Forum Archive :
Cube Handling in Races
Alexander Nitschke wrote:
> One additional comment: In money game 15 checkers each on the ace point
> is an initial double. So the trailer must be more reluctant to double
> than in money game.
I believe you, but this surprises me. I thought the "rule" was to wait
until it's five roll vs. five roll for the initial double, and four roll
vs. four roll for the redouble. Could someone please elaborate?


Sam Pottle writes:
Actually, Kleinman discusses this in _Vision Laughs At Counting_, so it's
not exactly new theory. Until I read Kleinman I thought as you did,
probably because Robertie's treatment in _Advanced Backgammon_ only went up
to the 5roll v. 5roll position. Robertie never actually says, though,
that the 5 roll position is the earliest initial double.
How should we approach the double/nodouble decision in a pure 8roll v.
8roll position?
Well, first of all, we're a solid favorite (about 67% cubeless), so an
initial double is at least worth thinking about. We lose our market
whenever we roll doubles and our opponent rolls nondoubles (about 14% of
the time). These are big market losers, too  our winning chances in the
resulting 6roll v. 7roll position are over 90%.
Average sequences (both players roll doubles, or both roll nondoubles)
wash; the resulting 6roll v 6roll or 7roll v. 7roll positions are still
double/take.
What about our poor sequences? We'll be sorry we doubled whenever we roll
nondoubles, opponent rolls doubles, and we roll nondoubles (about 11.5%).
Opponent will then be on roll in a 6roll v. 6roll position, leaving her
with a strong advantage, but still a roll or two away from a recube. In
fact, if we've doubled, opponent will hold the 2cube, while if we haven't,
opponent will give us an initial double (which we'll take), so that we will
be holding a 2cube. So in this variation, having doubled costs us
precisely the value of cube ownership of a 2cube in a 6roll v. 6roll
position.
How much is that cube ownership worth? One clue that it's pretty valuable
is the fact that opponent shouldn't redouble before reaching a 4roll v.
4roll position, while she has a correct initial double already. Also,
that recube is pretty efficient when it comes, and it frequently does come.
So, which costs more, the big market loser, or the big loss of cube
ownership? Let's assume that they're about the same. Then the fact that
our bad sequences are only 5/6 as numerous as our good ones tips the
balance in favor of doubling.
Let's put some numbers to these notions. Our equity after a good sequence
is about .80 (1.60 on a 2cube), so losing our market costs us about .60
when it happens. Our equity with opponent on roll in a 6roll v. 6roll
position is about .30 (.60 on a 2cube) if we own the cube, and about
.65 (1.30) if opponent owns the cube. So having doubled costs us about
.70 in these variations. Since .60 is bigger than 5/6 of .70, doubling
turns out to be correct by a small margin.
The margin is indeed quite small, as David Montgomery pointed out: .452 if
you double, .450 if you don't. But the double is technically correct.
Bonus questions:
(Assume we're allowed more than 15 checkers per side.)
 What is the smallest number n for which an nroll v. nroll
position is not a proper initial double?
 What is the smallest number n for which an nroll v. (n+1)roll
position is a take?


Steven Turner writes:
> What is the smallest number n for which an nroll v. nroll
> position is not a proper initial double?
It's n = 9 (the smallest n which can't occur in real backgammon).
33 v. 33 is a beaver.
> What is the smallest number n for which an nroll v. (n+1)roll
> position is a take?
n = 18.
25 v. 26 is an initial double but not a redouble.
30 v. 31 isn't even an initial double.
193 v. 194 is a beaver.
Of course anyone who plays backgammon with 387 chequers per side deserves
everything they get.


Jason Lee writes:
193 rolls vs. 194 rolls is a beaver.
What seems counterintuitive at first does make sense, I think ... in such a
looooooooong race, ownership of the cube is extra valuable, since over that
long period, you're likely to get a pretty efficient cube at some point. Do
I have that right?
JLee


Sam Pottle writes:
Yes. A 200 turn race is much closer to the continuous model of backgammon
than any normal position. In the continuous model, you can beaver with
cubeless chances of 40% (halfway between zero and the 80% cashpoint). With
194 vs. 193 rolls, the beavering side has a little under 41%, and the race
is so "smooth" that this is enough.


Tom writes:
Enjoyed this immensely, but can you provide some insight as to the
methodology you used to arrive at these figures, ie. how did you come up
with the numbers?


Stephen Turner writes:
I just wrote a little program to calculate them:
#include <stdio.h>
#define N (500)
int main(){
int i, j, t;
double eq1[N][N], eq2[N][N], eq0[N][N];
for (t = 0; t < N; t++) {
for (i = 0; i <= t; i++) {
j = t  i;
if (j == 0) {
eq0[i][j] = 1.0;
eq1[i][j] = 1.0;
eq2[i][j] = 1.0;
}
else if (i <= 1) {
eq0[i][j] = 1.0;
eq1[i][j] = 1.0;
eq2[i][j] = 1.0;
}
else {
eq0[i][j] = 5 * eq0[j][i  1] / 6  eq0[j][i  2] / 6;
eq1[i][j] = 5 * eq2[j][i  1] / 6  eq2[j][i  2] / 6;
eq2[i][j] = 5 * eq1[j][i  1] / 6  eq1[j][i  2] / 6;
}
if (2 * eq2[i][j] > eq1[i][j]) {
if (2 * eq2[i][j] > 1.0) {
printf("%3d%4d [%7.4f:%7.4f:%7.4f] redouble/drop\n",
i, j, eq0[i][j], eq1[i][j], eq2[i][j]);
eq1[i][j] = 1.0;
eq0[i][j] = 1.0;
}
else {
printf("%3d%4d [%7.4f:%7.4f:%7.4f] redouble/take\n",
i, j, eq0[i][j], eq1[i][j], eq2[i][j]);
eq1[i][j] = 2 * eq2[i][j];
eq0[i][j] = 2 * eq2[i][j];
}
}
else if (2 * eq2[i][j] > eq0[i][j]) {
printf("%3d%4d [%7.4f:%7.4f:%7.4f] initial double/take\n",
i, j, eq0[i][j], eq1[i][j], eq2[i][j]);
eq0[i][j] = 2 * eq2[i][j];
}
else if (eq2[i][j] >= 0)
printf("%3d%4d [%7.4f:%7.4f:%7.4f] no double/take\n",
i, j, eq0[i][j], eq1[i][j], eq2[i][j]);
else
printf("%3d%4d [%7.4f:%7.4f:%7.4f] no double/beaver\n",
i, j, eq0[i][j], eq1[i][j], eq2[i][j]);
}
printf("\n");
}
return(0);
}




Cube Handling in Races
 Bower's modified Thorp count (Chuck Bower, July 1997)
 Calculating winning chances (Raccoon, Jan 2007)
 Calculating winning chances (OpenWheel+, Nov 2005)
 Doubling formulas (Michael J. Zehr, Jan 1995)
 Doubling in a long race (Brian Sheppard, Feb 1998)
 EPC example (adambulldog+, Jan 2011)
 EPC example: stack and straggler (neilkaz+, Jan 2009)
 EPC examples: stack and straggler (Carlo Melzi+, Dec 2008)
 Effective pipcount (Douglas Zare, Sept 2003)
 Effective pipcount and type of position (Douglas Zare, Jan 2004)
 Kleinman count (Øystein Johansen+, Feb 2001)
 Kleinman count (André Nicoulin, Sept 1998)
 Kleinman count (Chuck Bower, Mar 1998)
 Lamford's race forumla (Michael Schell, Aug 2001)
 Nroll vs nroll bearoff (David Rubin+, July 2008)
 Nroll vs nroll bearoff (Gregg Cattanach, Nov 2002)
 Nroll vs nroll bearoff (Chuck Bower+, Dec 1997)
 Near end of game (Daniel Murphy, Mar 1997)
 Near end of game (David Montgomery, Feb 1997)
 Near end of game (Ron Karr, Feb 1997)
 One checker model (Kit Woolsey+, Feb 1998)
 Pip count percentage (Jeff Mogath+, Feb 2001)
 Pipcount formulas (Tom Keith+, June 2004)
 Thorp count (Chuck Bower, Jan 1997)
 Thorp count (Simon Woodhead, Sept 1991)
 Thorp count questions (Chuck Bower, Sept 1999)
 Value of a pip (Tom Keith, June 2004)
 Ward's racing formula (Marty Storer, Jan 1992)
 What's your favorite formula? (Timothy Chow+, Aug 2012)
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