won but now with a much greater proportion of gammons and single wins. This is why I think that it is unlikely that Too Good positions with winning probability of less that 59% are extremely unlikely.

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