Match Equities

Forum Archive : Match Equities

On calculating match equity tables

From:   Neil Kazaross
Date:   5 July 2004
Subject:   Re: OLM 4 July 04 -- cube (Very long)
Forum:   GammOnLine

For those who continue to cling to old METs that use 20% gammon rates and
don't have any bias for checker play at scores like GG and are calculated
via mathematical methods, I'll make a final and long winded attempt to
enlighten them.

1) My understanding is that the 20% gammon percentage accepted as gospel
for so many years comes from an offhand comment by Magriel back in the
70's. Some one asked him how often a gammon can be won and he said that he
thought about 20% of all games end in gammons. (But he didn't have any

2) Every bot rollout played cubeless for money from the starting position
(not allowing starting doubles, off course) that I've ever seen results in
26.7 or 26.8 % of all games ending in gammons or BGs. This is really some
pretty powerful data especially as more and more players learn and play
more like the bots do. This includes JF, Snowie, GNU, and even TD gammon
way back. The assumption that current humans somehow should win
considerably fewer gammons than the bots do is just plain wrong.

3) I don't recall all the factors and stats from Hal's database that
resulted in the W-H table. The W-H table was revolutionary for its time
since it showed somewhat more chances to come back when trailing by a
large ammount than was previously believed (based upon calculated tables
with a 20% gammon rate). I recall a 22% gammon figure from Hal's database
but cannot recall whether that figure was from Crawford/Post Crawford
scores where G's mattered or was from the entire database. (I seem to
recall 26% from the database for all games and that the sample size of
Crawford/PC games wasn't enormous but said 22%) One cannot rely on gammon
percentages from games where cubes were used since we don't know whether
gammonish games were doubled early or passed. As for 22% percentage in
one-sided gammon situations, I don't recall that database being large
enough for statistical significance, and just as important is the fact
that most players a decade ago, didn't come close to bearing off
aggressively enough to win more gammons. Also many players didn't blitz
enough. Anyhow, even if 22% is correct, it is more than 20% and doesn't
factor in what I call "checker play bias". What would you believe, a
database of a few thousand matches from a decade ago, or millions of games
played by bots today ?

4) Checker play bias at GG and GS scores comes in several ways. The
trailer who cannot lose a G can simply fall back and play whatever
defensive ace point game or backgame that he is able to get into. He never
has to worry about running to save the gammon and will, therefore get a
few more wins than he would if he had to run. The trailer doesn't have to
worry about losing a gammon and can simply pound away very aggressively
and also doesn't have to wimp it up and sit on anchors like the leader
does. The leader needs to think twice anytime he's considering leaving
several blots as that might lead to a gammon loss. When I made my first
equity table over 25 years ago, I thought about how obvious checker play
bias is. I read somewhere that Magriel said that 20% of all games end in
G's so I used that figure to get a GG ME of 30% but since I felt that
"checker play bias" meant so much, I raised that figure to 31% and used
that figure for the next 20 years. I now think that adding 1% is a bit too
much, noting that a fine balance exists at GG/GS where the leader can also
play to simply and make an advanced anchor. However, checker play bias
does exist, IMHO, and if we assume only a 26% gammon percentage we get an
equity of 31.5% for GG and I now think that GG equity is at least 32%,
partly due to checker play bias, partly due to gammon percentage over 26%.

5) Calculating an MET via mathematical methods was the only way in the PB
era (Pre-Bot). Even with a huge sample of matches played, one had to take
into account that the player leading in the match was often slightly
better than his opp and that would skew the stats and make comebacks less
likely than they truly are. One problem with many calculated MET's (Snowie
and Kleinman) is that they incorrectly assume perfect cube efficiency for
both sides. Obviously, cube use will not be perfectly efficient. My own
method to calculate a MET didn't assume perfect cube efficiency, and I
also believe that Trice's method doesn't assume perfect cube efficiency.
However, cube efficiency is not the same for both players at every score
so even that is an approximation.

6) The method used to calculate ME's is based upon the chances of each
side becoming good enough to cube the other player out. However, this
doesn't take into account factors such as the leader playing for a gammon
with a centered cube at -2-3, for example. A calculated MET is likely very
good at a score like -7-8, but for scores with only a few points to go and
for somewhat lopsided scores, it may be inaccurate even if it uses 26.7%
Gs and a proper percentage of BGs.

7) METs need to have 3 significant digits to allow for accurate TP
calculations at scores like we have in today's OLM. The fact that this is
too much memory and too much math for most humans to use doesn't really
matter since they can do the TP calculations at home and recall the
principles involved at relevant scores.

8) A MET rolled out by GNUBG is one that is rolled out to statistical
significance by a bot that plays as well as the best humans do (if not
better) It plays the checkers according to the match score very well and
has very good mathematical algorithms for determining cube use at various
scores. The games are played vs an equally skilled opponent, GNUBG so,
unlike a database there are no effects from the fact that the leader is
more likely to be the better player.

I will continue for the rest of my playing career to use rolled out METs
and g11 [] is the best that we
have right now. For those who don't want to learn a new MET, I understand,
and simulations show that it doesn't make a huge difference in their match
winning chances, however, there is no cost to using a more accurate MET for
their analysis with bots.

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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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