I thought that some r.g.b. readers might be interested.
Title: Solving the Race in Backgammon
Arthur Benjamin is the Smallwood Family Professor of
Mathematics at Harvey Mudd College. In 2020, he won
the inaugural American Backgammon Tour Online (ABTO)
with the best overall performance in a series of 17
national tournaments. He has written several books that
present mathematics in a fun and magical way. He earned
his PhD in Mathematical Sciences from JHU in 1989.
Didn't Axel already prove this through his experiment here
even about positions prior to losing contact and, in fact,
starting from the opening roll on...?
On 2/28/2022 6:03 AM, MK wrote:
Didn't Axel already prove this through his experiment hereThere is a potentially interesting question along these lines
even about positions prior to losing contact and, in fact,
starting from the opening roll on...?
that I don't think has been investigated before, and Axel might
be in a good position to do so. Start with some racing position
where the player on roll is just barely an equity favorite; e.g.,
XGID=--ABCDE------------ddcbb--:0:0:1:00:0:0:0:0:10
X:Player 1 O:Player 2
Score is X:0 O:0. Unlimited Game
+13-14-15-16-17-18------19-20-21-22-23-24-+
| | | O O O O O |
| | | O O O O O |
| | | O O O |
| | | O O |
| | | |
| |BAR| |
| | | X |
| | | X X |
| | | X X X |
| | | X X X X |
| | | X X X X X |
+12-11-10--9--8--7-------6--5--4--3--2--1-+
Pip count X: 70 O: 66 X-O: 0-0
Cube: 1
X on roll, cube action
The player on roll adopts an equilibrium strategy. The opponent
does not necessarily play an equilibrium strategy. Now we ask,
how many times do they have to play out the position before the
player on roll has a 90% chance of coming out ahead? Call this
number N.
I'd expect that if the opponent's goal is to maximize N, she can
do better than adopt an equilibrium strategy, but this would be
interesting to investigate, and I don't think it has been done
before. If beavers and raccoons are allowed then naturally one
would expect that N can be made even larger.
It might make sense to start with simpler racing positions than
the one above, where one can exhaustively explore the space of
strategies and find what the optimal strategy is (meaning the
one that maximizes N).
Start with some racing position where the player on roll is just
barely an equity favorite
The player on roll adopts an equilibrium strategy. The opponent does
not necessarily play an equilibrium strategy. Now we ask, how many
times do they have to play out the position before the player on roll
has a 90% chance of coming out ahead? Call this number N.
I'd expect that if the opponent's goal is to maximize N, she can do
better than adopt an equilibrium strategy
The need to maximise N gives a positive (rather than neutral) value to increasing the variance. So, if a pass is sufficiently marginal, it
becomes a take and if a hold is sufficiently marginal it becomes a
cube.
Why do you think this is interesting? I'm not sure anyone else is interested.
On 2/28/2022 8:24 AM, peps...@gmail.com wrote:
Why do you think this is interesting? I'm not sure anyone else is interested.You're probably right that very few others are interested; otherwise
it would probably have been done already.
Ultimately, I can only say that I'm following my personal instincts
about what constitutes an interesting research direction. Those
instincts have generally served me well in the past. Finding tractable
but nontrivial problems whose solution answers simple questions about fundamental mathematical objects is part of that. In this case, this
is one of the first questions I would want to understand when trying
to study non-equilibrium strategies.
But backgammon concepts could hardly be further from fundamental mathematical objects.
"Mathematical objects?" -- a definite yes.
But "Fundamental"?? --- Surely you're joking, Mr. Chow?
I'll make up a simple game right now called TossCoin to illustrate my point.
On 3/1/2022 9:22 AM, peps...@gmail.com wrote:
But backgammon concepts could hardly be further from fundamental mathematical objects.
"Mathematical objects?" -- a definite yes.
But "Fundamental"?? --- Surely you're joking, Mr. Chow?
I'll make up a simple game right now called TossCoin to illustrate my point.I don't mind asking the same questions about an even simpler game
than backgammon. But I don't think that games that people actually
play "could hardly be further from fundamental mathematical objects." Mathematics, in the end, is a human activity. What we regard as
fundamental is ultimately based on human preferences. Now I do agree
that a game with lots of finicky, ad hoc rules doesn't really merit
being called fundamental, but backgammon races are pretty simple from
a mathematical point of view.
On 3/1/2022 9:22 AM, peps...@gmail.com wrote:
But "Fundamental"?? --- Surely you're joking, Mr. Chow?
I'll make up a simple game right now called TossCoin
..... backgammon races are pretty simple from
a mathematical point of view.
("What are the properties of the sum of the digits?" etc.) then most mathematicians (including me)
don't think of them as "real mathematics".
Now let me offer a few arguments of my own.
in your example, X will need 70/8.167=8.571 rolls and O will need 66/8.167=8.081 rolls
let's say O is the "mutant" and start bearing of pieces with actual
dice numbers adding up to 8, occasionally editing the position so that
the next turn won't hit an empty point, (it's okay to do this for the
sake of the argument here, since we'll do it for both sides).
We know that in this situation O will never get ahead and thus never
double.
X's cube actions will be all No-double/Take until the 6th roll
MK <mu...@compuplus.net> writes:They are also foobar in my opinion.
Now let me offer a few arguments of my own.These arguments are FUBAR.
On 3/2/2022 3:32 AM, peps...@gmail.com wrote:
("What are the properties of the sum of the digits?" etc.) then most mathematicians (including me)It's true that this is the majority opinion. I don't entirely
don't think of them as "real mathematics".
agree. I partially agree, because questions of that sort *tend* not
to lead very far, and if a line of investigation leads to a dead end,
then I agree that it is not terribly interesting. But I don't for
that reason regard them as "not real mathematics."
It's my belief that non-equilibrium strategies in games involving a
doubling cube are a fruitful topic of research. The question I
suggested is a concrete question to focus on. Its answer is admittedly
not extremely interesting for its own sake, but I have faith that
trying to answer it will lead to interesting insights that will suggest further lines of investigation.
Art Benjamin sent the announcement below to a bunch of people, and
I thought that some r.g.b. readers might be interested.
---
Dear friends in Math and Backgammon,
This Thursday, March 3, at 4 PM Pacific (7 PM Eastern) I will be
giving a (virtual) presentation on Solving the Race in Backgammon
to the Johns Hopkins University Applied Mathematics Community (organized
by HUSAM, the Hopkins Undergraduate Society for Applied Mathematics).
The talk will be aimed at undergraduate applied math students who do not necessarily have a background in backgammon, but I will quickly get them
up to speed with concepts like the pip count and the doubling cube.
For those who saw my presentation in Claremont earlier this month, this
talk will be more streamlined to allow more time for the important backgammon results at the end.
Title: Solving the Race in Backgammon
Abstract: Backgammon is perhaps the oldest game that is still played
today. It is a game that combines luck with skill, where two players
take turns rolling dice and decide how to move their checkers in the
best possible way. It is the ultimate math game, where players who
possess a little bit of mathematical knowledge can have a big advantage
over their opponents. Players also have the opportunity to double the stakes of a game using something called the doubling cube, which—when
used optimally—leads to players winning more in the long run. Optimal
use of the doubling cube relies on a player's ability to estimate their winning chances at any stage of the game.
When played to completion, every game of backgammon eventually becomes
a race, where each player attempts to remove all of their checkers
before their opponent does. The goal of our research is to be able to determine the optimal doubling cube action for any racing position,
and approximate the game winning chances for both sides. By calculating
the Effective Pip Count for both players and identifying the positions' Variance Types, we arrive at a reasonably simple method for achieving
this which is demonstrably superior to other popular methods.
Arthur Benjamin is the Smallwood Family Professor of Mathematics at
Harvey Mudd College. In 2020, he won the inaugural American Backgammon
Tour Online (ABTO) with the best overall performance in a series of 17 national tournaments. He has written several books that present
mathematics in a fun and magical way. He earned his PhD in Mathematical Sciences from JHU in 1989.
Zoom Link for talk:
HTTPS://WSE.ZOOM.US/J/95972756601
MK <mu...@compuplus.net> writes:
in your example, X will need 70/8.167=8.571 rolls and
O will need 66/8.167=8.081 rolls
No. There is wastage involved. But you will get the benefit
of the ignorant, since that point CAN be repaired.
We know that in this situation O will never get ahead and
thus never double.
True, but a mute point, since the outcome is clear anyway
if all rolls are 8 pips and no wastage/miss can occur.
X's cube actions will be all No-double/Take until the 6th roll
No. X should double immediately and O should pass,
the outcome exactly 9 rolls later is deterministic. Because of
the fixed roll and the no-miss shuffling (= one checker race)
nothing can go wrong. There are simply no probabilities
involved, and that is the core of the cube skill.
On Wednesday, March 2, 2022 at 12:33:26 PM UTC, Axel Reichert wrote:
MK <mu...@compuplus.net> writes:
Now let me offer a few arguments of my own.
These arguments are FUBAR.
They are also foobar in my opinion.
He tends to reproduce almost identical arguments
whenever the doubling cube is discussed.
So the arguments function similarly to "foo", "bar"
when used software development texts to indicate
placeholders.
But the standard research methodology (at least in mathematics) is to look for the
simplest unsolved problems, rather than jump straight into backgammon.
So we'd look at the simplest possible games for which a doubling cube makes sense
(which is what I tried to do earlier in the thread).
Perhaps you and I differ in opinion, as to how complex backgammon races are.
Will the event be recorded, do you know?
On 3/2/2022 4:53 PM, peps...@gmail.com wrote:
Will the event be recorded, do you know?I don't know.
On 3/2/2022 11:48 AM, peps...@gmail.com wrote:
But the standard research methodology (at least in mathematics) is to look for theFocusing on the simplest unsolved problems is certainly a
simplest unsolved problems, rather than jump straight into backgammon.
So we'd look at the simplest possible games for which a doubling cube makes sense
(which is what I tried to do earlier in the thread).
Perhaps you and I differ in opinion, as to how complex backgammon races are.
sound general principle.
But there is also value in articulating problems that are
slightly more of a stretch, if they have some kind of sex appeal.
For example, the twin prime conjecture does not meet the criterion
of "simplest unsolved problem" but it is appealing.
On Thursday, March 3, 2022 at 5:15:57 AM UTC, Tim Chow wrote:
On 3/2/2022 4:53 PM, peps...@gmail.com wrote:
Will the event be recorded, do you know?
I don't know.
That's a very ambiguous response, Tim.
Do you mean that you don't know if the event will be
recorded?
Or do you mean that you don't know whether you
know that the event will be recorded?
(J/k of course).
But the twin prime conjecture proves my point, I think.
The interested mathematicians don't immediately try to prove the conjecture. They focus on simpler (previously) unknown questions, related to the conjecture.
Such as "Does there exist any N such that there are an infinite number of prime pairs (p, p + N)?
(Yes, there are.)
On 3/3/2022 3:17 AM, peps...@gmail.com wrote:
But the twin prime conjecture proves my point, I think.Sure. I'm not prohibiting anyone from tackling my proposed
The interested mathematicians don't immediately try to prove the conjecture.
They focus on simpler (previously) unknown questions, related to the conjecture.
Such as "Does there exist any N such that there are an infinite number of prime pairs (p, p + N)?
(Yes, there are.)
challenge by solving simpler problems first---that would be
absurd, and unenforceable anyway.
But it doesn't follow that it's a mistake to pose the problem
in the first place.
On Thursday, March 3, 2022 at 12:54:02 PM UTC, Tim Chow wrote:
On 3/3/2022 3:17 AM, peps...@gmail.com wrote:
But the twin prime conjecture proves my point, I think.Sure. I'm not prohibiting anyone from tackling my proposed
The interested mathematicians don't immediately try to prove the conjecture.
They focus on simpler (previously) unknown questions, related to the conjecture.
Such as "Does there exist any N such that there are an infinite number of prime pairs (p, p + N)?
(Yes, there are.)
challenge by solving simpler problems first---that would be
absurd, and unenforceable anyway.
But it doesn't follow that it's a mistake to pose the problem
in the first place.
It's not a mistake to pose the problem. It's a mistake to work on the problem.
Don't run before you can walk, and all that.
But that's my opinion.
Feel free to work on it anyway.
On 3/3/2022 7:56 AM, peps...@gmail.com wrote:
On Thursday, March 3, 2022 at 12:54:02 PM UTC, Tim Chow wrote:
On 3/3/2022 3:17 AM, peps...@gmail.com wrote:
But the twin prime conjecture proves my point, I think.Sure. I'm not prohibiting anyone from tackling my proposed
The interested mathematicians don't immediately try to prove the conjecture.
They focus on simpler (previously) unknown questions, related to the conjecture.
Such as "Does there exist any N such that there are an infinite number of prime pairs (p, p + N)?
(Yes, there are.)
challenge by solving simpler problems first---that would be
absurd, and unenforceable anyway.
But it doesn't follow that it's a mistake to pose the problem
in the first place.
It's not a mistake to pose the problem. It's a mistake to work on the problem.
Don't run before you can walk, and all that.
But that's my opinion.But working on your suggested problem *is* working on my problem,
Feel free to work on it anyway.
just as Yitang Zhang *did* work on the twin prime conjecture. It
seems you are posing a false dichotomy.
after 4 billion games wastage will also be the same for both
players
Just add "bullshit" before the final period in your paragraph
and people may think your are Murat's sock puppet... ;)
On 3/2/2022 7:38 PM, MK wrote:
Just add "bullshit" before the final period in your paragraph
and people may think your are Murat's sock puppet... ;)
No sock puppet of yours could make as much sense
as Axel does.
On March 3, 2022 at 1:13:25 AM UTC-7, peps...@gmail.com wrote:
That's a very ambiguous response, Tim.
.....
.....
"Club" is spelled with a "c" not with a "k", Mr. J/c (i.e.
the "jackass of clubs")...
MK <mu...@compuplus.net> writes:
after 4 billion games wastage will also be the same
for both players
No. But it is probably wastage to try to explain. Read
Walter Trice's article:
https://www.bkgm.com/articles/EffectivePipCount/
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