Match Equities

Forum Archive : Match Equities

Which match equity table is best?

From:   Martin Krainer
Date:   14 October 2003
Subject:   Question to different MET's
Google:   3f8bc29b$0$24880$

I know that Kit's MET is very popular, but there are some other
interesting ones like from Jacobs, Kazaross, Zadeh and so on.
Well, I know they differ not very much in the most cases, but for example
in 1away, 2away there are differences up to 1,5-3% (many have 68,5 there,
and Kit has 70). That could make a big difference in an analyze.
What causes these differences? (just the estimated chances for gammons?)
What MET do you prefer?


Douglas Zare  writes:

It can make a big difference relative to the value of one point in
the match. However, this only happens when you hit certain
decisions at certain scores. Most of the time, your decisions
should not depend on which MET you use, and when they do, it
is rare that the difference will decide the match.

Several empirical tests by Joseph Heled have found that modern
tables only outperform the Woolsey-Heinrich table by roughly
50.1-49.9 in 7 point matches, or about 1 elo point.

Nevertheless, all common match equity tables agree that gammons
are more valuable at 5-away 5-away and 4-away 5-away than for
money. Particularly if you use a bot to analyze match play, you
need to understand what the bot's match equity table says about
the match score, or else you will get noisy feedback from the bot.

> What causes these differences? (just the estimated chances for
> gammons?) What MET do you prefer?

In constructing match equity tables, some people use more or less
complicated mathematical models. The mathematical models
do not provide enough complexity, sometimes ignoring imperfect
cube efficiency, sometimes ignoring the possibility of becoming too
good to double, and sometimes assuming that the checker play is
the same as for money play. Sometimes the mathematical models
produce silly results like a lower equity for the trailer at 2-away
4-away than at Crawford 2-away. However, these errors may
cancel. That the model was too simple doesn't mean the MET is

Some people used empirical evidence to construct match equity
tables. There was not enough data collected to give accurate
estimates for many scores. I believe the Woolsey-Heinrich
table used results of over a thousand matches just for the 5x5
table, and the uncertainties were in cases greater than a percent.
There wasn't enough data to give an accurate, consistent match
equity table for longer matches. Further, much of the evidence
collected for the Woolsey-Heinrich table was collected from
people who might have been viewed as experts at the time, but
whose play was far below what we expect of experts today,
particularly with regards to match play. Also, I believe Kit
Woolsey stated that he smoothed out the entries to make them
easier to remember, but I think later analysis shows the 3-away
4-away entry in the W-H MET is wrong, and there should _not_
be a smooth progression of 2-away 3-away, 3-away 4-away,
4-away 5-away, and 5-away 6-away.

To me, it is important that a match equity table give values in
tenths of a percent, rather than just to the nearest percent. It is
not that I usually judge positions within tenths of a percent,
but roundoff errors in the entries get magnified when you use
the entries to compute take points and gammon prices. An
error of 0.5% may get magnified to 4%, worth about 2 pips
in the race and 0.200 after D/T, at some match scores. This is
far more important for longer matches and lopsided match
scores than for scores within the 5-point match.

The gammon rate assumed in the creation of a MET is
important. A higher gammon rate tends to favor the trailer,
with some exceptions such as 4-away 5-away and a decision
at 2-away 3-away. A table based on a higher gammon rate
will tend to have lower gammon prices for the trailer (e.g.,
post-Crawford), since the trailer will have less urgency to
win a gammon now if a later gammon is likely. I think the
26% gammon rate assumed by modern tables such as
Snowie's table is much more accurate than the ~21% assumed
for the W-H table.

The W-H table has all of these faults: A bad entry at 3-away
4-away, only 2 digits of precision, an assumed gammon rate
of 20%. However, it was not convincingly beaten in tests,
appearing only 1 elo point worse, which is impressive to no
one. Since the more modern tables disagree with each other far
less than they disagree with the W-H table, I think it is not of
practical value to look for improvements on the modern
METs. It is far more important to start using some MET than
to switch to a better one. So, I start my backgammon students
off with the W-H MET, albeit with the 3-away 4-away entry
crossed out.

Douglas Zare

Martin Krainer  writes:

Hello Douglas,

Thanks much for this great answer. Very interesting was, that you said
that the 26% gammon rate is more accurate than 20 or 21%.

Why dont we know already exactly about the gammon chances? Arent the
PC's already fast enough, that we could say: Its nearly for sure a
number between 25 and 27 %?

Douglas Zare  writes:

There are a couple of issues.

First, we can determine the gammon rate for any particular bot
and setting, but they don't have to agree with each other. Gregg
Cattanach posted a rollout on Gammonline suggesting that the
gammon rate for Snowie 4 2-ply standard is about 27.2%+-0.3%.
This would not be the same for other bots, though. Snowie 4 has
a style that leads more backgames and to fewer blitzes than
Snowie 3. Jellyfish, on the other hand, avoids backgames even
more, but it bears off too aggressively after closing a checker out.
These style differences may make a large difference in the
gammon rate without making a large difference in the playing

Second, the gammon rate in money play may not be the same
rate in match play, so it might not be the right thing to use when
making a match equity table. For example, I think it is clear to
play very aggressively for the gammon trailing Crawford 2-away.
The leader doesn't need to trade any wins for gammons, and can
concentrate on winning and saving the gammon. Does the net
effect of the trailer's altered strategy and the leader's altered
strategy favor one side or the other? This is not something to
guess a priori. Some rollouts suggest that the trailer wins
slightly less than 50%, but so many more gammons that the trailer
wins more than 32% of the matches, which is what you would get
if you assumed no alteration, but a 28% gammon rate.

I'm not sure what happens at other scores, but I am very
suspicious of the assumption that the bg/g/w distribution from
money play carries over to match play.

Here is an example: Snowie's MET says the trailer wins 25.22%
from Crawford 3-away. This appears to come from ignoring the
value of the free drop, and assuming that the trailer wins a
backgammon 0.9% of the time. The free drop is certainly worth
something, and so many METs say the equity should be less than
25%. However, at Crawford 3-away, gammons are so
unimportant that the checker play is close to that at DMP, when
one more frequently plays backgames and ace point games, which
lead to many more backgammons. At Crawford 3-away, the
backgammons are worth something, and they should not happen
only 0.9% of the time, even though the leader need not risk a
backgammon to save the gammon. I think the net effect is larger
than the value of the free drop.

So, in conclusion, we don't really know how many gammons ought
to be won in money play, but if we did, it would not necessarily
let us improve our match equity tables.

Douglas Zare
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Match Equities

Constructing a match equity table  (Walter Trice, Apr 2000) 
Does it matter which match equity table you use?  (Klaus Evers+, Nov 2005)  [GammOnLine forum]
Does it matter which match equity table you use?  (Achim Mueller+, Dec 2003) 
Does it matter which match equity table you use?  (Chuck Bower+, Sept 2001)  [GammOnLine forum] [Long message]
ME Table: Big Brother  (Peter Fankhauser, July 1996) 
ME Table: Dunstan  (Ian Dunstan+, Aug 2004)  [GammOnLine forum]
ME Table: Escoffery  (David Escoffery, Nov 1991) 
ME Table: Friedman  (Elliott C Winslow, Oct 1991) 
ME Table: Kazaross  (Neil Kazaross, Dec 2003)  [GammOnLine forum]
ME Table: Kazaross-XG2  (neilkaz, Aug 2011) 
ME Table: Rockwell-Kazaross  (Chuck Bower+, June 2010) 
ME Table: Snowie  (Chase, Apr 2002) 
ME Table: Snowie  (Harald Retter, Aug 1998) 
ME Table: Woolsey  (Raccoon, Apr 2006) 
ME Table: Woolsey  (Kit Woolsey, May 1994) 
ME Table: Woolsey  (William R. Tallmadge, Jan 1994) 
ME Table: Zadeh  (Jørn Thyssen, Mar 2004)  [GammOnLine forum]
ME Table: Zorba  (Robert-Jan Veldhuizen+, Dec 2003) 
ME at 1-away/2-away (crawford)  (Fabrice Liardet+, Nov 2007)  [GammOnLine forum]
ME at 1-away/2-away (crawford)  (Ian Shaw+, Apr 2003)  [GammOnLine forum]
Match equities--an alternate view  (Durf Freund, Oct 1994) 
Neil's new numbers  (neilkaz, Aug 2011) 
Neil's numbers  (Kit Woolsey+, Oct 1994) 
On calculating match equity tables  (Neil Kazaross, July 2004)  [GammOnLine forum]
Turner formula  (Gregg Cattanach, Feb 2003) 
Turner formula  (Stephen Turner, June 1994) 
Using a match equity table  (Michael J. Zehr, June 1992) 
Value of free drop  (Neil Kazaross, Oct 2002)  [GammOnLine forum]
Which match equity table is best?  (Martin Krainer+, Oct 2003) 
Which match equity table is best?  (Ian Shaw+, Dec 2001) 
Why use a match equity table?  (Kit Woolsey, Feb 1999) 
Worth memorizing?  (Alef Rosenbaum+, Feb 2003) 

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