Rollouts

 Settlement limit

 From: Michael J. Zehr Address: michaelz@michaelz.com Date: 2 April 1998 Subject: Re: Settlement limit Forum: rec.games.backgammon Google: 35233295.372392A@michaelz.com

```> Could somebody explain me how to use the "settlement limit" when
> making rollout with JellyFish ?

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
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
```

Rollouts

Cautionary tale  (Kit Woolsey, Sept 1995)
Combining rollouts  (Gregg Cattanach+, Dec 2003)
Confidence intervals  (Bob Koca, Nov 2010)
Confidence intervals  (Timothy Chow, May 2010)
Confidence intervals  (Gerry Tesauro, Feb 1994)
Cubeless vs centered-cube rollouts  (Ron Karr, Dec 1997)
Duplicate dice  (David Montgomery, June 1998)
How reliable are rollouts?  (David Montgomery, Aug 1999)
Level-5 versus level-6 rollouts  (Michael J. Zehr, June 1998)
Level-5 versus level-6 rollouts  (Chuck Bower, Aug 1997)
Positions with inaccurate rollouts  (Douglas Zare, Oct 2002)
Reporting results of rollouts  (David Montgomery, June 1995)
Rollout settings  (Lokicol+, Apr 2010)
Settlement limit  (Michael J. Zehr, Apr 1998)
Settlement limit  (Kit Woolsey, Dec 1997)
Settlement limit in races  (Alexander Nitschke, Dec 1997)
Some guidelines  (Kit Woolsey, Apr 1996)
Standard error and JSD  (rambiz+, Feb 2011)
Standard error and JSD  (Stick+, Oct 2007)
Systematic error  (Chuck Bower, Oct 1996)
Tips for doing rollouts  (Douglas Zare, June 2002)
Truncated rollouts  (Gregg Cattanach, Oct 2002)
Truncated rollouts: pros and cons  (Jason Lee+, Jan 2006)
What is a rollout?  (Gregg Cattanach, Dec 1999)

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