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Beaver calculus
Posted By: Rick Janowski
Date: Tuesday, 25 June 2013, at 7:53 a.m.
In Response To: Beaver calculus (leobueno)
There are formulae covering correct beavering and raccoon cube action in my paper “Takepoints in Money Games” < http://www.bkgm.com/articles/Janowski/cubeformulae.pdf >. Refer to the Cube Action Formulae Table on page 9 of the document. There also tabulated values in tables A2 and A3 on page 22. Essentially what you are looking at here is the raccoon point, which regardless of if raccoons are being used, is the point of last beaver for your opponent. In order words, the minimum equity or probability you need in the given position to avoid being correctly beavered. The formula is:
Raccoon Point, RP = (L+0.5*x) / (W + L + 0.5*x)
Where W and L are the average cubeless values of wins and losses respectively (e.g., W = 1.0 were all wins are single games, L= 1.25 were 25% of losses gammons, assuming no backgammons) and x is a cube efficiency ranging from minimum of 0.0 for a fully dead cube and 1.0 for a theoretical fully live cube. For doubling/no doubling decisions, x will be typically about 0.7, and for take decisions typically about 0.6. For beaver/racoon decisions an intermediate value of 0.65 looks realistic.
In practice what does this mean?
Let’s consider the zero cubeless equity situation, ie, where neither player is favourite with the cube still in the middle (or dead). The probability here may be expressed as L /(W +L), which is clearly 50% where W = L, i.e., where there are no gammons, or the gammon proportions are the same. Whatever the probability is for the zero equity of the particular situation the raccoon point will have a higher probability by typically 0.07 (i.e. 7%) where all the opponent wins are single games, but about half that value, 0.03 when all his wins are gammons. I realise that this may seem counterintuitive but cube mathematics have a few surprises.
Another way of looking at this problem is by considering the change in cubeless equity at the raccoon point. Here for all positions where W = L, then the cubeless equity is about 0.14 (consistent with 50 +7% for gammonless games). This can then vary within the range between 0.10 (where W=1, L=2.ie, all wins are gammons but losses all gammons) to 0.20 (the converse, where W=2, L=1). Such extreme situations will very rarely occur in practice, and in my view, the typical cubeless equity range for the raccoon point is about 0.12 to 0.17. In other words, whenever then cubeless equity is less than 0.12 beavering is nearly always correct (last roll positions would be an exception), but when the cubeless equity exceeds 0.17 beavering is nearly always wrong). In between 0.12 and 0.17 is a grey area dependent of the relative gammon proportions of the two players.

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