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What is the lowest winning % for Too good?
Posted By: Rick Janowski
Date: Sunday, 7 July 2013, at 10:12 a.m.
In Response To: What is the lowest winning % for Too good? (Mochy)
Summary: Although 44.4% (= 4/9) are the theoretical minimum cubeless winning chances required for a position to be too good (in money games), the extreme requirements of 100% backgammon wins combined with a perfectly live cube make this value unrealistic from a practical perspective. The example provided by Xavier with winning chances of 61.3% looks to be very close to the practical limit. Perhaps positions can be found which are too good with winning chances of less than 59 or 60%, but this seems unlikely. A detailed explanation is provided below for those interested:
Assuming money games where Jacoby is not in effect, so gammons are active, then too good points (expressed in terms of cubeless probability) will fall between the extreme limits of dead cube (no cube vig) and live cube (maximum theoretical cube vig) as follows:
When the cube is dead, the Too Good point, TG (dead) = (L + 1) / (W + L)
When the cube is live, the Too Good point, TG (Live) = (L + 1) / (W + L + 0.5),
Where W is the average cubeless value of games won (eg, W = 2 where all wins are gammons but no backgammons) and L is conversely the average value of games lost. Notice that the only difference in the two equations is the addition of 0.5 in the denominator which effectively models maximum cube vig. Unlike for takes, where take points are generally nearer to the live limit than the dead limit, the converse is generally true of too good points (this is because the gradient of cubeowned equity line tends to reduce considerably for probabilities above the cashpoint).For typical positions, the Too Good point could be estimated from the following expression, intermediate between the dead and live limits:
For normal positions, TG (normal) = (L + 1) / (W + L + 0.125).
Considering the two examples provided by Ray, where W = 2 or 3,and L =1:
(1) Where W = 2 (fairly extreme case) and L =1, the Too Good Window is between 66.7% (dead) and 57.1% (live) with 64% typical. If one were to consider a less extreme case where W = 1.5 and L =1, the window would be 80% to 66.7%, with 76.2% typical.
(2) Where W = 3 (ultraextreme case) and L =1, the Too Good Window is between 50% (dead) and 44.4% (live) with 48.5% typical. Although this case is of course unrealistic it does represent the absolute lowest limit for too good points. The absolute extreme case which considers a perfectly live cube and all backgammon wins backgammons combined with all single game losses is a cubeless probability of 44.4% ( = 4/9).
In Mochy’s example, where W = 1.97 and L = 1, this is a fairly extreme case, where the Too Good Window is 67.5% (dead) to 57.7% (live), with 64.7% typical. The actual probability of 66.8% is greater than both the dead and live cube limits so is independent of cube vig.
In Xavier’s example, where W = 2.20 and L = 1, this is an even more extremes case because of the very high proportion of backgammons. Here the Too Good Window ranges from 62.4% (dead) to 54% (live) with 60.1% typical. The actual probability of 61.3% is greater than both the typical and live cube limits but not the dead cube limit, so it is reliant on cube vig.
Conclusion: Although 44.4% (= 4/9) are the theoretical minimum cubeless winning chances required for a position to be too good, the extreme requirements of 100% backgammon wins combined with a perfectly live cube make this value unrealistic from a practical perspective. The example provided by Xavier with winning chances of 61.3% looks to be very close to the practical limit. Perhaps positions can be found which are too good with winning chances of less than 59 or 60%, but this seems unlikely.

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