> I think that for a given LONG race , in a money game against a PERFECT
> opponent, in which we can know the exact cubeless winning chances (and
> also the volatility) we should define a mathematical double (redouble)
> Am I in error?
In this post I hope to describe the state of the art in racing models, so
you can see where we stand. Possibly you can supply some of the missing
pieces in our understanding.
It should be clear that in a short race (say once we get to a bearoff)
no simple formula will work. An easy way to see this is that there will be
some cutoff involved (e.g. CWP > 0.4) whereas we know that bearoff
positions exist in which doubling is correct with a CWP almost equal to 0.
But you specified a LONG race, and that changes things quite a bit.
In a LONG race the usual caveats and exceptions (listed by other
respondents to this thread) apply with greatly reduced force.
The usual rule of thumb (i.e. take if your pip deficit is less than 10%
of the opponent's total) are derived from exhaustively enumerating a
simplified racing game in which each player has just one checker, and
advances his man in accordance with the rules of backgammon. This
game doesn't feature the bearoff tactics of backgammon (e.g. wastage),
but it is a terrific approximation to actual LONG races.
In fact, if you could reasonably approximate the "wastage" involved in
bearing off, then this model would be correct for any practical purpose.
And this is where complicated models of pip-counting come in.
The problem is to adjust the actual pip count by some amount intended to
account for inefficient bearoffs. The Thorp count, for example, adds 1 for
every man on the ace point, since such men are likely to waste pips, and it
adds 1 for every empty point in the inner table, since missing involves
wastage, and it adds 2 for every man that you have to bear off, since races
with high pip counts and fewer men to bear off are most efficient. Every
pip counting method makes adjustments of this nature, with the goal of
My backgammon program pushes this approach to the limit with
a more-or-less exhaustive enumeration of wastage. The estimator
can estimate pip counts with an average error less than 0.1 pips,
and a worst-case error of 1.0 pips in the extreme position (which for this
method is to have 15 men on the 7 point). So it is possible to push this
approach a very long way.
The next problem is volatility. The 10% rule is predicated on races that
have the usual volatility. But the volatility of some positions differs
because of "speed boards." A speed board is is crunched on the
lower points. Any doublet bears off 4 men, so this board is the cause
of many surprising come-from-behind victories. Other positions
(i.e. those maximally spread out) have lower volatility than usual, since
few doublets bear off 4 men.
To my knowledge, no published rules adjust for volatility in any way, let
alone a theoretically satisfying way. So we are theoretically
far away from having the practically perfect rule that you seek.
To some extent, LONG races preclude speed boards.
Yet it is still possible for volatility to affect equity. For example, I
believe that JellyFish spreads its men too much when bearing them in
prior to a race. The ideal is to have men on the 4, 5, and 6 points in ratio
3 to 5 to 7 men respectively. I consistently see men on the 1, 2,
and 3 points when JF prepares for a race. (Particularly when
JF is playing an "almost race," like midpoint-only contact.)
So rollouts are the only theoretically sound method of determining
whether to double in a race. In practice even rollouts are not
theoretically sound, since in theory checker play is imperfect, and
doubling during the rollout is imperfect. But that level of accuracy is not
required for practical understanding of a particular position.
I hope that this post has shown the gaps in our understanding. If you want
to extend the state-of-the-art, you can work on two areas. Probably most
important is to model racing equity in bearoffs. Another opportunity is to
tell us how to model volatility.
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