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Mar 17, 1995, 5:42:45 AM3/17/95

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Hello !

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

Mar 20, 1995, 12:31:22 PM3/20/95

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In article <1995Mar17...@sara.cc.utu.fi> john...@sara.cc.utu.fi (MIKA JOHNSSON) writes:

>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*

Mar 21, 1995, 6:28:50 PM3/21/95

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MIKA JOHNSSON (john...@sara.cc.utu.fi) wrote:

: Hello !

: 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

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