Backgammon paradoxes
Paul Epstein
positions illustrating three of the four. All paradoxes refer to
money play.
Paradox 1: The Jacoby Paradox. When considering a redouble, you need
to consider what the cube would be worth to your opponent. If, on the
next roll, you bear off 19/36 of the time, you redouble if your
opponent is certain to bearoff immediately upon rolling. However, you
shouldn't redouble if your opponent is a 29/36 favourite to bearoff on
the next roll because you hand over a valuable cube. Second scenario
is better (for the on-roll person) than the first. However, the first
is a redouble and the second isn't.
Paradox 2: The Kauder Paradox. Double/beaver is sometimes the correct
cube action.
Paradox 3: The Latto Paradox. In money play, some checker positions
are such that the cube action is redouble/take if the roller has the
cube, but a hold if the cube is in the middle.
Paradox 4: Assume that the position is a race -- no possible contact
in other words. In money play, I roll such that each die allows me to
bear off a checker. (In other words, I've either rolled a non-double
which allows the removal of two checkers or a double which allows the
removal of four.) The paradox is that sometimes this obvious
maximum-removal play is a mistake.
Can anyone give such an example where maximum removal is wrong? (I
don't know the name of this paradox.)
Thank you,
Paul Epstein
Gregg Cattanach
> Paradox 4: Assume that the position is a race -- no possible contact
> in other words. In money play, I roll such that each die allows me to
> bear off a checker. (In other words, I've either rolled a non-double
> which allows the removal of two checkers or a double which allows the
> removal of four.) The paradox is that sometimes this obvious
> maximum-removal play is a mistake.
>
> Can anyone give such an example where maximum removal is wrong? (I
> don't know the name of this paradox.)
>
> Thank you,
>
> Paul Epstein
Here's one example: The correct play is not to bear off a 4 with your 4,
rather play 6/2 with the 4. Most of these have to do with major overstacked
points and gaps on the low points. If you subscribe to GammonLine there is
an article about some of these unusual bearoffs where it's not right to bear
off the max # of checkers. (June 2003 back issue). This isn't a 'paradox',
just a situation where it's better to fill a gap than to bear off a checker,
(both of which are laudable goals.)
Money session. Score X-O: 0-0
X to play (6 4)
+24-23-22-21-20-19-------18-17-16-15-14-13-+
| O O O O O | | |
| O O O O O | | |
| O O O | | | S
| | | | n
| | | | o
| |BAR| | w
| 6 | | | i
| X X | | | e
| X X X | | |
| X X X | | |
| X X X X | | |
+-1--2--3--4--5--6--------7--8--9-10-11-12-+
Pipcount X: 44 O: 53 X-O: 0-0/Money (1)
Men Off X: 1 O: 2
CubeValue: 1
1. D 6/off 6/2 Eq.: 0.460
0.0% 0.0% 65.8% 34.2% 0.0% 0.0%
2. D 6/off 4/off Eq.: 0.438 (-0.022)
0.0% 0.0% 65.1% 34.9% 0.0% 0.0%
Hardy Hübener
> Paradox 4: Assume that the position is a race -- no possible contact
> in other words. In money play, I roll such that each die allows me to
> bear off a checker. (In other words, I've either rolled a non-double
> which allows the removal of two checkers or a double which allows the
> removal of four.) The paradox is that sometimes this obvious
> maximum-removal play is a mistake.
Hi Paul,
as an example is already give just a short remark from me: I am not sure, if
I would call it a "paradox". I found the following definition for a paradox:
"A paradox is an apparently true statement that seems to lead to a logical
self-contradiction, or to a situation that contradicts common intuition. Put
simply, a paradox is 'the opposite of what one thinks to be true.'"
As I have the feeling, that in few cases its better not to bear off, in my
point of view its not a paradox, but just a fact. Well, but thats my humble
opionion ;-)
Yours,
Hardy ;-)
--
Hardy_whv on FIBS :-) www.hardyhuebener.de (Last Update: August 2004)
Paul Epstein
Thanks but I do quibble with your quibble with my use of the word
"paradox." Yes, it's not literally a paradox, but this seems to be a
standard word for unusual backgammon positions.
After all, Latto's paradox, Jacoby's paradox, and Kauder's paradox are
not paradoxes either but they are still referred to as paradoxes in
the literature, and I specifically remember Robertie using the word
"paradox" to describe some of these three positions.
Why is the situation I describe less of a paradox than the Latto,
Jacoby and Kauder positions? Or are they not paradoxes either?
Paul Epstein
Paul Epstein
Perhaps. But then, surely you should also take issue with Latto,
Kauder, and Jacoby (or perhaps others who cited their positions and
referred to them as "paradoxes".) Neither you nor Gregg appear to
have any problem when the word "paradox" is used to describe positions
named after the above three.
Are you sure you're not just singling me out for criticism because I'm
an unknown player?
If you're implying that positions such as I ask for happen often, then
you're very much mistaken. It does indeed happen often that, in a
non-contact bearoff, a player fills in gaps with both dice and
declines to remove a single man. But that's not what I asked. It's
extremely rare that a player rolls non-doubles and eschews (correctly)
an opportunity to remove two men.
Paul Epstein
Hardy Hübener
> Perhaps. But then, surely you should also take issue with Latto,
> Kauder, and Jacoby (or perhaps others who cited their positions and
> referred to them as "paradoxes".) Neither you nor Gregg appear to
> have any problem when the word "paradox" is used to describe positions
> named after the above three.
> Are you sure you're not just singling me out for criticism because I'm
> an unknown player?
Your thin is much too thin! As I said ...
>>Well, but thats my humble opionion ;-)
.. NO CRITICISM, just an opinion. Okay? Calm down again. And for sure, I
don't care if you are a known player or not.
> If you're implying that positions such as I ask for happen often, then
> you're very much mistaken. It does indeed happen often that, in a
> non-contact bearoff, a player fills in gaps with both dice and
> declines to remove a single man. But that's not what I asked. It's
> extremely rare that a player rolls non-doubles and eschews (correctly)
> an opportunity to remove two men.
Well, the question is how "frequent" is "frequent". Especially if you are an
extreme underdog during bearoff, only few rolls left (1-2), you have to
optimize for your joker rolls. Those are doubles. In that case its quite
freqently correct, to optimize your man for those jokers.
If thats paradox ("a situation that contradicts common intuition") probably
depends on the "common intuition". Perhaps mine is not common enough ;-)
Gregg Cattanach
news:9b562672.0408...@posting.google.com...
The redouble but not initial double is a paradox because the math in almost
every 'regular' position indicates this is impossible. Filling a gap or
bearing off a checker are both constructive actions during a bearoff, it's
just an assumption (not valid) that bearing off a checker with one die is
always best.
Gregg C.
Gregg Cattanach
>
> Are you sure you're not just singling me out for criticism because I'm
> an unknown player?
>
> If you're implying that positions such as I ask for happen often, then
> you're very much mistaken. It does indeed happen often that, in a
> non-contact bearoff, a player fills in gaps with both dice and
> declines to remove a single man. But that's not what I asked. It's
> extremely rare that a player rolls non-doubles and eschews (correctly)
> an opportunity to remove two men.
>
> Paul Epstein
We're not picking on you, it's just semantics. If you want to call the
two-dice, two checker thing a paradox, that's just fine, you're not far off.
par·a·dox , n.
1. a statement or proposition that seems self-contradictory or absurd but
in reality expresses a possible truth.
2. a self-contradictory and false proposition.
3. any person, thing, or situation exhibiting an apparently contradictory
nature.
4. an opinion or statement contrary to commonly accepted opinion.
Gregg C.
Carlo Melzi
"There exist some checkers position where, after rolling two dices
that allow you to make the 5 point, it is better not to make the
point."
With such a large definition of paradox, we could find plenty in
backgammon.
I guess what's really interesting in a REAL paradox is the underlying
mathematic that seems impossible (i.e. double/take and no double/take
in a slightly better position is my favourite).
Carlo Melzi
tallrock
the well known match doubling window equation ( risk/ risk + gain). There
are situations where it is correct to double (when gammons aren't an issue)
with a game winning chance below the minimum threshold suggested by the
equation.
Pop quiz folks... who can suggest a sample position and score, and provide
the rationale for this apparent "paradox?"
Paul Epstein
Carlo,
Forgoing an opportunity to make the 5 point is absolutely a
non-paradox because it happens all the time -- in perhaps 50% of
games.
The scenario I asked for definitely happens in fewer than 1 in 10,000
games and is far, far rarer than the second example of your posting --
your "favourite" paradox. [The example demonstrated by Gregg gives a
position that is very unlikely to occur in practice.]
There are (at least) two respects in which the scenario I asked for
satisfies a reasonable definition of a backgammon paradox.
1) The object is to remove all your checkers. So, forgoing an
opportunity to remove checkers is already very slightly "paradoxical"
in a non-contact position.
2) The scenario I asked for is actually very rare indeed. Robertie
estimates that it would be unlikely to occur in a person's entire
lifetime of playing backgammon. (And I'm sure he's thinking in terms
of people who play an enormous amount).
Paul Epstein
Bob Koca
From 1995 on this group:
http://groups.google.com/groups?selm=9509221927.ZM6062%40bobrae.bd.psu.edu&output=gplain
Related to this I've noticed that the Trice - Jacobs match equity
table for a 50 FIBS point advantage gives the underdog less chance of
winning at 4 away 2 away than it does at Crawford 2 away 1 away.
However the clearly suboptimal strategy of always doubling at the
first opportunity when down 4 away 2 away achieves the same equity as
being down Crawford 2 away 1 away.
Perhaps their algorithm for cube actions doubles too slowly when the
opponent is 2 away. I will guess that preventing the 2 away player
from eventually claiming with a cube is not adequately accounted for.
,Bob Koca
tallrock
"Bob Koca"
>
> Related to this I've noticed that the Trice - Jacobs match equity
> table for a 50 FIBS point advantage gives the underdog less chance of
> winning at 4 away 2 away than it does at Crawford 2 away 1 away.
> However the clearly suboptimal strategy of always doubling at the
> first opportunity when down 4 away 2 away achieves the same equity as
> being down Crawford 2 away 1 away.
> Perhaps their algorithm for cube actions doubles too slowly when the
> opponent is 2 away. I will guess that preventing the 2 away player
> from eventually claiming with a cube is not adequately accounted for.
>
> ,Bob Koca
I believe you are on the right track with your conclusion... but it is not
true that "always doubling at the
first opportunity when down 4 away 2 away achieves the same equity as
being down Crawford 2 away 1 away." The "match underdog" may already be a
substantial "game underdog "(or favorite) by the time the cube may be
turned. Imagine your opponent opens with 3-1, and now you cube...
Hugh McNeil
Michael Sullivan
> Paradox 3: The Latto Paradox. In money play, some checker positions
> are such that the cube action is redouble/take if the roller has the
> cube, but a hold if the cube is in the middle.
Does anyone have an example of the Latto paradox that could plausibly
arise from correct play by botth sides?
The basic idea is that you get to a position where you have a double,
but can still lose some gammons -- it's correct to redouble, but it's
not correct to initial double, because you activate gammons that way.
Trivial to construct, but all the ones I've seen are the result of
backgames getting a good result. In every case, it's hard to imagine
that the opponent did not have an opportunity to make a *clear* double
beforehand (remember it would *activate gammons* and they could not be
too good) -- thus the initial double position would never arise in
actual play.
I would feel so much more satisfied if a Latto paradox position could be
found which included possible paths where correct play would yield
either a centered cube or a 2+ cube
I've tried to produce such a beast, and been so far unable to do so (not
that I've tried particularly hard).
Michael
Roland Scheicher
> in other words. In money play, I roll such that each die allows me to
> bear off a checker. (In other words, I've either rolled a non-double
> which allows the removal of two checkers or a double which allows the
> removal of four.) The paradox is that sometimes this obvious
> maximum-removal play is a mistake.
>
In "The Backgammon Book" by J. Crawford and O. Jacoby the authors
describe a simple race position, where you should not bear off the
maximum of checkers.
Suppose you have six checkers left: two on the one point, two on the
two point and two on the three point. Your opponent has four checkers
on his one point. You roll double one.
1) If you decide to bear off the maximum of checkers, you remove the
two checkers from your one point and one from your two point. You win,
if and only if your opponent does not throw a double and you throw
double 3 or better on your next roll, thus your probability to win is
5/6 * 4/36 = 20/216 = 9,26%.
2) If you decide to bear off the two checkers from the one point and
move the two checkers from your three point to your two point, you
will win, if and only if your opponent does not throw a double and you
throw double 2 or better on your next roll, thus your probability to
win is 5/6 * 5/36 = 25/216 = 11,57%.
Remark 1: If your opponent had five or six checkers on his one point
you obviously should bear off three checkers, but since your opponent
has only four checkers left you should bear off only two - sort of a
"paradox".
Remark 2: This position does neither deal with overstacked points nor
with the question of filling gaps as other examples do.
Yours
Roland
Gregg Cattanach
Your example isn't really the same as the OP was thinking of. He was asking
about situations where you can bear two checkers off directly with the 2
numbers on the dice. It is extrodinarily rare to have this be incorrect.
Your example takes the last checker off with 2 dice, (the last two ones).
There are many situations in quite 'normal' positions where it is better to
shuffle the checkers around with 2 dice instead of combining them to take
one checker off.
Here's an excerpt from that article that was on Gammonline that lists
several of the OP's original 'paradoxes'.
=============================================
- 2 - DECLINING TO BEAR A CHECKER OFF
This is an extremely rarely correct play.
Walter TRICE states that such plays must be made only with at least 10
checkers on the board, when being ahead in the race (actually we bring the
following correction : being on the favorable side of a break-even point,
the break-even point depending on the position of the player on move) and
necessarily with the idea of filling a gap.
As we shall see, there exist another family of positions.
With a 2, one can fill a gap on the 2-point ...
5 1 5 1 0 0 / 5 4 4 0 0 0 (an example familiar to MAGRIEL's readers)
The right 2 is : 4/2
because of the sequences 43/53/63/54/64/65 followed by two 2s.
... or on the 3-point...
5 1 0 5 1 0 / 5 4 4 0 0 0
The right play for a 2 is : 5/3 , not 2/Off
because of the subsequent rolls containing 5s or 6s followed by various
rolls containing 2s and 3s
... or even on the 4-point
5 1 2 0 5 1 / 0 5 5 5 0 0
4 1 6 0 0 2 / 0 2 3 2 3 3
The right play for a 2 is 6/4, not 2/Off in both positions.
With a 3, one can fill a gap on the 2-point, if the 3-point is also
sensitive ...
5 0 1 5 1 0 / 5 4 4 0 0 0
... and also if it is not ...
6 0 6 0 3 0 / 4 3 0 1 0 7
32 should be played 5/2 5/3
2 0 2 0 2 0 / 0 0 1 3 2 7
31 should be played 1/0ff 5/2 !!!
This is incredible, and naturally the obvious 3/Off is correct in front of
almost every opposite position. Here the reason is a subsequent 42 which
could possibly played 5/3/Off.
After 1/Off the non bearing off play is made with only 5 men on the board,
which contradicts seriously Trice's first statement
(Note : this position has been found by ourselves many years ago).
One can also fill a gap on the 3-point :
4 1 1 6 0 2 / 0 2 2 3 4 4
The right play for 3 is 6/3, not 3/Off
With a 4, one can fill a gap on the 2-point :
4 0 6 1 0 2 / 0 2 3 3 2 3
The right play for 4 is 6/2, not 4/Off
--
Gregg C.
Bob Koca
The only way your objection is valid is if the market has been
lost when the cube is turned. If your opponent wins the first roll and
you cube, clearly it is a take at 4away 2 away. Suppose you have won
the first roll, your opponent rolls and then you cube. What is the
best sequence for you? A 31 followed by a 54 perhaps? This is still an
easy take.
Note that I was not advocating doubling immediately at the score
4away 2away if your opponent starts with a 31. I was pointing out that
if one were to use the strategy of always immediately doubling then
one's equity at 4away 2away is equal to that of 2away 1away Crawford.
Since that strategy is suboptimal, the actual equity when down 4 away
2away must be larger than 2 away 1 away Crawford.
Here is a proof:
THEOREM: Assuming that there is no market losing sequence in the
first 2 rolls then the strategy "double at the first opportunity"
gives the 4away 2away player the same match equity as though the score
were Crawford 2away 1 away.
PROOF:
Case 1: Suppose you lose the game. Then starting at either
score you have lost the match.
Case 2: Suppose you win a single game. Then the score would
either be 2away 2away or 1away 1away. This gives the same equity
because the next game will be for the match either way. (Comment:
Note that equal strength opponents was not assumed here).
Case 3: Suppose you win a gammon or bg. Then you have won the
match starting at either score.
,Bob Koca
Walter Trice
...
>
>
> Related to this I've noticed that the Trice - Jacobs match equity
> table for a 50 FIBS point advantage gives the underdog less chance of
> winning at 4 away 2 away than it does at Crawford 2 away 1 away.
> However the clearly suboptimal strategy of always doubling at the
> first opportunity when down 4 away 2 away achieves the same equity as
> being down Crawford 2 away 1 away.
> Perhaps their algorithm for cube actions doubles too slowly when the
> opponent is 2 away. I will guess that preventing the 2 away player
> from eventually claiming with a cube is not adequately accounted for.
>
> ,Bob Koca
I don't quite see the problem. The tables presume that players make
mistakes with the cube as well as in the play of the checkers. These
errors are incorporated into the tables via adjustments to cube
efficiency that differ for the two unequal players.
At Crawford -2/-1 the rules do not allow either player to drop a take,
to double early, to cash when too good, or simply to forget about the
cube. At -4/-2 the rules permit all of these, and I have seen opponents
make all kinds of mistakes at the -4/-2 score.
In the 50-rating point table we show the stronger player with 71.0% at
-2/-4 and 70.2% at c/-2, and with 33.1% at -4/-2 and 32.3% at -2/c. Thus
the higher-rated is always doing better when his opponent has the chance
to make a mistake with the cube. This seems to me to be both logically
consistent and, in practice, reasonably realistic. Of course it is
possible to encounter an opponent who might realize that his judgment is
so bad that he would do better to double at the first opportunity at
-4/-2, but then I never have.
-- Walter Trice
Paul Epstein
even with accurate play, to the cube being in the middle or on the
side of the back-gamer. It is possible, for example, that the
not-on-roll side got into a mess by rolling 44 on the last roll. And
it's also possible that the not-on-roll side never had a double
because of all the opportunities to leave double and triple shots.
Paul Epstein
m...@panix.com (Michael Sullivan) wrote in message news:<1gizub0.kn3s481h7ock5N%m...@panix.com>...
Michael Sullivan
> Actually, I think that the checker position already given could lead,
> even with accurate play, to the cube being in the middle or on the
> side of the back-gamer. It is possible, for example, that the
> not-on-roll side got into a mess by rolling 44 on the last roll. And
> it's also possible that the not-on-roll side never had a double
> because of all the opportunities to leave double and triple shots.
Remember that the first of these is not an option -- the Latto paradox
only happens in money play with Jacoby. If the not-on-roll side had a
great position and got slaughtered by a root on the last roll, then they
were pretty stupid not to double, given that too good is impossible.
Even if they had a poor-ish posittion right through the leaduup to the
backgame and were worried about the possibilities of double and triple
shots, it's very hard to imagine that the advantages of activating
gammons with such a huge lead in the race and so many opposing checkers
back wouldn't at some point, have pushed things over the top into a
double.
Michael
Paul Epstein
"...very hard to imagine." Perhaps, and that's why Latto paradox
positions are very rare.
"Remember that the first of these is not an option." Not sure what
that sentence of yours means. In a previous thread, a Latto paradox
position was found. Imagine that the not-on-roll side just rolled a
44. We can then reconstruct the previous position. You now appear to
say that, if this previous position occurs, the cube doesn't stay in
the middle with best play. However, I disagree with you. Take the
Latto paradox position. Assume it arose from a 44 and retract that
44. My assessment of that position is: not-good-enough-to-double
(whether the double is initial or re-). This is not a case of a great
position being slaughtered by a root number. The player defending
against a back game never had a great position to begin with. In any
case, I don't think the concept of a "root" applies here. A root
involves restrictions on one's options because of the opponents
points. However, in the two positions I'm referring to (before and
after the 44), the back-gamer doesn't have any points. [However,
that's a quibble -- the main point is that 44 is a horror shake or, in
Snowie language, an "anti-joker."]
Paul Epstein