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Chuck Bower wrote:
> For many years, Thorp was the best available method for 'typical'
> bearoffs. Compared to guessing, and even vs. percentage methods of
> the day, it was pretty good.
I think the main reason the Thorp count is better than guessing is
that it contains the 10%+-2 pips rules:
A lead of 10%-2 pips is a borderline initial double.
A lead of 10%-1 pip is a borderling redouble.
A lead of 10%+2 pips is a borderline take.
This is pretty accurate from about 50 to 110 pips. On either side, the
trailer must be more conservative.
This much is good. However, in addition to these, Thorp contains a lot
of adjustments. I think these adjustments were chosen to agree roughly
with a few calculable positions at the end of the bearoff. However,
some of these adjustments are bad in general. I don't believe they
were tested on typical positions, since it is quite boring to roll out
a bear-in position enough to figure out whether it wins 24% or 21%.
With computers, this is easy, and it is time to move on from the Thorp
count to simpler, more accurate systems.
Effective Pipcount and Wastage
The effective pip count is the average number of rolls needed to bear
off times the average roll (8.167 pips). This assumes that the goal is
to bear off in as few rolls as possible on average, which usually
coincides with the correct strategy for beating your opponent.
XX | | | |
XXX | | | |
XXX | | | |
XXX | | | |
XXX | X | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+
OFF 1 2 3 4 5 6 7 8 9 10 11 12
This position takes an average of 1.25 rolls to bear off.
The effective pip count is 1.25 x (49/6) = 10.208.
Since the actual pip count is 6, the average wastage is 4.208 pips.
Some of the potential wastage comes on the first roll, if it is more
than 6 pips. If the first roll is less than 6 pips, it is likely that
most of the second roll will be wasted.
| | | |
| | | |
XX | X X | | |
XXX | X X | | |
XXX | X X X | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+
OFF 1 2 3 4 5 6 7 8 9 10 11 12
This position takes an average of 3.97 rolls.
The effective pip count is 32.425.
The actual pip count is 20, so the average wastage is 12.425.
Several programs have databases of effective pip counts: gnubg,
Trice's Bearoff Quizmaster, and Zbot (alpha).
Jørn Thyssen asked:
> How do you calculate effective pip counts over the board?
You add an estimate of the wastage to the nominal pip count. The
wastage is a bit more than 7 for an efficient position with 15
checkers. To that you add the basic linear EPC adjustments:
1 for the second checker on the ace point.
2 for each additional checker on the ace point.
1 for the third and additional checkers on the deuce point.
Usually, this means not to make any adjustment, unlike the count of
checkers and points in the home board for each side, which are
positive.
In most positions, I just add the excess of the wastage over that of a
typical efficient position to the pip count, e.g., I might count 70
pips plus 1 pip of extra wastage.
I believe Walter Trice discovered that a pure n-roll position has an
epc of very close to 7n+1, e.g., an 8-roll position has an epc of 57.
This is comparable with an efficient position with a nominal pip count
of 50, such as the one from the online match, 6^4 5^3 4^2 3.
Very efficient positions with only a few checkers on low points are
easy to understand. So, too, are pure n-roll positions. Between the
two, it is tougher to estimate the epc accurately. I estimate the
"synergy," the nonlinear part of the wastage, using reference
positions and experience. The synergy is typically small because the
basic adjustments are good.
Example A:
OFF 24 23 22 21 20 19 18 17 16 15 14 13
+---+---+---+---+---+---+---+---+---+---+---+---+---+
OOO | O O O O | | |
OOO | O O | | |
OO | O | | |
| | | |
| | | |
| | | |
| X | | |
| X | | |
XX | X X X | | |
XXX | X X X X X | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+
OFF 1 2 3 4 5 6 7 8 9 10 11 12
A) 1111223445 vs. 2445666: O has a pip count of 33, and a fairly
nice distribution. I figured the wastage was 6.5 pips, for an epc of
39.5. A database says 6.774, so 39.774 effective pips. X has 10
checkers, so is at best a 5-roll position, which would be 7*5+1 = 36
effective pips. Each miss and nonworking double (out of 36) on each
roll costs 0.2 effective pips, and there are a lot of those: 3-3 is a
double miss, while 2-2 and 4-4 are not working doubles this turn, and
may not be for a few turns. 1-1 would take off 4 checkers, but would
set up many future misses. Repeated 3s would miss. I figured that this
was about 2.5 pips worse than a 5-roll position, but a database says
it is 3.213 pips worse, for a total of 39.213 effective pips. O trails
by the roll and about a half pip, and wins 30.3% of the time.
An exact cubeful database gives the following equities:
ND: 0.622 NR: 0.661 DT: 0.656
This is an initial double, but not quite a redouble, although it is
very close. OTB, my calculations should have led me to redouble, but I
didn't since I thought it was too easy a take, and a particular crew
member might take way too deeply.
Example B:
OFF 24 23 22 21 20 19 18 17 16 15 14 13
+---+---+---+---+---+---+---+---+---+---+---+---+---+
OOO | O O | | |
OOO | O O | | |
OOO | O | | |
O | | | |
| | | |
| | | |
| X | | |
X | X | | |
XXX | X X | | |
XXX | X X X X | | |
+---+---+---+---+---+---+---+---+---+---+---+---+---+
OFF 1 2 3 4 5 6 7 8 9 10 11 12
B) 11112234 vs. 44666: Now, X is much closer to a 4-roll position.
2-2 and 3-3 don't work, but they might work next turn, and it is hard
to miss without rolling doubles. I guessed X is 0.8 pips worse than
a 4 roll position, and a database says 0.936 pips, for an epc of
29.936. O's position is efficient as before, wasting 6.527 pips, for
an epc of 32.527. Now that X leads by 2.5 pips, the pass is clear,
and O only wins 19.5% of the time.
ND: 0.830 NR: 0.851 DT: 1.159
OTB, the field passed.
I presented these positions together because I didn't find it
intuitive that more than a third of O's winning chances had evaporated
on the 5-4 5-2 exchange. X only gained 2 nominal pips, and 2
effective pips, but this was the distance between a borderline
redouble and a solid 1.159 pass.
Douglas Zare
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