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# Expectation Relative to Opponent Strength (was: cube decision)

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### Darse Billings

Jul 25, 1994, 12:39:53â€¯PM7/25/94
to
[follow-ups to rec.games.backgammon]

In rec.games.backgammon kwoo...@netcom.com (Kit Woolsey) writes:

>Wayne King (wk...@ritz.mordor.com) wrote:

>: 13 18 19 24
>: +-----------------------------------------+ X
>: | . X . . . O | | X X X X X . |
>: | X O | | X X X X |
>: | | | X X X |
>: | | | | ---
>: | | | | | 1 |
>: | |BAR| | ---
>: | | | |
>: | | | |
>: | | | O |
>: | | | O O O O O |
>: | X . . . . O | | O O O O O O |
>: +-----------------------------------------+ O
>: 12 7 6 1

>: Pipcounts: O = 86, X = 86

>: Match Score O-3 away
>: X-4 away

>: O on roll. Cube action?

>: The following position came up in my 2nd round match of Snoopy's
>: tournament. As O this position puzzles me. O has many market losers which
>: would seem to point to doubling, yet X has a very efficient re-double if
>: O misses. I would appreciate any thoughts as to (1) what is the correct
>: cube action and (2) how one in general would take into account an opponents
>: re-double when deciding whether or not to double.

>: Wayne King (wking on FIBS)

>Very interesting problem. I could write a book about match cube
>decisions like this (come to think of it, I *have* written such a book).
>The score 3 away, 4 away is perhaps the trickiest match score of all.

>The first step toward attacking any decision on whether or not to double
>should always be to put yourself in the other guy's shoes and ask if he
>should take. Sometimes this solves the problem quickly. If the answer
>is no, or maybe not, then the double is automatic. If he has a monster
>take then probably you shouldn't be doubling unless the position is
>super-volatile (which admittedly this one is). In this sort of position
>if your opponent has a Big take you probably should not be doubling if
>you are ahead in the match, since he has more cube leverage. However if
>you are behind in the match then you should be inclined to double if
>there is a sufficient chance you will lose your market. However if his
>take is fairly close, then it is usually correct to double even if you

>Now, to the actual position. In order to avoid overly complex math, I'm
>going to make some rough assumptions -- the way I would do at the table.
>The idea is to make these assumptions about equally favorable for both
>players, so you can get a good approximation of the equity in the
>position. Let's suppose that if O hits the shot he always wins a single
>game (he might not, but that may be compensated for by a possible
>gammon). If O misses the shot X does have huge cube leverage, since O's
>take point is 40%. Let's assume that X will redouble, O will take, and
>that X will win 60% of the time. O has 16 hitting numbers out of 36.
>Therefore: 4/9 of the time O will be ahead 1 away, 4 away, for 83%
>equity. 5/9 of the time they will play for the match, with O's equity
>40%. Averaging these out, O's equity is about 64%. Since if X passed
>O's equity at 4 away, 2 away is 68%, it looks like X has a fairly close
>decision, so it might be right for O to double since a lot rides on the
>next roll.

>Let's see if this turns out to be correct. Suppose O doesn't double. If
>he hits by our assumptions he will win and be ahead 2 away, 4 away for
>68% equity -- this happens 4/9 of the time. If he misses X won't be
>flinging the cube over automatically, so let's just assume that X will
>win a 1-game 60% of the time and O will win a 1-game 40% of the time.
>Since O is missing 5/9 of the time, he will win 2/5 of these, so he will
>win 2/3 of the time for 68% equity and lose 1/3 of the time for 50%
>equity. This averages to 62% equity. Since we already estimated O's
>equity in the match at 64% if he doubles, it looks like it is correct to
>double.

(There is a minor error in the above arithmetic, but this has been
pointed out and is not relevant to the overall method).

>I make no claims that this is the correct answer. I made several rough
>and questionable assumptions, and if you change some of them it might
>well change the final result. Also I give no guarantees that my
>arithmetic is correct. However, this is the approach one should take at
>the table.
> Kit

Thank you for demonstrating this method of analysis, Kit!

This leads me to a question I have been pondering for some time...

To what degree do you take the strength of your opponent into account
in cube decisions (and, to a lesser extent, checker playing decisions)?

In poker, a strong player (call her SP) will frequently pass on small
positive expectation plays, in order to reduce variance. For example,
in a no-limit poker tournament, SP will often fold a hand rather than
committing her entire stack of chips, even though she is certain that
calling is a positive expectation play -- and she is not wrong to do so!

The reason is that, as a strong player, SP has an inherent advantage
(positive expectation) over her opponents in each hand that is dealt.
Before committing everything, SP must be sure that her expectation for
the play *exceeds* this inherent advantage. Otherwise, she is much better
off waiting for a more profitable situation in which to invest her money.
Against weak opposition, such fine opportunities present themselves with
dependable frequency; and SP cannot take advantage of them if she has
already been eliminated by an unlucky turn of the cards.

More generally, a strong player always faces a trade-off between these
relative positive expectations and the evil of high variance. Accepting
a high variance proposition effectively reduces the number of games
played, and thus short-term luck will have an increased influence on the
outcome, at the expense of the game's skill element.

Beyond these considerations, simply avoiding early elimination may have
additional value. In tournaments where prizes are awarded for high
finishes other than first place, there is an intrinsic value in just
surviving, which again favours more conservative strategies.

Given this, it should come as no surprise that a good player may make
different decisions based on the strength of the immediate opponent, and
(in poker) the strength of the rest of the field. Here is an example:

Suppose I hold a good draw to a winning hand, with a one-in-four chance
of succeeding; but my opponent bets everything I have, which is nearly
twice the size of the current pot. (This roughly corresponds to a
narrow take of a double in backgammon). Early in the poker tournament,
I will pass on this borderline situation, even if there is as much as a
five percent overlay in my favour. Later in the tournament, if only
players of equal caliber remain, I might still pass provided I had
enough chips to survive a while longer. But if I am in a desperate
situation, or if I feel the rest of the field are superior players, I
will happily call this all-in bet. In fact, I will *seek out* this type
of opportunity even when the pot odds are not quite sufficient, in order
to allow the high variance to work in my favour.

Analogously, in order to reduce variance in backgammon, the stronger
player should be somewhat less inclined to accept narrow takes, or to
double in highly volatile positions.

In backgammon problems like the one above, an additional question
exists whenever there is a significant difference in the strength of
the combatants: Is it worth the 2% increase in equity to probably
determine the result of the match on the next roll of the dice?

If O is the stronger player, then he should not double into a probable
redouble unless the increase in equity exceeds his inherent advantage.
On the other hand, if O fears X, he may double even if it means *losing*
equity, provided that loss is smaller than his inherent disadvantage.

There are further implications of this principle that apply to playing
strategy in most games, including poker and backgammon, and even chess
and Go. (Chess and Go are deterministic perfect information games, but
are played between imperfect opponents...)

In particular, it may be *correct* for a weaker player to choose
slightly "inferior" moves which lead to highly volatile (high variance)
positions. A common example in backgammon might be choosing to slot a
checker on the five-point, rather than splitting the back men, with
opening rolls of 4-1 or 2-1. Conversely, it may be incorrect for the
stronger player to make an objectively "best" move if there is a good
alternative which is not as sharp...

In general, the implication is to play a more boring style against
weaker opponents, and a sharper style against superior players. In
other words, you try to grind down a lesser opponent, but go for the
quick kill against someone stronger.

However, it is easy to take this rule too far, which is very dangerous.
Keep in mind that the actual differences are not very significant in
practice. It is almost never correct to pass on a large positive
expectation play merely to reduce variance; nor is it correct to play
recklessly in the hope of trapping a superior opponent.

Cheers, - Darse.

--

char*p="char*p=%c%s%c;main(){printf(p,34,p,34);}";main(){printf(p,34,p,34);}

### Kit Woolsey

Jul 25, 1994, 8:25:36â€¯PM7/25/94
to
Darse Billings (da...@cs.ualberta.ca) wrote:

: This leads me to a question I have been pondering for some time...

: Cheers, - Darse.

You bring up a very interesting point which has been debated for several
years. In the 1970's when *nobody* knew how to play the game the
"grind'em out" style might have been effective (although even then it was
debatable). Nowadays when most intermediate players play a quite
reasonable game, it is not a good idea at all. In fact, I tend to be
more aggressive with my doubles and take against weaker opposition.
There are two reasons for this:
1) If a double is taken, the game has to be played to conclusion. This
gives the weaker player more opportunities to make mistakes. Anybody can
play the first few rolls decently; it is in the later stages of the game
where the really costly blunders in checker play occur. By being
conservative with the cube hoping to grind them out, you don't give your
weaker opponent a chance to make these mistakes.
2) I don't mind jacking the cube up to the 4-level, even when it will put
one of us way ahead or behind in the match. The reason is that weak
players are more likely to make serious cube errors when the match score
is lopsided. If the score is close then proper cube strategy is roughly
equivalent to money play, so the weaker player's lack of understanding
match strategy won't matter -- his normal cube action is likely to be
correct. If the score is lopsided (either whether I am well ahead or
well behind), my opponent is more likely to make a serious cube error.

The exception to all this occurs when, if I make the aggressive cube
action, the match will be decided on that game and there is no or little
skill left in the game. This is somewhat analogous to your all-in
example in the no-limit poker tournament. For example suppose the
situation is such that if I pass a double I will be behind 2 away, 5
away, while if I take it is for the match (this could occur if I am
behind 4 away, 5 away, and my opponent redoubles to 4). Let's also
assume it is a straight race, so there is little or no skill left.
Against an equal opponent my chances behind 2 away, 5 away are 25%, so I
would act accordingly. However if my opponent is weaker I might up my
estimate of my chances at that score to, say, 35%, so I would take the
redouble only if I had better than 35% chances to win the race. I
emphasize that this is true only in a no-skill position. If the position
is still complex there will be plenty of chances for my opponent to screw
up the play, which means my probability of winning the game in question
might be considerably better than the position itself would indicate.

To summarize: I don't really take my opponent's skill level into account
on my cube decisions unless if the cube is taken it is the last game of
the match. I strongly believe that players who are overly conservative
against weaker opponents are throwing away much of their (imagined?)

Kit

### Joshua E. Randall

Jul 25, 1994, 7:32:23â€¯PM7/25/94
to
In a previous article, <310psp\$o...@scapa.cs.ualberta.ca> wrote:

> To what degree do you take the strength of your opponent into account
> in cube decisions (and, to a lesser extent, checker playing decisions)?
>

> [ remainder of article deleted for brevity ]

This is an extremely interesting topic, and one on which I have not read
nearly enough. A (relatively cheesy and unknown) backgammon book I read
recommended the same conclusion as this poster: play conservatively
against a weak opponent, and opportunistically against a strong one.
My question, though, regards whether this is a valid (mathematically and
"feel"-based) conclusion.

As an aside, a situation came up recently in Snoopy's tournament. I was
playing the third round against an opponent whom I felt was significantly
weaker than me (no offense to that person if he's reading this). After
some initial games I was up 3-1 in a 7 point match. A situation arose
in which he offered me a close double. I took, and ended up winning the
game. But should I have dropped, expecting that I could wear him down
by superior skill?

The next game (I'm now up 5-1, or 2-away, 6-away) after a bad start, I
got my back game going and managed to trap him on the bar. He had
already born off two checkers, though, so my gammon chances were gone.
I foolishly offered a double, which he redoubled. Should I have taken
this redouble? If he won, the match would be tied, and a few lucky
breaks for him could win him the whole thing. If I had dropped, I still
would have had a 5-3 advantage. (In actuality, I took and won.)

Comments? Did I do the right thing, or was I dumb but fortunate? (Hey,
as I always say, "better to be lucky than good" :-)

--
--
Joshua E. Randall ran...@minerva.cis.yale.edu

### Christopher Yep

Jul 26, 1994, 12:57:25â€¯AM7/26/94
to
In article <1994Jul25....@news.yale.edu>,

Joshua E. Randall <ran...@minerva.cis.yale.edu> wrote:
>In a previous article, <310psp\$o...@scapa.cs.ualberta.ca> wrote:
>
>> To what degree do you take the strength of your opponent into account
>> in cube decisions (and, to a lesser extent, checker playing decisions)?
>>
>> [ remainder of article deleted for brevity ]
>
>This is an extremely interesting topic, and one on which I have not read
>nearly enough. A (relatively cheesy and unknown) backgammon book I read
>recommended the same conclusion as this poster: play conservatively
>against a weak opponent, and opportunistically against a strong one.

I disagree with their conclusions. See below.

>My question, though, regards whether this is a valid (mathematically and
>"feel"-based) conclusion.

It depends what assumptions you wish to make.

I believe that if one complicates a position even slightly (similar to
"playing opportunistically"), a weaker player will butcher the position -
he will make much more serious mistakes, and he will have more time to
make them, because the game will usually last longer.

In a game that is more or less a "Mediterranean-style" racing game, a
weaker player usually won't make many serious mistakes, provided he can
at least make reasonable cube decisions based on races (not that hard to do).

I think that in general the skill difference between two players can best
be seen by how they play complicated positions. Mistakes involving huge
chunks of equity can occur in these positions.

I would suggest either playing roughly the same, or striving slightly to
complicate the position against a weaker opponent. There will be more
gammons, should you decide to complicate the position, and thus
potentially more variance (which is bad for the better player), but I think
the additional mistakes (and often more severe mistakes) that your
opponent is likely to make, will more than offset the fact that the games
may be more mutually gammonish.

As for cube decisions, I wouldn't change your strategy too much, except
in games in which there is very little skill left. Against a weaker
opponent, you might think that you should pass "marginal take positions,"
and hold the cube in "marginal double positions." Remember, though, that
if you really are better than your opponent, that your built-in skill
advantage will show itself during the remainder of the game, so that your
position really is better than it looks!

So the 2 considerations,

(1) play more conservatively with the cube to make it more likely that
(2) you are the better player, so your winning chances are greater than
they appear

more or less cancel each other out.

[Note: I'm assuming that "you" are the better player.]

For positions with at least a small amount of skill left in them (most
positions, other than races):

The above ideas are generally valid for your take/pass decisions.
For your double/no double decisions, note that most weaker players will
accept positions that are actually drops, sometimes even intentionally
(i.e. the weaker player knows that it "should" be a drop). By my above
comments, I feel that the weaker player has actually erred - he would
normally have a better chance at winning the match if he just played as
well as possible. Also, my personal opinion is that it is embarrassing
to knowingly play incorrectly against a better opponent, although this is
another topic. Of course, if the cube has reached a high level (4,8+),
especially in a very gammonish position, than it may very well be okay
for the weaker player to have "incorrectly" accepted. If you have a
suspicion that your opponent often accepts clear drops, then you may be
able to gain a little more equity by trying to wait to double until the last
possible moment when he will still accept. Use your judgement here, and
I would suggest being conservative (i.e. double when you think you
"should," as if you were playing an equal opponent).

For positions in which there is little skill remaining (races, a few other
positions):

Here, you can be conservative in your doubling decisions. However,
don't be too conservative... Consider this:

Backgammon is a game of little things, with a significant amount of
luck. Since you play on FIBS, I'll try to relate this in FIBS-terms:
A rating difference of 133 points shows a large difference in skill (in
my opinion). However, in a 7 pt. match, the weaker player can be
expected to win 40% of the time, according to the rating system.
Considering that a 7 pt. match typically lasts between 4 and 10 games,
40% is quite a lot for such a significant difference in skill between the
two players. This means that in general you should play as you would
against an equal player, and only occasionally, should you drop a very
close take, or hold the cube instead of doubling in a very close double/no
double decision.

The above is only for positions of no (or very little) skill. For
positions with a significant amount of skill remaining, see above (my
conclusion was to not alter your strategy, in general).

Note: A rating difference of 203 pts. (a very large difference in skill)
corresponds to the weaker player having 35% match chances - still quite
high.

>
>As an aside, a situation came up recently in Snoopy's tournament. I was
>playing the third round against an opponent whom I felt was significantly
>weaker than me (no offense to that person if he's reading this). After
>some initial games I was up 3-1 in a 7 point match. A situation arose
>in which he offered me a close double. I took, and ended up winning the
>game. But should I have dropped, expecting that I could wear him down
>by superior skill?
>
>The next game (I'm now up 5-1, or 2-away, 6-away) after a bad start, I
>got my back game going and managed to trap him on the bar. He had
>already born off two checkers, though, so my gammon chances were gone.
>I foolishly offered a double, which he redoubled. Should I have taken
>this redouble? If he won, the match would be tied, and a few lucky
>breaks for him could win him the whole thing. If I had dropped, I still
>would have had a 5-3 advantage. (In actuality, I took and won.)

I hope that my interpretation (assumption) of the events is correct:

1. You were playing a backgame, when your opponent left a shot.
2. You hit the shot, and later closed out your opponent, with him
having only borne off 2 men, freeing your back men at some point in the
process.
3. You either are ready to bear off, or will be, once you bring in your
stragglers.
4. Since your opponent was playing against your backgame, he probably has
gaps in his board, probably gaps on some of the lower numbers.

If 1-4 are correct, then I would estimate your opponents winning
chances between 5 and 10%, depending on the positioning of your 3 spares,

5. You doubled your opponent, expecting him to drop. [Were you
expecting him to drop?] When he accepted, you were so surprised that you
6. Your opponent redoubled to 4 on his (next) turn. Against an equal
opponent, you realized that this was of course a trivial take, but
against your opponent, you thought that maybe you could try to make up
for your "mistake" of doubling him in the first place, and try to "grind
him down" from 2-away/4-away, by dropping his redouble.

My first comment is that if you are 2-away, you should think very
redouble you to 4 on his next turn, and think about whether you want the
cube to be at 4. [99.9%+ of the time, your opponent should redouble you
on his turn. < 0.1% of the time (probably), his position will have
gotten so good during that half-turn (i.e. since you rolled so badly),
that he is now playing on for gammon, instead of redoubling you to 4.
However, I've never seen this happen in any game [has anyone else?]) If
you're uncomfortable with the cube being at 4, then it's probably not a good
idea for you to be doubling to 2 in the first place!

However, in the actual position (which is a gammonless position btw) your
opponent was the one who made a big error. His take point was 20% (I'll
throw some numbers at you, don't worry too much about them, since I think
the ultimate conclusion is clear by a large margin - i.e. he should have
chosen to accept it (as in the actual match), it's virtually assured that he
wouldn't make any future doubling mistakes in the game (i.e. since he would
instantly redouble to 4), and the cube would be at a high value (4).
Furthermore, *his* position is relatively easy to play. These criteria argue
for a slightly more liberal take, ... but not by very much. (Reread
everything I've written above). Your opponent would have to be unbelievably
worse before taking would be profitable for him.

Since your opponent was almost surely wrong to accept your double, then
naturally you would accept his redouble (since his take point of 20% is
based on the assumption that you are essentially offering to make the
cube a "dead" 4-cube). Being a 90-95% favorite, it's almost unimaginable
to drop your opponent's double, even though it's at the 4-level.

If you like numbers, I'll throw another one out for you:

If you think that you might have erred in doubling to 2, then when your
opponent offers you the cube at 4, against an equal player you should
accept if you think that you can win 36% of the time. Against a weaker
player, you may sometimes adjust 36% upwards very slightly, depending on
the position. However, being a 90-95% favorite, it is almost unthinkable
to drop the redouble. I think that you should accept the redouble even
against a non-human (a dog, cat, mouse, bacteria, etc.) [presumably humans
are better players than non-humans] :-)

>
>Comments? Did I do the right thing, or was I dumb but fortunate? (Hey,
>as I always say, "better to be lucky than good" :-)
>
>
>
>
>
>--
>--
>Joshua E. Randall ran...@minerva.cis.yale.edu

Something to remember: In a gammonless game, if the cube is in the middle
(at 1 of course), and the player on roll is 2-away, with > 83% chance to
win the game, then the "correct" cube-action is always double-drop
(assuming the game isn't crawford).

--
Christopher Yep chri...@soda.berkeley.edu
as soda.berkeley.edu is being physically moved, and renamed this week.

### Darse Billings

Jul 26, 1994, 3:00:27â€¯PM7/26/94
to
>Darse Billings (da...@cs.ualberta.ca) wrote:

>: This leads me to a question I have been pondering for some time...

>: To what degree do you take the strength of your opponent into account
>: in cube decisions (and, to a lesser extent, checker playing decisions)?

>: [...]

>: In backgammon problems like the one above, an additional question
>: exists whenever there is a significant difference in the strength of
>: the combatants: Is it worth the 2% increase in equity to probably
>: determine the result of the match on the next roll of the dice?

>: In general, the implication is to play a more boring style against
>: weaker opponents, and a sharper style against superior players. In
>: other words, you try to grind down a lesser opponent, but go for the
>: quick kill against someone stronger.

>: However, it is easy to take this rule too far, which is very dangerous.
>: Keep in mind that the actual differences are not very significant in
>: practice. It is almost never correct to pass on a large positive
>: expectation play merely to reduce variance; nor is it correct to play
>: recklessly in the hope of trapping a superior opponent.

First, I should add a couple of points for clarification.

The above analysis is based on only *one* aspect of the game -- theoretical
expectation and variance. The statements are mathematically valid, but not
necessarily *relevant* to any particular game situation, which is why I was
careful to qualify each conclusion.

There are many other factors involved in each checker playing decision or
doubling decision. These other factors may be in conflict with the above
directions, and may well take precedence over them. That is the thrust of
my question -- to what degree is the above logic applicable to backgammon
*in practice*?

One factor beyond expectation and variance is playing to exploit the
opponent's weaknesses. If your opponent is weaker precisely because he
cannot handle sharp positions very well, then obviously it will be
counter-productive to avoid sharper positions. Conversely, if your
strength is in dull technical positions, then you are unlikely to be
successful by suddenly opening up against a better player.

None of this contradicts what I have said -- it only puts it in proper
perspective. Based on Christopher Yep's comments, it may be the case that
"strength" in backgammon is most often exhibited by the ability to handle
sharp positions and make the appropriate cube decisions. If this is true,
then the mathematical facts I have presented may be of limited usefulness
to backgammon.

It may be that in backgammon, it is difficult for the stronger player to
reduce variance without also sacrificing much of the playing advantage
she enjoys. In other games, such as chess, it may be easier for a strong
player to control the nature of the position without reducing the winning
chances.

When I played competitive chess, I was often able to control the tempo of
the game to suit my preferences, but I was also rather one-dimensional.
I usually employed a slow, strategic, "positional" style, which usually
resulted in dull, technical positions. I might have been described as a
"low-variance" chess player. Consequently, I seldom lost to a lower
rated player, but in turn rarely defeated a stronger opponent. That
style also resulted in a relatively large number of draws, against both
stronger and weaker opposition. One downside of this approach was that
weaker players were sometimes "forced" to make decent moves, because the
bad moves were always obvious. With a more balanced approach to the
game, I might have adjusted to both the strength and the style of the
opponent, and given the weaker players "enough rope to hang themselves".

A flexible chess player will, of course, adapt her style to play against
an opponent's weakness. But all else being equal, she should attempt to
play a long, grinding game against a weaker player; and a sharp, tense
game (leading to complex positions where neither side can see all the
outcomes) against a superior opponent...

kwoo...@netcom.com (Kit Woolsey) writes:

>You bring up a very interesting point which has been debated for several
>years. In the 1970's when *nobody* knew how to play the game the
>"grind'em out" style might have been effective (although even then it was
>debatable). Nowadays when most intermediate players play a quite
>reasonable game, it is not a good idea at all.

Keep in mind that I am not suggesting an *extreme* shift in styles...
Indeed, most positions may not even have valid alternative moves worth
considering. I would never advocate playing only dry running games
against weaker players, simply because those positions also make it less
likely that your opponent will make serious mistakes.

The changes would be more subtle than that, and might occur infrequently.
Perhaps you have a choice between two equally good moves, one of which
adds strength to a prime, while the other hits two men but leaves blots
in your home board. Or perhaps you can chose to leave only one loose
checker instead of two. All else being equal, you want to chose the
alternative which will force the largest number of skillful moves to be
made over the remainder of the game. That is, you want to avoid the
option which will (more or less) decide the game with one roll of the
dice. Beyond this, you want to chose the position which will keep the
cube on a smaller number, in order to increase the number of decisions
to be made later in the match.

I have heard that in backgammon, a strong player has little advantage
over an intermediate player based solely on checker play. Clearly then,
the theoretical ideas I have offered should have little influence on
checker playing decisions against those opponents (but the same ideas
might still be relevant to cube decisions, where the stronger player's
advantage actually lies). In this case, changes in checker play would
only be useful against particularly weak opponents.

> In fact, I tend to be
>more aggressive with my doubles and take against weaker opposition.
>There are two reasons for this:
>1) If a double is taken, the game has to be played to conclusion. This
>gives the weaker player more opportunities to make mistakes. Anybody can
>play the first few rolls decently; it is in the later stages of the game
>where the really costly blunders in checker play occur. By being
>conservative with the cube hoping to grind them out, you don't give your
>weaker opponent a chance to make these mistakes.

This is consistent with the idea of maximizing the number of difficult
moves in the game. So we have a trade-off situation, and I'll gladly
trust your judgement regarding which factor is the most important, and
which is merely compensation.

>2) I don't mind jacking the cube up to the 4-level, even when it will put
>one of us way ahead or behind in the match. The reason is that weak
>players are more likely to make serious cube errors when the match score
>is lopsided. If the score is close then proper cube strategy is roughly
>equivalent to money play, so the weaker player's lack of understanding
>match strategy won't matter -- his normal cube action is likely to be
>correct. If the score is lopsided (either whether I am well ahead or
>well behind), my opponent is more likely to make a serious cube error.

This is a good point which also arises in the poker tournament setting.
Weaker players are much more apt to call an all-in bet from a strong
opponent if there is a chance of eliminating that player. On the other
hand, they are also more prone to panic in a difficult or desperate
situation when they have the short stack. Correct decisions should be
based primarily on the expectation (and variance) of the play, without
much regard to potential elimination, so the advantage for the strong
player tends to be greater in these lop-sided situations.

>The exception to all this occurs when, if I make the aggressive cube
>action, the match will be decided on that game and there is no or little
>skill left in the game. This is somewhat analogous to your all-in
>example in the no-limit poker tournament. For example suppose the
>situation is such that if I pass a double I will be behind 2 away, 5
>away, while if I take it is for the match (this could occur if I am
>behind 4 away, 5 away, and my opponent redoubles to 4). Let's also
>assume it is a straight race, so there is little or no skill left.
>Against an equal opponent my chances behind 2 away, 5 away are 25%, so I
>would act accordingly. However if my opponent is weaker I might up my
>estimate of my chances at that score to, say, 35%, so I would take the
>redouble only if I had better than 35% chances to win the race. I
>emphasize that this is true only in a no-skill position.

Your match equity computations are based on having an even 50% chance of
winning any given game. Can those equations be modified to account for
opponents of different strength? That is, can you use a base level of,
say, 0.6 equity per game, to account for your superior skill, and thereby
dispense with the need for estimates? I would find an example of those
calculations very interesting.

Your example also has some rather extreme conditions -- a match going
double and a no-skill position. But if it is true for this case, is it
not also true, to a lesser degree, for large swings in match score and
positions requiring a minimal amount of skill? I take it you feel that
in common practice, the factor of opponent strength is quickly dominated
by other considerations. Again, I must defer to your wisdom, and thank

> If the position
>is still complex there will be plenty of chances for my opponent to screw
>up the play, which means my probability of winning the game in question
>might be considerably better than the position itself would indicate.

You correctly point out that the potential increase in the number of
future games must be balanced against the added number of difficult
decisions in the current game. In a highly complex position (presumably
within your own capacity but beyond that of the opponent), maximizing the
number of difficult decisions in the current game may well be worth more
than playing additional games. Again, it is a judgement call.

Nevertheless, there is always *some* value in playing a larger number of
games against a weaker opponent, so future opportunities should at least
be considered. If a modest change in style can significantly increase the
average number of games in a match, then the consequent reduction in
variance should be worth a slight loss in immediate expectation.

>To summarize: I don't really take my opponent's skill level into account
>on my cube decisions unless if the cube is taken it is the last game of
>the match. I strongly believe that players who are overly conservative
>against weaker opponents are throwing away much of their (imagined?)

Yes, it would seem that for backgammon, many factors are working against
the general principle. And as I have already warned, the theory can
actually be very detrimental if taken too far. Perhaps it is best
forgotten by backgammon players, except those who are very experienced
and wish to use it as a subtle refinement for borderline decisions...

Cheers, - Darse.
--
Go is better than Chess. Poker is more lucrative. Sex is more fun.

Darse Billings, 7 kyu; 2065 CFC; meaningless IRC sb/hand ratios:
(rayzor on IRC) Hold'em +0.22 ; HiLo Omaha +0.76

### Joshua E. Randall

Jul 26, 1994, 3:46:38â€¯PM7/26/94
to
In a previous article, <31253l\$s...@agate.berkeley.edu> wrote:

>I believe that if one complicates a position even slightly (similar to
>"playing opportunistically"), a weaker player will butcher the position -
>he will make much more serious mistakes, and he will have more time to
>make them, because the game will usually last longer.

An excellent point which Kit echoed in a later article. This is,
indeed, the OPPOSITE of the recommendation I had read. But, as I said,
this book was "cheesy and unknown" -- well, at least we now know why it
was the latter. :-)

>As for cube decisions, I wouldn't change your strategy too much, except

>in games in which there is very little skill left. [...]

You don't think that one should offer a weaker opponent doubles which
SHOULD be drops, but which he will probably take because he doesn't know

I wrote:

>>As an aside, a situation came up recently in Snoopy's tournament. [...]

I remembered the full details of situation. (It's so ugly, I must have
blocked it out of my memory. :-) Here it is:

I'm up 5-1 against a perceived weaker opponent. He has me on the bar,
where I flunk a few times, and he begins bearing off. However, bad
rolls force him to leave blots, and I finally hit one. After some
milling around (I blot him, he re-enters and blots me, I re-enter and
blot him, etc.) I finally close him out. My stragglers are free to move
(i.e. he doesn't have any sort of prime and I roll well).

Immediately after closing my board, I double, expecting him to drop. He
takes. The game continues. I bear almost all my men off and have two
left on my 1 point; he has 3 left, 2 on his 2 point and 1 on his 1
point. He doubles, to 4. I saw it as: he needs to rolls double 2's or
higher, so slightly less than 1 in 6. If he fails I automatically win
the game and the match. If he succeeds the match is tied at 2-away.
What should I have done? (I took, he didn't roll doubles, so I won.)

>My first comment is that if you are 2-away, you should think very

>carefully about whether you want to double. [...]

I DIDN'T think very carefully, and that is why I'm not rated higher. :-)

>However, in the actual position (which is a gammonless position btw) your

>opponent was the one who made a big error. [...]

Hmm, so I inadvertantly proved my point, that weaker players will take
when they should drop, thus benefitting the stronger player.

### Lee Schumacher

Jul 26, 1994, 5:27:38â€¯PM7/26/94
to
ran...@minerva.cis.yale.edu (Joshua E. Randall) writes:

>Immediately after closing my board, I double, expecting him to drop. He
>takes. The game continues. I bear almost all my men off and have two
>left on my 1 point; he has 3 left, 2 on his 2 point and 1 on his 1
>point. He doubles, to 4. I saw it as: he needs to rolls double 2's or

You mean he waited to redouble !?! And you're worried about losing
from 5-5 ? oh my ...

>Joshua E. Randall ran...@minerva.cis.yale.edu

Lee.

### Anthony R Wuersch

Jul 26, 1994, 7:04:16â€¯PM7/26/94
to
In article <313mgb\$a...@scapa.cs.ualberta.ca>,
Darse Billings <da...@cs.ualberta.ca> wrote:

>I have heard that in backgammon, a strong player has little advantage
>over an intermediate player based solely on checker play.

Here in Boston, I know many checker snobs who would strongly disagree with
this statement.

Kit Woolsey just published a piece which claims that the revamped TD Gammon
(expert strength) beat the old TD gammon (intermediate strength) hands down,
I think in cubeless games.

There are cases, too, where stronger players will take cubes in back games
because they expect misplays.

Cheers,
Toni
--
Toni Wuersch
a...@world.std.com {uunet,bu.edu,bloom-beacon}!world!arw

### Christopher Yep

Jul 27, 1994, 4:31:15â€¯AM7/27/94
to
In article <1994Jul26....@news.yale.edu>,

Joshua E. Randall <ran...@minerva.cis.yale.edu> wrote:
>In a previous article, <31253l\$s...@agate.berkeley.edu> wrote:
>
>>I believe that if one complicates a position even slightly (similar to
>>"playing opportunistically"), a weaker player will butcher the position -
>>he will make much more serious mistakes, and he will have more time to
>>make them, because the game will usually last longer.
>
>An excellent point which Kit echoed in a later article. This is,
>indeed, the OPPOSITE of the recommendation I had read. But, as I said,
>this book was "cheesy and unknown" -- well, at least we now know why it
>was the latter. :-)
>
>>As for cube decisions, I wouldn't change your strategy too much, except
>>in games in which there is very little skill left. [...]
>
>You don't think that one should offer a weaker opponent doubles which
>SHOULD be drops, but which he will probably take because he doesn't know
>

I think that if you are playing a weaker opponent, you should still
double at approximately the same point that you would against an equal
player. Against an equal player, if you overshoot the mark, and suddenly
have a position which is untakeable for your opponent, then you would
double him out of the game (unless your position is now too good - so
that it is more profitable to play on for a gammon). I would recommend
the same against a weaker player, and if he happens to take, then you
gain a little.

If from experience, you know that a player takes too much, then ideally
you would want to wait until the last possible moment to double, at the last
possible moment when he will still take your double. But, when in doubt
I would recommend leaning toward the conservative side, meaning don't try
for "too much," before doubling him into an incorrect take. It's often
hard to predict an opponent's "breaking point," because he may understand
certain types of positions better than others, and you may not have
played him enough times to know for sure. If you have an opportunity to
double, in which you're confident that your opponent will incorrectly
take, and you decide to try to be greedy and hold the cube even longer,
if you overshoot your mark, you'll have wasted an opportunity to gain
some equity. With most opponents, it's probably better to just use
"normal" doubling strategies. If you think your opponent takes with too
few chances, you can wait a little longer, but not much more. I'm sorry
that this paragraph might be difficult to understand, but it's late at night.

>
>I wrote:
>
>>>As an aside, a situation came up recently in Snoopy's tournament. [...]
>
>I remembered the full details of situation. (It's so ugly, I must have
>blocked it out of my memory. :-) Here it is:
>
>I'm up 5-1 against a perceived weaker opponent. He has me on the bar,
>where I flunk a few times, and he begins bearing off. However, bad
>rolls force him to leave blots, and I finally hit one. After some
>milling around (I blot him, he re-enters and blots me, I re-enter and
>blot him, etc.) I finally close him out. My stragglers are free to move
>(i.e. he doesn't have any sort of prime and I roll well).
>
>Immediately after closing my board, I double, expecting him to drop. He
>takes. The game continues. I bear almost all my men off and have two
>left on my 1 point; he has 3 left, 2 on his 2 point and 1 on his 1
>point. He doubles, to 4. I saw it as: he needs to rolls double 2's or
>higher, so slightly less than 1 in 6. If he fails I automatically win
>the game and the match. If he succeeds the match is tied at 2-away.
>What should I have done? (I took, he didn't roll doubles, so I won.)

You still have a 86% (31/36) chance of winning the game (both players
will play perfectly). With such a high chance of winning, you shouldn't
think twice about accepting the double.

If you want some numbers, consider what has to happen for your opponent
to win the match:

If you take:

Opponent has to win the game (14% chance), then win from 2-away/2-away
match score (say 35% - 50%). Thus, his chance of winning the match is 5-7%.

If you drop, your opponent has to win from 4-away/2-away match score,
which is almost surely better then the 5-7% figure above. (Between equal
players, the 4-away player is thought to have about a 32% chance of
winning the match).

>--
>Joshua E. Randall ran...@minerva.cis.yale.edu

Chris

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