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Apr 1, 1998, 3:00:00 AM4/1/98

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I'm sorry but I have another question for you.

Could somebody explain me how to use the "settlement limit" when making

rollout with JellyFish ?

I only know it should keep in count the take-points of the players but I

don't understand how it works.

Thanks again,

Carlo Melzi

Apr 2, 1998, 3:00:00 AM4/2/98

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There's how it works and there's why it's done that way, and while the

first question is fairly simple on a technical level, to fully

understand it requires some historical perspective and overall

understanding of rollouts.

The answer is related to what Chuck Bower posted in his problem "So Much

for the Good Old Days":

"2) No matter how rabid of a player, you were limited in the number of

games played. Computers play games in a few seconds. Humans play games

that take minutes. That's a factor of 60 (or 100--commonly referred to

as two orders of magnitude). Computers can play 24 hours a day. Even

the most stalwart human surely plays no more than 50% of the time. In

summary, a computer playing for a month is about equivalent to an adult

human playing most of his/her active BG lifetime."

In the pre-computer days, people would rarely do more than 108 rollouts

of a position (three times for each starting roll) becase even such a

"small" rollout might take 3-4 hours. When doing rollouts, people

wouldn't use a live cube. Instead if a player with access to the cube

got to a position that was a huge double and maybe a drop, they'd score

it as a drop and continue. If it was an obvious take, they'd keep

playing without scoring it as a double/drop.

The reason for this is that if you're doing a 108-game rollout and you

get one game that has a double/take followed by a redouble/take (so the

cube is on 4), and all the other games reach a double/drop position

(with the cube ending on 1), then the results of that single game will

cause a 7-8% swing in the winning percentage for one side! This is

making way too much of the rollout depend on a single game, possibly

even a single lucky shot in that game. To keep this from happening (in

statistical terms, to keep the variance low so that the results converge

to the correct value faster), a rollout would be played until one side

faced a huge double with a close take/drop decision and it would be

scored as a drop. This is what the settlement limit of JF rollouts is

duplicating.

How does this work, and why .550?

If you can't ever redouble, you need to have an equity of no less then

-.5 in order to take a double. (An equity of -.5 on a 1-cube

corresponds to an equity of -1.0 on a 2-cube after taking, which is

exactly equal to the -1.0 equity you would have from dropping, so this

is the break-even point. Any lower equity and you would drop, any

higher and you would take.) This corresponds to a 25% winning chance if

there are no gammons.

But you do have the option of redoubling (depending on the score if it's

a match game of course) and this gives you additional equity. The

commonly accepted average is that you can take with 22% winning chances,

or an equity of -.56. If -.56 is the approximate drop point, then .56

equity is the approximate cash point for your opponent.

The first idea might be to use this as the settlement limit -- if your

equity is over .56, record it as a double/drop. The problem with this

is that sometimes you'll reach an equity of .550, and have a HUGE

double, and next roll will shoot to an equity of .8, and lose your

market by a mile. This means that the side with access to the cube ends

up losing a lot of equity by always losing their market. To balance

this, it's been found that setting the settlement limit a little bit

lower seems to work well on average. Sometimes the leading side gains

equity by cashing when the position is really a take, and sometimes the

leading side loses equity by not being able to turn the cube with an

equity lower than .550, even if the double is correct.

One has to approach using the settlement limit very carefully,

especially if you're rolling out a position from a match. If the cube

is centered then the cashpoint for the two sides might be different.

Furthermore, the correct settlement limit varies by position, and thus

varies during the course of a single game. (This is a topic for its own

posting.)

JF 3.0 doesn't allow rollouts with a live cube yet. With computers

getting faster and faster, the argument that it might take longer to

converge on the correct equity is less of an obstacle. Furthermore,

there are ways around this. One could do a rollout until a double/take

is reached, then play the position out twice from that point, valuing

each result as 1/2 what it would otherwise be. If the cube turned

again, the game would be played out twice from that position as well,

with each result contributing 1/4. This would be one way of preventing

a single game with a large cube from dominating the results of the

rollout.

Here's an example of how to use it for a match score:

The score is -4:-5 and the leader is contemplating taking a double.

(For simplicity, assume a race with no gammon potential.) First step is

to compute the drop point:

Drop for -4:-4 or .50 match equity

Take and win for -2:-5 or .75 match equity

Take and lose for -4:-3 or .41 match equity

You're risking .09 to gain .25, so the take point is 9/34, or 26.5%.

Suppose you roll out the position cubeless and get 24% winning chances.

Does this mean it's a drop? Not yet -- the taker might get enough

equity from holding the cube that it's a take.

The next step is to do a level 5 rollout using the cube and a settlement

limit and look only at the results for the one cube position. First

determine the drop point on a potential redouble. Drop to be at -5:-2

or .25 match equity, or take the cube at 4 and redouble to 8 to put the

match on the line. So the drop point is 25%.

25% corresponds to an equity of .500. As a first approximation, set the

settlement limit to .500 and do a level 5 rollout. Assuming that the

level 5 cubeless results match the level 6 cubeless results (so you know

level 5 is playing well enough), you can now see if the side taking the

initial cube got 26.5% wins using the cube. If so, then you know it's a

take. If not, we're still not quite finished.

Setting the settlement limit to .5 means that the side taking the first

cube always waits until he loses his market and then cashes. Certainly

he can do a bit better by doubling earlier. But what is the correct

settlement limit to use if you're giving your opponent a dead cube and

they have a drop point of 25%? There's no answer to this that will

apply to every position. Maybe .45 is right, maybe .49 is right. If I

rolled it out with a settlement limit of .45 and it the taking side

still won less than 26.5% I'd say it was a drop. But if a limit of .45

indicated take and a limit of .5 indicated drop, I'd say that

theoretically it was too close to call.

Not in a nutshell, but that's what a settlement limit is, where the idea

came from, and how to use it.

-Michael J. Zehr

Apr 4, 1998, 3:00:00 AM4/4/98

to

Michael J. Zehr (mich...@michaelz.com) wrote:

: JF 3.0 doesn't allow rollouts with a live cube yet. With computers

: getting faster and faster, the argument that it might take longer to

: converge on the correct equity is less of an obstacle. Furthermore,

: there are ways around this. One could do a rollout until a double/take

: is reached, then play the position out twice from that point, valuing

: each result as 1/2 what it would otherwise be. If the cube turned

: again, the game would be played out twice from that position as well,

: with each result contributing 1/4. This would be one way of preventing

: a single game with a large cube from dominating the results of the

: rollout.

This is one of the most clever ideas I have ever seen. It appears to me

to be mathematically sound, easy to implement, and the results should be

far more trustworthy than using a settlement point or risking letting the

cube get very high for a couple of rollout trials. The number of trials

would increase, of course, but since a redouble/take scenario is

relatively rare the increase would probably be a factor of 2 or so.

Congratulations, Michael, on a brilliant suggestion. Hopefully we will

see it implemented in the next Jellyfish update.

Kit

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