Forum Archive :
Match Equities
Does it matter which match equity table you use?

The standard Match Equity Table (MET) is the Woolsey (or Woolsey
Heinrich) table which has been around for around 10 years now.
Many others exist and have their followers: JacobsTrice and Snowie 2.1,
for example. Because of algorithms developed and memorization already
implemented there is a lot of inertia among individuals (myself
included) to just stick with what we already have. But is there a better
way? Some questions:
1) How much does the choice of MET matter?
2) Does the choice of MET depend upon the two players? If so, how much?
3) If none of the current MET's is optimal, how does one go about
building/choosing the proper one?
4) Should one memorize/formulize tables of doubling points and
takepoints instead of or in addition to memorizing an MET/algorithm?
5) What are the practical constraints? (For example, Kleinman argues for
3 significant figure tables, but are there very many who can actually
accurately memorize these, and are they worth it?)
6) How big of a table should one memorize/formulize? Sure, ideally you
would have a 25point match table in your head, but that doesn't seem
practical, so what is a good cutoff?


Douglas Zare writes:
> 1) How much does the choice of MET matter?
For most scores, there is little difference between the recommended
actions of different METs. There are some exceptions, most notably those
involving large cubes, positions within 2 points of the end of the game,
and 3away 4away, at which I think the WH table is wrong. There has
been some testing of gnu versus gnu with different match equity tables,
producing differences of about 50100 15 point matches in 100,000, IIRC.
The variance reduction system used was good enough that this might be
statistically significant, but it simultaneously suggests that it might
not matter much if you use the wrong table well.
> 2) Does the choice of MET depend upon the two players?
It may depend quite a bit on the playing styles of the players. Some
people just don't know how to play for the gammon. If two of those play
each other, a small lead means more, and leading Crawford 4away is a
much larger advantage than between bots. Again, though, the main cube
decisions affected are when the cube is large or one player is close to
victory.
> 3) How does one go about building/choosing the proper MET?
I don't think it is worth it for most people. I'm debating whether it is
worth it for me! (I'm doing it anyway, but not with as much effort as if
I thought it really important.) I start with DMP = 50% mwc, and work
backward using real cubeful backgammon play based on the earlier parts
of the table, simultaneously tweaking the match play algorithm. This
takes a lot of time. Using a bot that would not understand how the cube
should be used is little better than using a bad model of backgammon.
> 4) Should one memorize/formulize tables of doubling points and
> takepoints?
I think it is much better to memorize or have a feel for cubeful take
points and gammon prices, and not doubling points or exact MET entries.
For the racing take points on 4cubes and 8cubes, I ran a contest to
find a formula that best approximates these for Snowie's table, and the
winners were close fits and not too complicated although I don't
remember them. Huge cubes arise in tournament play, and it is valuable
not to blunder with them. I recommend learning a rough approximation so
that if you need to work things out for a large cube you can.
> 5) What are the practical constraints?
Three significant figures are essential for calculations done far from
the end of the match, or else the roundoff error swamps everything.
I don't think they are worth memorizing, and they are harder for most
people to work with than whole number percentages. I think the solution
is to do the work ahead of time, and remember that at this patch of
match scores the racing take point is x% higher than for money. You
won't figure this out with much confidence at the table, partly because
it helps to interpolate between results at neighboring match scores.
> 6) How big of a table should one memorize/formulize?
That depends on the lengths of matches you play most often. I play a lot
of 3, 5, and recently 7point matches, so I have most of those
situations memorized now. Before a serious tournament, I review take
points and gammon prices outside the part of the MET I know well.


Chuck Bower writes:
For 100,000 matches between nearly equal opponents, the standard
deviation [ sqrt(0.5*0.5*100,000) ] is 158. So I would interpret 50100
matches as not statistically significant. Or are you saying that a
difference of 50100 wasn't the actual result of the competition, but
instead a bot analysis of who should have won?
But regardless, if 300 matches is two standard deviations, then with 95%
confidence we can say that the choice of MET helps no more than
300/100,000 = 3 parts in 1000 = 0.3%. This assumes you know which MET to
use, and that the advantage is this much regardless of the opponent. It
looks like when you actually get a position where two different MET's
disagree on the proper cube action, the *error* made in choosing the
wrong cube decision must be rather small.


Douglas Zare writes:
That would be the standard deviation without any variance reduction.
However, the variance reduction used was to play pairs of matches with
an identical stream of rolls with the opponents (METs) reversed (rather
than hedging).
In some matches, there would be no decision that depended on the MET,
and these would be split exactly evenly. These would not contribute to
the variance. When there was a divergence, the rest of the match might
not notice the parallel dice, but the divergence may well mean that one
side had a 90% chance to win in one match and an 11% chance to win in
the other. Only two wins or two losses would cause a deviation from 50%.
By some rough calculations, I figured that the differences were indeed
statistically significant. At the same time, they were a statistically
significant rejection of the proposition that there was a huge
difference in the playing strength from using one of those METs rather
than the other.
It is possible that using another MET would produce different results,
perhaps producing a significant advantage or a significant disadvantage.
I think that the latter would result if one blindly used a MET which
rounded to the nearest percent. What happens in real life is that people
don't consult a MET rounded to the nearest percent to determine the take
point on an initial double at 13away 15away, but you have to be
careful what you tell a bot to do.
There is a strange MET proposed by Ortega and Kleinman whose methodology
is absurd. They interpolate between Kleinman's table and the WH table,
but by uneven amounts. I think the result looks to the naked eye as
though it should have the smoothness of a 3digit table (and it makes
the memorization and calculations about as hard), while having only the
precision of the 2digit table.


Kit Woolsey writes:
My personal opinions are:
> 1) How much does the choice of MET matter?
It doesn't matter much at all. All of the MET tables will be within 1 or
2 percent of each other for any given score, which is quite adequate to
come to any reasonable cube decision. If only we could estimate our
winning chances in a position so accurately.
> 2) Does the choice of MET depend upon the two players? If so, how
> much?
Jake Jacobs has written an excellent book "Can a fish taste twice as
good" which goes into this topic. My feeling is that most cube decisions
shouldn't be dependent upon perceived relative skill (particularly since
many players may have a warped perception about the relative skill of
them and their opponent in the first place). Differing skill level is
usually relevant only if the cube decision may be the final decision of
the match. Then one may make a revised estimate of matchwinning chances
if the conservative route is taken. However, this doesn't depend on the
MET used.
> 3) If none of the current MET's is optimal, how does one go about
> building/choosing the proper one?
It isn't easy. The difficulty involves assigning proper cube leverage at
different match scores. When I developed my table several years ago, I
used a combination of empirical results of over 1000 matches, some
mathematical analysis, and a lot of judgment. The methodology could
hardly be called rigorous, but the results have proven to be quite
practical.
> 4) Should one memorize/formulize tables of doubling points and
> takepoints instead of or in addition to memorizing an MET/algorithm?
I have never done so. The problem with trying to assign doubling points
and takepoints is that they may depend upon the potential recube vig,
so a purely mathematical approach can lead to some ridiculous results.
I find it more meaningful to just use the match equities along with with
logic of the position to derive my own doubling points and take points
when I need them.
> 5) What are the practical constraints? (For example, Kleinman argues
> for 3 significant figure tables, but are there very many who can
> actually accurately memorize these, and are they worth it?)
It definitely isn't worth it. To begin with, any attempt to construct a
MET will have some flaws which are greater than 3 significant figure
accuracy. Furthermore, even if someone had access to a perfect MET with
3 figure accuracy, the extra accuracy wouldn't be worth anything unless
he were able to estimate the equity of the position in question with the
same accuracy. Since most of us are quite happy if we can get within 2%
on that estimate, finetuning the MET to 3 significant digits simply
isn't worth anything as a practical matter.
> 6) How big of a table should one memorize/formulize? Sure, ideally you
> would have a 25point match table in your head, but that doesn't seem
> practical, so what is a good cutoff?
For sheer memorization, learning all the equities for a 7point match
will be sufficient for almost all problems you will find in actual play.
In addition, there are easy to use formulas which require no
memorization at all. I think the best of these is Neil's numbers.
Personally, I have memorized my MET only for the scores where the leader
is 1 away or 2 away, since the Neil's numbers formula is sometimes
inaccurate for those scores. For any other scores I rely on the formula
when I need it in actual play, which takes a few seconds to apply and is
as accurate as the MET itself.




Match Equities
 Constructing a match equity table (Walter Trice, Apr 2000)
 Does it matter which match equity table you use? (Klaus Evers+, Nov 2005)
 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)
 ME Table: Big Brother (Peter Fankhauser, July 1996)
 ME Table: Dunstan (Ian Dunstan+, Aug 2004)
 ME Table: Escoffery (David Escoffery, Nov 1991)
 ME Table: Friedman (Elliott C Winslow, Oct 1991)
 ME Table: Kazaross (Neil Kazaross, Dec 2003)
 ME Table: KazarossXG2 (neilkaz, Aug 2011)
 ME Table: RockwellKazaross (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)
 ME Table: Zorba (RobertJan Veldhuizen+, Dec 2003)
 ME at 1away/2away (crawford) (Fabrice Liardet+, Nov 2007)
 ME at 1away/2away (crawford) (Ian Shaw+, Apr 2003)
 Match equitiesan 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)
 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)
 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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