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# Early/Late Cost Ratio

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### Martin Krainer

Sep 25, 2003, 6:32:40â€¯AM9/25/03
to
Hello all,
Ive read now a very nice article from Tom Keith (Backgammon Match play
Strategy).
I have a question to the Early/Late Cost Ratio.
I understand, that when the ratio is 0.0 I can double anytime Im inbetween
the market window.
But i dont know exactly how to manage the other numbers.
Lets say Im 2 away and opponent is 8 away. When there are no gammon chances
I would like to have 83% to double.
Now the Cost ratio says at this score a number of 1.7.
How do I manage this number now?
Regards
Martin Krainer

### Tom Keith

Sep 25, 2003, 1:52:12â€¯PM9/25/03
to
Hi Martin.

First a brief history of the terminology. The "early/late cost ratio"
is a term I used the 1996 article you are referring to
(http://www.bkgm.com/articles/mpd.html). When the computer program
Snowie came out (1998), it included the ability to report this ratio
for a given match score. Snowie called it the "early-late ratio."

Douglas Zare has written a column on early-late ratio in the July 2002
issue of GammonVillage (http://www.gammonvillage.com/), which explains
the concept very nicely. (GammonVillage is available only by
subscription, but you'll find the \$20 for 3 months well worth your

One way to view the doubling decision in backgammon is that it depends
on four things:

1. Your current game-winning changes (GWC).

2. Opponent's take point (the point at which his equally well of

3. The volatility of the position (how much can change between now

4. The early-late ratio.

Your goal is to double as close to the opponent's take point as
possible. A double made at exactly the opponent's take point is said
to be an "efficient" double. Early doubles (where opponent has a
clear take) and late doubles (where the opponent has a clear pass) are
"inefficient." But inefficiency is not equal on both sides of the
take/drop line. That's the purpose of the early-late ratio, to
compare the inefficiency of doubling early to the inefficiency of
doubling late.

For example, in money play it turns out that the early-late ratio of
an INITIAL DOUBLE (when the cube is centered) is 0.5 -- an early
double is only half as inefficient as a late double. If opponent's
take point is, say, 75%, then doubling at 74% GWC is half as
inefficient as doubling at 76% GWC. Or, another way of looking at it,
doubling 2% early is just as good as doubling 1% late, and vice versa.

For a REDOUBLE (when you own the cube), the early-late ratio is 1.0.
An early redouble is just as inefficient as a late redouble. You are
equally well off redoubling 1% early as you are redoubling 1% late.

Why the difference? The reason is that owning the cube has value; as
long as you hold the cube your opponent can't use it. This fact has
recognized by players for a long time; the early-late ratio just puts
a number to it.

In match play things are more complicated. The early-late ratio
depends on not only on who owns the cube, but also on the score of the
match, the level of the cube, and the fraction of wins in the current
game that will be gammons. These are a lot of factors, and
unfortunately a single table doesn't encompass them all, so here are a
few tables for various match situations.

Early/Late Cost Ratio, Cube at 1, No Gammons

-2 -3 -4 -5 -6 -7 -8 -9 -10
----- ----- ----- ----- ----- ----- ----- ----- -----
-2: 0.00 0.84 1.72 1.35 0.89 1.02 1.04 1.05 0.93
-3: 0.29 0.85 1.42 0.94 0.76 0.77 0.83 0.79 0.77
-4: 0.36 0.74 1.06 0.88 0.71 0.72 0.74 0.74 0.72
-5: 0.35 0.51 0.76 0.64 0.61 0.62 0.65 0.65 0.65
-6: 0.22 0.37 0.56 0.56 0.56 0.58 0.61 0.62 0.63
-7: 0.27 0.33 0.48 0.47 0.50 0.52 0.56 0.57 0.58
-8: 0.22 0.29 0.40 0.42 0.45 0.48 0.51 0.53 0.55
-9: 0.25 0.28 0.39 0.40 0.43 0.46 0.49 0.50 0.52
-10: 0.19 0.26 0.36 0.38 0.42 0.44 0.47 0.49 0.50

Early/Late Cost Ratio, Cube at 1, 25% Gammons

-2 -3 -4 -5 -6 -7 -8 -9 -10
----- ----- ----- ----- ----- ----- ----- ----- -----
-2: 0.00 0.57 1.03 1.14 1.14 1.56 1.83 1.70 1.43
-3: 0.22 0.64 0.96 0.84 0.89 1.06 1.30 1.14 1.09
-4: 0.27 0.58 0.80 0.76 0.77 0.90 1.06 1.03 0.96
-5: 0.32 0.41 0.56 0.54 0.61 0.71 0.83 0.82 0.82
-6: 0.26 0.34 0.43 0.46 0.52 0.61 0.71 0.74 0.76
-7: 0.35 0.32 0.39 0.40 0.47 0.53 0.61 0.64 0.67
-8: 0.30 0.31 0.36 0.36 0.42 0.47 0.54 0.58 0.61
-9: 0.33 0.28 0.34 0.34 0.39 0.43 0.49 0.52 0.56
-10: 0.24 0.25 0.30 0.31 0.36 0.40 0.46 0.49 0.53

Early/Late Cost Ratio, Cube at 2, No Gammons

-3 -4 -5 -6 -7 -8 -9 -10
----- ----- ----- ----- ----- ----- ----- -----
-3: 0.48 0.94 1.41 1.79 2.48 3.21 2.83 2.62
-4: 0.39 0.91 1.29 1.63 2.15 2.75 2.60 2.38
-5: 0.41 0.78 1.04 1.32 1.70 2.18 2.06 2.07
-6: 0.30 0.63 0.85 1.06 1.37 1.75 1.79 1.81
-7: 0.31 0.58 0.75 0.92 1.17 1.47 1.52 1.59
-8: 0.25 0.51 0.67 0.82 1.02 1.27 1.37 1.43
-9: 0.25 0.47 0.59 0.72 0.89 1.10 1.19 1.28
-10: 0.20 0.41 0.53 0.64 0.78 0.96 1.06 1.15

Early/Late Cost Ratio, Cube at 2, 25% Gammons

-3 -4 -5 -6 -7 -8 -9 -10
----- ----- ----- ----- ----- ----- ----- -----
-3: 0.32 0.57 1.05 1.46 2.04 2.64 2.60 2.72
-4: 0.27 0.56 0.92 1.30 1.75 2.25 2.30 2.35
-5: 0.28 0.49 0.77 1.05 1.40 1.80 1.84 1.98
-6: 0.21 0.41 0.62 0.84 1.13 1.45 1.57 1.69
-7: 0.22 0.38 0.56 0.73 0.97 1.24 1.34 1.47
-8: 0.17 0.34 0.49 0.65 0.85 1.07 1.20 1.31
-9: 0.18 0.31 0.44 0.56 0.73 0.92 1.03 1.15
-10: 0.15 0.28 0.38 0.49 0.63 0.79 0.91 1.02

> Lets say I'm 2 away and opponent is 8 away. When there are no gammon

> chances I would like to have 83% to double.
> Now the Cost ratio says at this score a number of 1.7.
> How do I manage this number now?

The table presented in the article your refer to assumes that 20% of
the wins in the current game are gammons, so we can't use that here.
Instead refer to the first table above (initial double, no gammons).

This table says that the early-late ratio is 1.04. 1.04 is higher
than normal (normal being 0.5) and reflects the fact that you are
ahead in the match and will have to be more careful than usual about
letting the cube go too high. An early-late ratio of 1.04 is actually
about the same as for a REDOUBLE in money play.

Going back to factors that influence the doubling decision, the
primary indicators are your current game-winning chances (GWC) and the
opponent's takepoint (TP). You've computed your opponent's TP as 83%.

As long as you have no market losers (no chance that before your next
turn your GWC will exceed opponent's TP), then you have no decision to
make and can leave the cube alone. But once you have market losers,
you have to decide (a) how many there are, (b) how big they are, and
(c) how costly they are.

How big your market losers are depends on how close you are to
opponent's TP. It also depends on the volatility of the position. If
you are getting close to opponent's TP and the position is volatile,
then you will certainly have some big market losers. High volatility
argues for doubling early.

How costly your market losers are depends on the early-late ratio. If
you have a low E/L ratio, then losing your market is more costly than
normal and you want to be extra sure to double while your opponent is
still willing to take.

Notes

The Snowie definition of early-late ratio is not quite the same as my
definition. If A is the cost of doubling early and B is the cost of
doubling late, then my early-late ratios are reported as A/B. Snowie
early-late ratios are A/(A+B). Either method works; in both cases a
low number means you should double earlier than usual.

In his article on early-late ratio, Douglas Zare offers the following
suggestion for how you can relate the early-late ratio to actual play.
"The [Snowie] early-late ratio tells you roughly what fraction of
doubles should be passed in nongammonish positions. That is, roughly
1/3 of initial doubles should be taken, while 1/2 of the redoubles
should be taken." Perhaps this is a good argument for using the
Snowie definition of early-late ratio, since it gives you something to
relate to actual play.

Tom

### Martin Krainer

Sep 26, 2003, 4:06:28â€¯AM9/26/03
to
Thank you very much for this detailed answer, Tom.
I think i got a good idea of this concepts now.
Best Regards
Martin Krainer

"Tom Keith" <tom...@ETEbkgm.com> schrieb im Newsbeitrag
news:3F732BC8...@ETEbkgm.com...

### Douglas Zare

Sep 26, 2003, 2:27:35â€¯PM9/26/03
to

Tom Keith wrote:

> In his article on early-late ratio, Douglas Zare offers the following
> suggestion for how you can relate the early-late ratio to actual play.
> "The [Snowie] early-late ratio tells you roughly what fraction of
> doubles should be passed in nongammonish positions. That is, roughly
> 1/3 of initial doubles should be taken, while 1/2 of the redoubles
> should be taken."

Sorry, the first sentence is correct, and the second is not.
"Taken" should be replaced with "passed." It is more
commonly correct to pass a redouble than an intial double.

Douglas Zare

### Albert Silver

Sep 26, 2003, 7:13:48â€¯PM9/26/03
to
Douglas Zare <za...@math.columbia.edu> wrote in message news:<3F7487E5...@math.columbia.edu>...

Hmm... So instead of 1/2 of the redoubles should be taken, it should
read 1/2 of the redoubles should be passed?

Albert Silver

P.S. I understood because of your follow-up sentence, but it came
across as such a 'glass is half-full, half-empty' comment that I
couldn't resist. :-)

### Hank Youngerman

Sep 26, 2003, 6:48:23â€¯PM9/26/03
to
For other purposes, I had Jellyfish Level 5 playing against Snowie 3
3-ply in 9pt and 35pt matches. The full results are on a computer I
initial cubes and 290 recubes, 70% of initial doubles and 40% of
redoubles were taken.

It does stand to reason that more redoubles would be passed, since the
take decision is the same, but you redouble less aggressively than you
make an initial double.

But there is also no reason to believe that the set of positions that
present an initial cube are the same as the set of positions that
present a recube. I don't have any statistics of any sort about that.

That raises an interesting thought (OK, I'm rambling here). I wonder
if the Next Great Bot - Snowie, GNU, or whatever (my money is on GNU)
could have a selective export in a standard position format. The
format most likely would be a 26-character string (the 24 points plus
the Black and White bars) for which each character would have 31
possibilities (1-15 black checkers, 1-15 white checkers, or empty).
You'd find, I'm sure, almost no initial doubles made with a closed
board (you doubled long before you closed your board) but many
redoubles made with a closed board (you hit out of a low-anchor game).
This would be part of a wish-list of course, but wouldn't it be
interesting to know, for example, how often a 2-1 slot leads to
various types of games? You could roll out a 2-1 slot and then look,
for example, for positions that have well-timed backgames (defined by
some parameters).

### CuriousFellow

Sep 27, 2003, 7:53:30â€¯AM9/27/03
to
If I recall correctly there was a brief discussion in the gnubg digest some
10-15 issues ago about providing the ability to export in database format,
SQL support, or something along those lines. Perhaps somebody on the
development team can confirm and, if my recollection is correct, what
priority that has on the wish list?

"Hank Youngerman" <red...@redtopbg.com> wrote in message
news:nn99nv00egm6unnpt...@4ax.com...

### Louis Nardy Pillards

Sep 27, 2003, 8:39:35â€¯AM9/27/03
to
Hank Youngerman wrote:

> That raises an interesting thought (OK, I'm rambling here). I wonder
> if the Next Great Bot - Snowie, GNU, or whatever (my money is on GNU)
> could have a selective export in a standard position format. The
> format most likely would be a 26-character string (the 24 points plus
> the Black and White bars) for which each character would have 31
> possibilities (1-15 black checkers, 1-15 white checkers, or empty).
> You'd find, I'm sure, almost no initial doubles made with a closed
> board (you doubled long before you closed your board) but many
> redoubles made with a closed board (you hit out of a low-anchor game).
> This would be part of a wish-list of course, but wouldn't it be
> interesting to know, for example, how often a 2-1 slot leads to
> various types of games? You could roll out a 2-1 slot and then look,
> for example, for positions that have well-timed backgames (defined by
> some parameters).

Python is integrated in the no-gui version of gnubg (not GNU, since
that covers a lot, a lot more than GNU Backgammon, said gnubg).

And a 'standard position format'... gnubg Position ID, Match ID is not
suited?

### Hank Youngerman

Sep 27, 2003, 2:26:03â€¯AM9/27/03
to
I please "guilty" to ignorance. I am not that familiar with gnubg and
maybe some of what I suggested is already in there. I'll also be more
careful to call it gnubg in the future, you make a good point.

On 27 Sep 2003 12:39:35 GMT, "Louis Nardy Pillards" <nardy dot

### Louis Nardy Pillards

Sep 27, 2003, 11:44:24â€¯AM9/27/03
to
Hank Youngerman wrote:

smile...

remembers me of pleading "unguilty", yet being convicted.

nardy

### Jim Segrave

Sep 27, 2003, 11:57:31â€¯AM9/27/03
to
In article <_Sedb.99201\$Lnr1....@news01.bloor.is.net.cable.rogers.com>,

CuriousFellow <Curiou...@here.now> wrote:
>If I recall correctly there was a brief discussion in the gnubg digest some
>10-15 issues ago about providing the ability to export in database format,
>SQL support, or something along those lines. Perhaps somebody on the
>development team can confirm and, if my recollection is correct, what
>priority that has on the wish list?

There is a lot of interest among the developers. It has a high
priority in the sense that several developers think it's a good idea
and some work has been done to identify what might or might not go
into a first cut at such a database.

But putting a date on it is much harder - I may be the one expected to
do this, and, at the moment, time is at a premium.

--
Jim Segrave j...@jes-2.demon.nl

### Paul Tanenbaum

Oct 9, 2003, 6:38:26â€¯PM10/9/03
to
Tom Keith <tom...@ETEbkgm.com> wrote in message news:<3F732BC8...@ETEbkgm.com>...

> First a brief history of the terminology. The "early/late cost ratio"
> is a term I used the 1996 article you are referring to
> (http://www.bkgm.com/articles/mpd.html). When the computer program
> Snowie came out (1998), it included the ability to report this ratio
> for a given match score. Snowie called it the "early-late ratio."
>
> Douglas Zare has written a column on early-late ratio in the July 2002
> issue of GammonVillage (http://www.gammonvillage.com/), which explains
> the concept very nicely.
>
> One way to view the doubling decision in backgammon is that it depends
> on four things:
>
> 1. Your current game-winning changes (GWC).
>
> 2. Opponent's take point (the point at which his equally well of
> taking or dropping your double).
>
> 3. The volatility of the position (how much can change between now
>
> 4. The early-late ratio.
>
> Your goal is to double as close to the opponent's take point as
> possible. A double made at exactly the opponent's take point is said
> to be an "efficient" double. Early doubles (where opponent has a
> clear take) and late doubles (where the opponent has a clear pass) are
> "inefficient." But inefficiency is not equal on both sides of the
> take/drop line. That's the purpose of the early-late ratio, to
> compare the inefficiency of doubling early to the inefficiency of
> doubling late.
>
> For example, in money play it turns out that the early-late ratio of
> an INITIAL DOUBLE (when the cube is centered) is 0.5 -- an early
> double is only half as inefficient as a late double. If opponent's
> take point is, say, 75%, then doubling at 74% GWC is half as
> inefficient as doubling at 76% GWC. Or, another way of looking at it,
> doubling 2% early is just as good as doubling 1% late, and vice versa.

Tom, I don't subscribe to Gammonvillage, but can you elaborate?
Because I don't see it.

Let me offer a counterexample argument: Assume the continuous
game model with no gammons. Then each player's optimal
doubling point is at 80% winning chances. At that point, the
monetary expectation is exactly +1. Now the most natural
definition of inefficiency (I) is simply the deviation from +1.

i) For example, place A as a 82% favorite (he missed his market).
would expect:

2 * (62 - 18)/80 = \$1.1

since A has to reach 100% to win (from 82%), whereas B only
needs to reach 80%.

Thus I = 1.1 - 1 = \$.1

ii) Now give A a 78% GWC. He doubles, B takes, and his
expectation becomes:

2 * (58 - 22)/80 = \$.9

and I = 1 - .9 = \$.1

N.B.: 78% and 82% are symmetrical around 80%, yet we see the
SAME inefficiency.

> For a REDOUBLE (when you own the cube), the early-late ratio is 1.0.
> An early redouble is just as inefficient as a late redouble. You are
> equally well off redoubling 1% early as you are redoubling 1% late.
>
> Why the difference? The reason is that owning the cube has value; as
> long as you hold the cube your opponent can't use it. This fact has
> recognized by players for a long time; the early-late ratio just puts
> a number to it.

Of course the doubling points change with cube possession, but I
don't see how it affects this ratio thing.

---
Paul T.

### Tom Keith

Oct 9, 2003, 10:56:19â€¯PM10/9/03
to
Paul Tanenbaum wrote:
> Tom, I don't subscribe to Gammonvillage, but can you elaborate?
> Because I don't see it.

The idea of early-late ratio is to answer the question: Which is a
better strategy, to aim a little bit early with the cube, or aim a
little bit late?

Let's compare. Assume an efficient doubling model and no gammons.
We'll take two players with different strategies: Player A always
doubles at 79% (1% too early); Player B always doubles at 81% (1% too
late). Who will do better in the long run?

PLAYER A

Player A doubles at 79% GWC. All of A's doubles are taken. After a
double, Player A goes on to win the game when his GWC reach 100%. He
goes on to lose the game when his GWC drop to 20% (opponent will
redouble him out). So he wins two points 59/80 of the time, and he
loses two points 21/80 of the time. A's equity after doubling is

\$2*(59/80) - \$2*(21/80) = \$.95

How does this affect A's overall equity? From the beginning of the
game (50% GWC), Player A reaches his doubling point (79% GWC) 30/59 of
the time. The other 29/59 of the time, he loses one point. So A's
average winnings of all the games he plays is:

\$.95*(30/59) - \$1*(29/59) = \$-.00847

PLAYER B

Player B doubles at 81% GWC. All of B's doubles are dropped. That
means at 81% GWC, B's equity is \$1. From the beginning of the game
(50% GWC), Player B reaches his doubling point (81% GWC) 30/61 of the
time. The other 31/61 of the time, he loses one point. So B's
average winnings of all the games he plays is:

\$1*(30/61) - \$1*(31/61) = \$-.01639

RESULT

Perfect play would yield an equity of \$0. A strategy of doubling a
little bit early (as Player A does) loses about half as much equity as
a strategy of doubling a little bit late (as Player B does).

---

> Of course the doubling points change with cube possession, but I
> don't see how it affects this ratio thing.

All of the above is for an initial double. For a redouble, the
opponent won't win until he reduces your GWC to 0, because he can't
double you out. Assume the players start off with 20% GWC (they just
took the cube). The equations above change as follows (still using a
1-cube):

Player A wins on average: \$.95*(20/79) - \$1*(59/79) = \$-.50633
Player B wins on average: \$1*(20/81) - \$1*(61/81) = \$-.50617

Perfect play owning a 1-cube from 20% GWC wins on average \$-.5. The
strategies of Players A and B are now about equally inefficient.

Does that explain it?

Tom

Oct 10, 2003, 4:35:20â€¯AM10/10/03
to
Tom Keith <t...@bkgm.com> wrote in message news:<3F86205E...@bkgm.com>...

Yes that explains the technicality of it all but can you show us some examples
of real life play where this information might be useful?

### Tom Keith

Oct 11, 2003, 9:50:23â€¯AM10/11/03
to
>
> Yes that explains the technicality of it all but can you show us some examples
> of real life play where this information might be useful?
>

Well, I'm certainly not suggesting that anyone go out and memorize
tables of early-late ratios. But it is useful to have a sense of when
your early-late ratio low and when it is high.

In money play, you should be more cautious about offering a recube
than an initial cube. For example, in a medium-length race you need
to have a one pip greater lead in the race to redouble than to offer
an initial double. (See http://www.bkgm.com/rgb/rgb.cgi?view+699)
The reason for the extra caution is the higher early-late ratio when
you own the cube.

In match play, there are some scores which have a very low early-late
ratio. Having a low early-late ratio means you want to be very
aggressive with the cube -- you want to try extra hard to get that
double in before you lose your market.

The most extreme example is two-away/two-away. The early-late ratio
at this score is zero -- so you should double as early as you can (or
at least before you have ANY chance of losing your market).

Early-late ratio takes on greater importance in high-volatility
positions. High volatility means that you may be forced to double
earlier than normal and factoring early-late ratio can help lead to
the right decision when the choice is close.

But early-late ratio is generally a secondary consideration. The most
important thing to know is your chances in the game and what your
opponent's take point is.

Tom

### Paul Tanenbaum

Oct 14, 2003, 4:10:39â€¯PM10/14/03
to
Tom Keith <t...@bkgm.com> wrote in message news:<3F86205E...@bkgm.com>...
> ...

> The idea of early-late ratio is to answer the question: Which is a
> better strategy, to aim a little bit early with the cube, or aim a
> little bit late?
>
> Let's compare. Assume an efficient doubling model and no gammons.
> We'll take two players with different strategies: Player A always
> doubles at 79% (1% too early); Player B always doubles at 81% (1% too
> late). Who will do better in the long run?
>
> PLAYER A ...
> So A's average winnings is: \$-.00847
>
> PLAYER B ...
> So B's average winnings is: \$-.01639

Tom, thanks for the response, your effort is appreciated.

Although our models differ, each is internally consistent. Mine
compares the equities after doubling, while yours looks at the
strategy before doubling, during the play, which is more
applicable to the game.

As an alternative formulation, rather than comparing doubling
points symmetrical around 80%, one might relate a set of
corresponding early vs late points that produce equivalent
equities. In your example, if A knows his GWC is going to jump
to 81%, he doubles at 78.1%, which results in the same equity (and inefficiency).

---
Paul T.

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