The book that Jake Jacobs and I co-wrote, called "Can A Fish Taste Twice As
Good?" contains, among other goodies, a match equity table for equal
players that goes out as far as 25 point matches. This table agrees
reasonably well with Kit Woolsey's 15 point match table, and needless to
say I have a certain amount of confidence in the Trice/Jacobs table, since
I invented the methodology behind it and did the programming.
Sam Pottle wrote:
> Would you be willing to share a thumbnail sketch of that methodology
> with us? I'm curious about the construction of match equity tables, and
> I've run across only two approaches in my reading.
> One is the Woolsey/Heinrich approach, which uses a database of human
> expert matches to estimate the distribution of outcomes for games at any
> given score (leader wins 1 point, trailer wins 2, etc.). This makes
> sense, but I can't go through it myself without access to the data.
> The other approach is used by Tom Keith, and I've also found it used by
> Kleinman. This model assumes (a) perfectly efficient cubes, and (b) a
> fixed gammon rate. The whole table can then be derived from these
> The first assumption is awful, of course, but by underestimating the
> gammon rate (Keith uses 20%), you can get pretty good results, because
> the errors tend to cancel. Still, it's not the most intellectually
> satisfying model.
> If you are doing something different I would be most interested to hear
> about it. If what I'm asking for is what you're selling (in the book),
> then I beg your pardon. Perhaps I should just go buy it.
Yes. There's nothing about it that I would consider a "trade secret," and
the mathematics, though complicated-looking when you put all the variables
in, is really elementary stuff.
In the special case where the players are equal in strength, all I am doing
amounts to a small modification of what Tom Keith describes. Your objection
was that he assumed perfect cube efficiency. Well, that can be fixed by the
simple expedient of adding "something" onto the cubeless probability of
winning that a player is assumed to need in order to acquire 1 incremental
match point worth of equity.
To make this concrete, let's briefly go way way back to Danny Kleinman's
analysis of 2-away/3-away. You can find this in his article "The Biased
Cube in the 3 Point Match", in "Vision Laughs at Counting With Advice to
the Dicelorn." Kleinman reasoned that with his assumed gammon rate of 20%
the leader needs a cpw of 1/3 to take, whereas the trailer needs 1/4. So
the leader would have to move from 1/2 to 3/4 to cash, but the trailer
would only have to move the smaller distance from 1/2 to 2/3. The ratio of
these "probability distances" is 2 to 3 -- that is (2/3 - 1/2) is 2/3 of
(3/4 - 1/2), so (invoking the "continuous model") the trailer is 3/2 as
likely as the leader to gain the equity equivalent of 1 point. If the
leader wins he'd be at 75%, whereas if he loses he goes to 50%, so the
leader's equity at the start of the 2-away/3-away game is .4*.75 + .6*.5 =
The modification amounts to assuming that a player always "loses his
market" by a certain amount of cpw. That is, the ratio (2/3 - 1/2) to (3/4
- 1/2) is replaced by (2/3 + x - 1/2) to (3/4 + y - 1/2). In theory, the
numbers x and y should differ from each other, and should vary in different
match and game circumstances, but I wound up concluding that in practice it
wouldn't hurt too much to make them a constant. I refer to this adjustment
as an "overshoot factor" -- i.e., the amount of cpw by which a player is
assumed to "overshoot" his cash-point. The method of table construction
described by Norman Zadeh in his 1977 article in the journal Management
Science uses the same method to adjust for "discontinuity." [Note: Zadeh's
method and mine differ more significantly in the general case of unequal
players, which we are not considering here.]
Your other comment on Tom Keith's approach was that he assumed a fixed
gammon rate. Naturally when you define a model or write a program you don't
have to do this -- you just make it a variable, or a "user input." Since
the gammon rate has more influence on the match equity numbers than
anything else, making it a variable is a smart thing to do! The additional
factors I take into account amount to "bells and whistles" -- they include
free drop vig, backgammon rate, and an assumed reduction in the gammon rate
whenever the cube level goes up (for reasons like, for instance, when the
cube goes to 4 there's a good chance somebody just hit a shot in a contact
bearoff and CAN'T win a gammon.)
The "tricky" part of writing a match equity table program is the recursive
procedure Tom Keith describes, where you start by calculating take-points
for the highest possible live cube, and use these to keep adjusting things
for recube vig while working your way down to the center-cube situation.
This is all a bit sketchy. Feel free to ask specific questions if you think
the answers would make things clearer.
-- Walter Trice