How diifferent gammonrates affect match equities ?
MIKA JOHNSSON
I was reading our finnish backgammon news and there was an article about
equities. The writer claimed that 12-away - 14-away is worth according Friedman
(36% gammons) 58% (this sounds ok to me, Woolseys table says the same also the
other formulas (Janowski and Kazaross) give about the same). Then the writer
claims that with 20% gammonrate this score is worth 63% !, I find this quite
high, can the gammonrate make so big difference ??
I was wondering anyone having a program that counts equities for different
gammonrates ?
What gammonrate is Kit Woolsey using in his table ?
What gammonrate does the formulas assume (or the underlying tables the formulas
estimate)?
I saw that in Roberties table (Genud vs. Dwek) 12-away - 14-away was worth 61%
What is the current estimate for gammonrate ?
-Mika
Michael J Zehr
>I was reading our finnish backgammon news and there was an article
>about equities. The writer claimed that 12-away - 14-away is worth
>according Friedman (36% gammons) 58% (this sounds ok to me, Woolseys
>table says the same also the other formulas (Janowski and Kazaross)
>give about the same). Then the writer claims that with 20% gammonrate
>this score is worth 63% !, I find this quite high, can the gammonrate
>make so big difference ??
>
>I saw that in Roberties table (Genud vs. Dwek) 12-away - 14-away was worth 61%
I have a model that I've been using to look at how varying gammon rates
and players of different skill levels affects the match equity tables.
The basis for the model is as follows: I started with a gammon rate of
20%. I worked out a formula to determine chance of winning 1pt, 2pt, or
4pt at a given score, depending on the droppoints for each player. I
tweaked it until it agreed with Kit's table for small values of points
left to play. Then I ran it out to -15:-15 and compared the results
with Kit's table. It agrees in most places to within 1%. There's a
systemic bais to underrate trailers chances at lopsided scores. This
might or might not mean anything. [*]
Kit's table says 58% for this score (as you mentioned), mine says 58.3%.
If I change the gammon rate from 20 to 36, my model says 57.8.
Does this have any bearing to actual play? I don't know, but you know
about as much as I do about how I came up with the figure, so you can
judge for yourself.
The figure of 63% doesn't sound very realistic. Consider that from
-14:-14, winning 2 points gains you 13%. In general with match equity
tables, until one side is close to winning, the gain per point won is
proportional to the points the trailer has left. So winning 2 more
points should put you close to 76%, and that's a score of -14:-10. Are
the first 4 points worth the same as the next 10? That seems very
counter intuitive, even with a very high gammon rate.
-michael j zehr
[*] kit's table came from extensive analysis of recorded matches. but
the matches were played without using the old tables. what kit found
was that when people played using the old tables, the trailer did quite
a bit better than the old tables predicted. hence clearly the old
tables were wrong. the real test of the new tables is if match results
using the new table are consistent with the equities in the new table.
i don't think the new tables have been around long enough for this to be
deomonstrated one way or another. this is an interactive process, and
kit's table are certainly one step closer to the "true" answers, but it
might or might not be the final step.
Lest I'm accused of being blind to the defects in my own work, I will say
that it was models *not* based on recorded match results that were found
to be wrong by Kit. Such is the model I have. On the other hand, it's
very close to Kit's table. So if a big change in the gammon rate has a
very small effect on the match equity of -12:-14 in my model, chances
are that would hold true in a "real" match, even if there's a
discrepancy as to the actual value of -12:-14. (I'm not sure what
exactly a real match is when you can vary the gammon rate. Is that a
new feature in FIBS? set gammon 40? *grin*
Kit Woolsey
: Hello !
First of all, let's see what we mean by gammon rate. If there is a
possibility that the game may end with a cube turn then gammon rate
really doesn't mean all that much. So, we are only looking at games
which must be played to conclusion -- and this can happen only at
Crawford or at Post-Crawford scores. Under these circumstances,
depending on the score, either the gammon will be of great value to one
side or it will be of no value to anybody.
Thus, it is clear that a change in the estimated gammon rate would only
have a major effect on Crawford and Post-Crawford equities. For other
scores the gammon rate would have less effect, decreasing as we got
farther away from the Crawford game. Thus the above claim that the
equity changes from 58% to 63% on the 12 away 14 away score as the
gammon rate changes is clearly wrong -- a change in the gammon rate will
have very little affect for that score.
When I first constructed my equity table I believe I used a gammon rate
of about 21%. This was consistent with my own estimates as well as the
results from a data base of several hundred games which had to be played
to conclusion due to the match score. Currently I think the
theoretically correct figure should be higher, perhaps 25%, with correct
play. However since most players are not sufficiently adept at gammon
collection the 21% figure appears to work out well in practice, and the
match equity table is consistent with real life results. Personally I
consider Friedman's 36% estimate way out to lunch. He has no evidence to
back this up other than his own, probably biased, rollouts. All three
computer programs (Expert Backgammon, TD-Gammon, and Jellyfish) come up
with a gammon rate somewhere in the mid 20's, and this would be what the
majority of experts would agree with also, I think.
Keep in mind that I did not construct my equity table using a precise
formula. Rather, I took a large data base of empirical results, molded
together some assumptions from these results, did the appropriate
fudging, and out came my match equity table. I don't claim it is
mathematically precise, but it does have one very important thing going
for it -- it appears to work!
When I first tried constructing a match equity table (about 15 years ago),
I also had the equity for 12 away, 14 away higher than the 58% I have now.
What happened was that I did not give sufficient weight to the trailer's
cube leverage at different match scores, so in general I kept coming up
with the leader having more of an advantage than he actually had in real
life. I believe Robertie made the same error, which is why his table in
Genud vs. Dwek is different from mine. I still don't know how to express
this cube leverage in mathematical terms -- perhaps someone who can find
an accurate way to do so can come up with a better table. However, I do
believe that my table does properly represent the cube leverage the
trailer has in real life.
Kit