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Feb 24, 1995, 8:03:30â€¯PM2/24/95

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I understand that, when 2-away, 2-away, one ought to double as soon as

one has an advantage. If this is to be taken literally, should the cube

always be turned on the 3rd or 4th move?

one has an advantage. If this is to be taken literally, should the cube

always be turned on the 3rd or 4th move?

For example, if I start with 1 3 and my opponent follows up with a

mediocre roll like 5 1, should I double?

--

James Eibisch

Reading, England Liar. What's in your toolbox?

Feb 26, 1995, 12:32:48â€¯PM2/26/95

to

jeib...@revolver.demon.co.uk (James Eibisch) writes:

>I understand that, when 2-away, 2-away, one ought to double as soon as

>one has an advantage. If this is to be taken literally, should the cube

>always be turned on the 3rd or 4th move?

The short answer is "yes". This has been discussed at length

previously, so I will try to be brief.

It can be shown that it is always theoretically correct to double at

2-away 2-away (assuming the two players are of comparable strength).

It can be proven mathematically, and is almost immediately obvious to

those familiar with game theory.

In practice, one need not double if there are no market losers

(sequences leading to a correct double/drop). If there is even one

sequence (of the 21*21 roll combinations) where your advantage could

grow to the point where your opponent has a valid drop, then you must

double, or risk a "mathematical disaster".

Strong players will often delay doubling in the hope that the weaker

opponent will incorrectly drop when doubled in a position that is

difficult to assess. This is a valid tactic, in practice.

There is also some theoretical justification for the stronger player to

delay doubling until her advantage is somewhat greater than a 50% chance;

but in that case, the weaker side should be doubling sooner (even when

at a slight disadvantage). So it could be argued that other factors may

warrant not doubling, even if there are a few market losers, but this

is usually a specious way of thinking.

Doubling at 2-away 2-away is almost never wrong, whereas not doubling

could be an enormous error. So unless you have no respect at all for

your opponent, just turn the cube and concentrate on more important

things, like correct checker play.

>For example, if I start with 1 3 and my opponent follows up with a

>mediocre roll like 5 1, should I double?

Absolutely. If you roll 66, for example, your opponent could have a

legitimate drop next turn. This would constitute a potentially large

loss in terms of expected value. That risk is much higher than any

possible gain from waiting, so you must double immediately.

Cheers, - Darse.

--

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

Feb 27, 1995, 4:49:21â€¯PM2/27/95

to

James Eibisch (jeib...@revolver.demon.co.uk) wrote:

: I understand that, when 2-away, 2-away, one ought to double as soon as

: I understand that, when 2-away, 2-away, one ought to double as soon as

: one has an advantage. If this is to be taken literally, should the cube

: always be turned on the 3rd or 4th move?

: always be turned on the 3rd or 4th move?

In fact you don't even have to have any advantage. You might just have

say 40 % and you should double anyhow.

The important thing here is whether you have a market looser sequence.

If you do - you should double.

If you don't - you can wait.

I practical term, if you are up agÃ¡inst a stronger player, and especially

if you are not so experienced, you should double at the first given

opportunity.

...... Yes, that means even after you opponent started with 3-1 !!

(deleted)

--

Erik Gravgaard ----------------------

Pres. of the NORDIC WIDE OPEN 1995

Danish Backgammon Federation April 14th to 17th

(erikg @ FIBS) Copenhagen SAS Hotel

Be there !

----------------------

Mar 1, 1995, 3:02:50â€¯AM3/1/95

to

Albert's explanation is a very accurate summary of what I do in

practice. If I am playing against someone who I know understands the

proper strategy at this match score I will double if any market loser is

possible (which essentially means one of us will be doubling almost

immediately). However if I am playing against an unknown opponent or an

opponent whom I do not think understands the proper strategy, then I will

often fail to make theoretically correct doubles with a tiny chance of

losing my market, since I expect he will be more likely to make even

larger errors. Of course when an opponent fails to double virtually

instantly, this is a good indication that he does not understand the

strategy fully.

practice. If I am playing against someone who I know understands the

proper strategy at this match score I will double if any market loser is

possible (which essentially means one of us will be doubling almost

immediately). However if I am playing against an unknown opponent or an

opponent whom I do not think understands the proper strategy, then I will

often fail to make theoretically correct doubles with a tiny chance of

losing my market, since I expect he will be more likely to make even

larger errors. Of course when an opponent fails to double virtually

instantly, this is a good indication that he does not understand the

strategy fully.

Kit

Mar 1, 1995, 7:32:14â€¯AM3/1/95

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From: jeib...@revolver.demon.co.uk (James Eibisch)

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Subject: 2-away, 2-away doubling strategy

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Subject: 2-away, 2-away doubling strategy

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James Eibisch (jeib...@revolver.demon.co.uk) wrote:

> I understand that, when 2-away, 2-away, one ought to double as soon as

> one has an advantage. If this is to be taken literally, should the cube

> always be turned on the 3rd or 4th move?

>

> For example, if I start with 1 3 and my opponent follows up with a

> mediocre roll like 5 1, should I double?

Well, you can't exactly take that advice literally--obviously, if after

any two-ply sequence (your roll followed by opponent's) your opponent

always will have a correct acceptance, you should hang onto the cube.

So each roll at 2-away, 2-away is something to think hard about.

Usually the decision to double or wait won't matter much, but every

little bit of equity is important in the long run.

A common misconception is that if you have any market-losing sequence

early in the game at that score, you should double. That's not exactly

true, because often such a sequence will leave you too good to double.

That is, if such a sequence comes up, you can afford to play on, hoping

for an undoubled gammon, and trying to balance the chance of that against

the chance of losing your market for the cube altogether.

2-away, 2-away is a lot more complicated a score than most people think!

Play well,

Marty

Mar 1, 1995, 2:12:13â€¯PM3/1/95

to

In article <1995Feb28.2...@k12.ucs.umass.edu> as...@k12.ucs.umass.edu (Albert Steg (Winsor)) writes:

[ stuff deleted ]

There has been no proof that doubling with any market loser is

the theoretically optimal strategy in a 2 point match against

a perfect opponent. The reason is that players have different

doubling opportunities - each player may double only prior to

his or her roll. In a modified version of backgammon in which

either player may turn a centered cube before each roll, so

that both players have the same doubling opportunities, then

it can be shown that you must double with any market loser.

>A general policy that occurs to me is that, if I am less likely to lose my

>market on the next parlay than my opponent is to lose _his_ market on his

>next turn, taking into consideration both the current position and my

>opponent's skills, then I should refrain from doubling. Is this sensible?

Actually, what matters is whether _you_ will lose _your opponent's_

market. That is, will you roll so badly that you will have to drop.

When you are on roll, you have (potentially) two sorts of market

losers: those that lose your market, and those that lose your

opponent's market. (Technically, your market losers consist of

2 roll sequences, because of the difference in doubling opportunities.)

By not doubling, you lose equity due to the times you lose your market

and subsequently double your opponent out, and you gain equity from

the times when you lose your opponent's market, and you subsequently

drop your opponent's double.

>I have a vague recollection on this topic a year ago that was very involved

>in "game theory" questions ---I don't mean to stir up those questions.

>Please make follow-up posting relevant to practical tournament play if

>possible.

Sorry, I didn't really reply as you requested. As a practical matter,

I believe that with perfect play, and even with the level of play

attainable by a strong player, the 2 point match should always be

a single game affair, and you should risk losing your market only

to the degree that you believe it is likely your opponent will

make a bigger mistake, or if there is a bigger danger from losing your

opponent's market.

Not everyone agrees with this. For instance, John (doc) Bazigos

has indicated to me that he believes that the discrepancy in doubling

opportunities does allow for 2 point match games that are played

to conclusion without a cube turn, even with perfect play, and I

believe that he feels this is not just theoretically, but also

practically, significant. I suspect that 2 point match doubling will

be one of the first topics featured in his publication, whenever it

comes out.

David Montgomery

monty on FIBS

Mar 1, 1995, 3:55:21â€¯PM3/1/95

to

In article <3j2gud$4...@twix.cs.umd.edu>,

David Montgomery <mo...@cs.umd.edu> wrote:

|>In article <1995Feb28.2...@k12.ucs.umass.edu> as...@k12.ucs.umass.edu (Albert Steg (Winsor)) writes:

|>

|>[ stuff deleted ]

|>

|>There has been no proof that doubling with any market loser is

|>the theoretically optimal strategy in a 2 point match against

|>a perfect opponent. The reason is that players have different

David Montgomery <mo...@cs.umd.edu> wrote:

|>In article <1995Feb28.2...@k12.ucs.umass.edu> as...@k12.ucs.umass.edu (Albert Steg (Winsor)) writes:

|>

|>[ stuff deleted ]

|>

|>There has been no proof that doubling with any market loser is

|>the theoretically optimal strategy in a 2 point match against

|>a perfect opponent. The reason is that players have different

I believe that it has been proved before, and can again...

|>doubling opportunities - each player may double only prior to

|>his or her roll. In a modified version of backgammon in which

|>either player may turn a centered cube before each roll, so

|>that both players have the same doubling opportunities, then

|>it can be shown that you must double with any market loser.

|>

|>>A general policy that occurs to me is that, if I am less likely to lose my

|>>market on the next parlay than my opponent is to lose _his_ market on his

|>>next turn, taking into consideration both the current position and my

|>>opponent's skills, then I should refrain from doubling. Is this sensible?

|>

|>Actually, what matters is whether _you_ will lose _your opponent's_

|>market. That is, will you roll so badly that you will have to drop.

|>When you are on roll, you have (potentially) two sorts of market

|>losers: those that lose your market, and those that lose your

|>opponent's market. (Technically, your market losers consist of

|>2 roll sequences, because of the difference in doubling opportunities.)

|>By not doubling, you lose equity due to the times you lose your market

|>and subsequently double your opponent out, and you gain equity from

|>the times when you lose your opponent's market, and you subsequently

|>drop your opponent's double.

But, if you have 1-roll sequences which lose your opponent's market, then

he should have doubled last roll, if he had been playing perfectly

(unless he too had a chance of losing your market). The point is that at

some point in the game, one player will have the situation where he has a

market loser(s) and no rolls which lose his "opponent's market." In this

situation, he must double, since he will lose equity if he does in fact

roll a market loser.

The exceptions are the border cases, where a player has only "optional"

market losers (i.e. 2-roll sequences which bring him to exactly -1:-2

equity (~70%), if such a thing is possible), or market losers which

consist of only gammons (ie, if he rolls the market loser, then he goes

on to win a gammon 100% of the time). In these cases, doubling is optional.

|>

|>>I have a vague recollection on this topic a year ago that was very involved

|>>in "game theory" questions ---I don't mean to stir up those questions.

|>>Please make follow-up posting relevant to practical tournament play if

|>>possible.

|>

|>Sorry, I didn't really reply as you requested. As a practical matter,

|>I believe that with perfect play, and even with the level of play

|>attainable by a strong player, the 2 point match should always be

|>a single game affair, and you should risk losing your market only

|>to the degree that you believe it is likely your opponent will

|>make a bigger mistake, or if there is a bigger danger from losing your

|>opponent's market.

If each player adopts the strategy of doubling when the chance of

losing his market is larger than the chance of losing his opponent's market,

then this is equivalent to the players doubling when any market loser exists,

since the first time in the game that there exists a market loser: this

market loser will not be accompanied with a chance of losing the other

player's market, otherwise this would not be the first time that the

chance of a market loser came up in the game.

Two assumptions I have made here: it is impossible to lose one's market

on the first roll of the game; both players can evaluate any position

perfectly (in terms of what their game winning chances are).

|>

|>Not everyone agrees with this. For instance, John (doc) Bazigos

|>has indicated to me that he believes that the discrepancy in doubling

|>opportunities does allow for 2 point match games that are played

|>to conclusion without a cube turn, even with perfect play, and I

|>believe that he feels this is not just theoretically, but also

|>practically, significant. I suspect that 2 point match doubling will

|>be one of the first topics featured in his publication, whenever it

|>comes out.

|>

|>David Montgomery

|>monty on FIBS

|>

I believe that if one's opponent adopts the strategy above, then the best

doubling strategy is to match his. Also, among perfect players, it is

only possible for a -2:-2 game to end with the cube at 1 in the border

cases that I have indicated above, in which a player has an optional

market loser, then when he does roll this optional market loser, he

doubles out, and his opponent passes (even though the pass was in fact

optional); or in the case where his market loser results in 100% gammons

(in this case he will win the match, although the cube will still be on 1).

Neither of these 2 cases may even be possible, but they are the only

exceptions possible.

If anyone does not believe this, then I would suggest playing mock

games, choosing the dice rolls as he goes along, and thinking very

carefully about the rolls leading up to any possible market loser...

There was a long thread about this topic some time ago (1 year ago?).

Maybe others have some of these articles (I lost most of them).

Chris

Mar 1, 1995, 6:09:50â€¯PM3/1/95

to

: When you are on roll, you have (potentially) two sorts of market

: losers: those that lose your market, and those that lose your

: opponent's market. (Technically, your market losers consist of

: 2 roll sequences, because of the difference in doubling opportunities.)

: By not doubling, you lose equity due to the times you lose your market

: and subsequently double your opponent out, and you gain equity from

: the times when you lose your opponent's market, and you subsequently

: drop your opponent's double.

: losers: those that lose your market, and those that lose your

: opponent's market. (Technically, your market losers consist of

: 2 roll sequences, because of the difference in doubling opportunities.)

: By not doubling, you lose equity due to the times you lose your market

: and subsequently double your opponent out, and you gain equity from

: the times when you lose your opponent's market, and you subsequently

: drop your opponent's double.

Interesting and refreshing new concepts, losing your opponent

market. But it won't work theoretically, because if u're in position

where u can roll a single roll which gives your opponent cash, then

your opponent, who follows the same strategy as u, has already doubled

u before his last roll, anticipating upcoming sequence ( or, if he

didn't, it means that it's your turn to open game, and it's well known

that u can't lose any1's market on very 1st roll of the game

). Remember , we're trying to derive an optimal strategy here, which would be

accepted by every1 by virtue of being OPTIMAL.

On the other hand, gammon rate at -2:-2 is .75. So whenever u

anticipate a sequence after which your gammon chances times 3 will

exceed your losses times 4, then it's better for u to hold off with a

cube. So the correct abstract method would be consider 21x21 sample of

2 ply sequences, evaluate each one as double/take, cash, or play-on

and go from there. U can't work with equity here 'cause cubeless

equity doesn't mean much in 2-ptr. So probably u have to compare

number of cash equences with number of play-on sequences ( which both

will be dominated by number of double/take sequences. ).

Let's pretend for a while gammons don't count. Player1 makes

opening move. If P2 sees a 2-ply market loser for himself, he should

double ( let's also assume both players follow identical cube

strategy, as if u were playing both sides ). If he doesn't see a

market loser, but can see P1 losing his market on 3rd ply, he should

double as well, since he's getting a cube next turn anyway. But, if

there's no market loser thruout 3 plies and P2 doesn't cube, then it's

P1's turn again, and he goes thru the same reasoning, cubing whenever

he sees market loser within 3 plies from him, which means P2 gained

nothing by holding cube. Et cetera. So it turnes out that if both

players start with "double whenever 2 ply market loser", it

mathematically can be extended to "double whenever n ply market

loser", i.e., always double ( please remind me if Kit's postulate

was as strong ). Bonus question: how would u go about constructing a

game where cubeless gammonless equity would be in (50+/-n)% range,

where n is small enough to assure retaining market?

So I've hopefully shown that it's always correct to double on

your turn IF gammons don't count. So the construction of

played-out-to-the-end game would have to utilize gammon vig. Now at

this particular instance I don't see how it can be done. Moreover, I

don't see developing a sound cube strategy for -2:-2 with gammons in

effect. Indeed, if u're using your fancy gammons-vs-losses strategy,

and I am using good old "let it go" strategy, then u can be sure to

have your gammons turned off right away. So, we have as given that

Kit's strategy effectively beats any other strategy. Beats in sense

that doesn't allow any equity gain that one could expect from his

strategy. The question is, can u increase your equity by prior

knowledge of your opponent strategy ( I assume opponent being of equal

strength, but using different cube strategy ). Thus every cube

decision of yours becomes an issue of figuring out opponent's

strategy. So if your opponent opens the game, it might be a good idea

not to double right away, trading your remote 2 ply market losing

sequences for a opportunity to adjust to his strategy and gain extra

equity from it. Besides, the only real market losing sequence is

5-2: 8 22 5-5: 3 3 1 1

0

Or maybe I am wrong? What do u think about folowing:

5-2: 8 11 6-6: 18 18 7 7

6-5: 5 8 Cube action?

Nah, it's can't be pass. Forget it :)

So I have convinced myself it's practically better not to double after

your opponent have opened the game, UNLESS u absolutely know that he

uses "double at 1st chance" strategy. Then u should double and gain

equity on 5-2/5-5/dance sequence. Besides, there's another subtle

advantage of holding cube. Suppose iether of 2 above sequences

occured. Now your opponent will either double ( which doesn't look

righ to me ) or will retain cube, hoping to be doubled out. Now P2

gets a cheap shot at gammon. Of course he still can have his gammons

turned off next turn, but his opponent will be a clear underdog. So

once we have established that it's better not to double immediately

following opening, we can easily have played-to-the end games, as doc

stated.

To summarize I see only 2 alternatives:

1) Every1 uses "double at 1st chance" strategy, no matter what.

2) Every1 tries to bluff his opponent, making him guess whether he

follows 1) or not

I prefer part 2, where u can gain on your opponent's wrong

guesses. Note: P2 gets some advantagee in 2). Indeed his decision not

to double is easy, since he doesn't lose much by a wrong guess. The

following P1's decision is harder, since his wrong guess can cost him

more, since he's much likelier to lose his market in 2 plies. Of

course. I'd say if he's a clear underdog, he should roll, hoping to be

able to drop correctly should things go bad on his current

roll. Otherwise he should keep guessing :).

And clearly tak/drope decisions are trivial since it right away

becomes a game of complete information ( of course ability to evaluate

your win % is a must, as in any other take/drop decision ).

Well, that was kinda long. But it's such a fascinating subject, I

couldn't stop and my attitude kept changing back and forth as I was

writing. I like my humble idea of thinking about the problem from the

point of incomplete information decision ( at least I never heard it

explicitely expressed before ).

Thanks for having me :)

Igor

Mar 1, 1995, 6:20:49â€¯PM3/1/95

to

: A common misconception is that if you have any market-losing sequence

: early in the game at that score, you should double. That's not exactly

: true, because often such a sequence will leave you too good to double.

: That is, if such a sequence comes up, you can afford to play on, hoping

: for an undoubled gammon, and trying to balance the chance of that against

: the chance of losing your market for the cube altogether.

Not true, Marty. If you know what you are doing, you would have doubled

before you got to such a situation where playing for an undoubled gammon

is correct. I'll say it again -- if there is any possible market losing

sequence, then it is theoretically correct to double. And, it is never

wrong to double at your earliest opportunity. The only mistake with the

cube one can make at this score is to risk losing one's market by failing

to double.

Kit

Mar 2, 1995, 12:10:20â€¯AM3/2/95

to

In article <3j2urv$7...@news.bu.edu>, Igor Sheyn <sh...@cs.bu.edu> wrote:

|>: When you are on roll, you have (potentially) two sorts of market

|>: losers: those that lose your market, and those that lose your

|>: opponent's market. (Technically, your market losers consist of

|>: 2 roll sequences, because of the difference in doubling opportunities.)

|>: By not doubling, you lose equity due to the times you lose your market

|>: and subsequently double your opponent out, and you gain equity from

|>: the times when you lose your opponent's market, and you subsequently

|>: drop your opponent's double.

|>

|>Interesting and refreshing new concepts, losing your opponent

|>market. But it won't work theoretically, because if u're in position

|>where u can roll a single roll which gives your opponent cash, then

|>your opponent, who follows the same strategy as u, has already doubled

|>u before his last roll, anticipating upcoming sequence ( or, if he

|>didn't, it means that it's your turn to open game, and it's well known

|>that u can't lose any1's market on very 1st roll of the game

|>). Remember , we're trying to derive an optimal strategy here, which would be

|>accepted by every1 by virtue of being OPTIMAL.

|>: When you are on roll, you have (potentially) two sorts of market

|>: losers: those that lose your market, and those that lose your

|>: opponent's market. (Technically, your market losers consist of

|>: 2 roll sequences, because of the difference in doubling opportunities.)

|>: By not doubling, you lose equity due to the times you lose your market

|>: and subsequently double your opponent out, and you gain equity from

|>: the times when you lose your opponent's market, and you subsequently

|>: drop your opponent's double.

|>

|>Interesting and refreshing new concepts, losing your opponent

|>market. But it won't work theoretically, because if u're in position

|>where u can roll a single roll which gives your opponent cash, then

|>your opponent, who follows the same strategy as u, has already doubled

|>u before his last roll, anticipating upcoming sequence ( or, if he

|>didn't, it means that it's your turn to open game, and it's well known

|>that u can't lose any1's market on very 1st roll of the game

|>). Remember , we're trying to derive an optimal strategy here, which would be

|>accepted by every1 by virtue of being OPTIMAL.

Exactly...

|> On the other hand, gammon rate at -2:-2 is .75. So whenever u

|>anticipate a sequence after which your gammon chances times 3 will

|>exceed your losses times 4, then it's better for u to hold off with a

|>cube. So the correct abstract method would be consider 21x21 sample of

|>2 ply sequences, evaluate each one as double/take, cash, or play-on

|>and go from there. U can't work with equity here 'cause cubeless

|>equity doesn't mean much in 2-ptr. So probably u have to compare

|>number of cash equences with number of play-on sequences ( which both

|>will be dominated by number of double/take sequences. ).

I have to disagree here. If your opponent is always going to double when

he has a chance of losing his market (except when you have made a mistake

previously, and now there exist chances of market losers and also chances of

losing _your_ market (ie "market losers" in both directions), such that

it is now profitable for him to hold the cube), then the gammon price or

gammon rate, etc.) isn't quite as relevant.

Below, assume that your opponent is a perfect player.

For example, suppose a market losing sequence occurs such that now you have

a 99.7% chance of winning a single game, with 0.2% gammons, and 0.1%

losses. Well, clearly it would have been better if you had doubled

during the previous roll. So perhaps you meant something else in your last

paragraph?

If after the market losing sequence you have: 99.8% gammons, 0.1% single

wins, 0.1% losses, you still should have doubled previously. You don't

gain anything from leaving the cube in the middle, since your opponent

can always turn it whenever it is to his advantage (he can even turn it

next roll if he wants). If this sequence were to actually happen, then:

usually after having rolled the market loser, you will want to continue

playing on for the gammon. ~99.8 of the time you will breathe a sigh of

relief. ~0.1% of the time you will have to double your opponent out

[in which case it would have been better to have doubled him

earlier]. Everything else will play the same.

Note that unless you make a mistake, every time that you lose, you lose the

match, since your opponent will always double you into the game (except

the rare border cases which I discussed in my previous post) when a

market loser exists.

|> Let's pretend for a while gammons don't count. Player1 makes

|>opening move. If P2 sees a 2-ply market loser for himself, he should

|>double ( let's also assume both players follow identical cube

|>strategy, as if u were playing both sides ). If he doesn't see a

|>market loser, but can see P1 losing his market on 3rd ply, he should

|>double as well, since he's getting a cube next turn anyway. But, if

If he *might* "get a cube next turn anyway," then there's no reason to

double this turn. You can't shake the logic that if a player doesn't

have a market losing sequence (of 2 rolls), then it is not _mandatory_ to

double. A player will never lose his market if he always doubles when he

has a market loser. The last sentence is a bit redundant, but certainly

true!

However, if a player has neither (2-ply) market losers, nor (1-ply)

opponent market losers, then he can optionally double (and his opponent

must then accept). But this double is not mandatory. (I wasn't sure how

to interpret your last paragraph. Maybe this is what you meant?)

As a corollary, a version of perfect play (i.e. no other strategy is

_better_) is to double on the first roll of the game.

|>there's no market loser thruout 3 plies and P2 doesn't cube, then it's

|>P1's turn again, and he goes thru the same reasoning, cubing whenever

|>he sees market loser within 3 plies from him, which means P2 gained

|>nothing by holding cube. Et cetera. So it turnes out that if both

|>players start with "double whenever 2 ply market loser", it

|>mathematically can be extended to "double whenever n ply market

|>loser", i.e., always double ( please remind me if Kit's postulate

I agree with the conclusion if you specify that the double really is

"optional."

|>was as strong ). Bonus question: how would u go about constructing a

|>game where cubeless gammonless equity would be in (50+/-n)% range,

|>where n is small enough to assure retaining market?

Not possible, because the final move of the game always brings one player

to the 100% mark, i.e. there *must* be at least one market losing

sequence in every completed game.

|>So I've hopefully shown that it's always correct to double on

|>your turn IF gammons don't count. So the construction of

|>played-out-to-the-end game would have to utilize gammon vig. Now at

|>this particular instance I don't see how it can be done. Moreover, I

|>don't see developing a sound cube strategy for -2:-2 with gammons in

|>effect. Indeed, if u're using your fancy gammons-vs-losses strategy,

See my remarks above. Or are you answering your own question below? I'm

not really sure...

|>and I am using good old "let it go" strategy, then u can be sure to

|>have your gammons turned off right away. So, we have as given that

|>Kit's strategy effectively beats any other strategy. Beats in sense

|>that doesn't allow any equity gain that one could expect from his

|>strategy. The question is, can u increase your equity by prior

|>knowledge of your opponent strategy ( I assume opponent being of equal

|>strength, but using different cube strategy ). Thus every cube

|>decision of yours becomes an issue of figuring out opponent's

|>strategy. So if your opponent opens the game, it might be a good idea

Definitely something to consider... I have rarely played a player who

plays perfectly (WRT cube strategy) at -2:-2! To counter this, I rarely

play perfectly WRT cube strategy at -2:-2, unless I am playing someone who

knows how to handle the cube properly at -2:-2 (and who is prepared to do

so... i.e. only if my opponent also knows that I know this strategy also,

etc.!)

Sounds right...

|>

|>I prefer part 2, where u can gain on your opponent's wrong

|>guesses. Note: P2 gets some advantagee in 2). Indeed his decision not

|>to double is easy, since he doesn't lose much by a wrong guess. The

|>following P1's decision is harder, since his wrong guess can cost him

|>more, since he's much likelier to lose his market in 2 plies. Of

|>course. I'd say if he's a clear underdog, he should roll, hoping to be

|>able to drop correctly should things go bad on his current

|>roll. Otherwise he should keep guessing :).

When I strongly suspect that my opponent will play imperfectly with the

cube, then this is also the strategy that I use. As a practical matter,

I follow this strategy almost all the time. Basically I'm comparing the

number of market losers (and their severity) with the chance that I think

he will err (and how badly it's possible for him to err) in the future

WRT doubling (or more precisely, with not doubling).

|>And clearly tak/drope decisions are trivial since it right away

|>becomes a game of complete information ( of course ability to evaluate

|>your win % is a must, as in any other take/drop decision ).

Yep, if cubeless % is >= ~30% (-2:-1 equity), then take, otherwise drop...

|>

|>Well, that was kinda long. But it's such a fascinating subject, I

|>couldn't stop and my attitude kept changing back and forth as I was

|>writing. I like my humble idea of thinking about the problem from the

Your opinions did change throught your post, but at the end, I agree with

you... : )

|>point of incomplete information decision ( at least I never heard it

|>explicitely expressed before ).

|>Thanks for having me :)

|>Igor

Chris

Feb 28, 1995, 4:53:49â€¯PM2/28/95

to

In a previous article, er...@inet.uni-c.dk (Erik Gravgaard) says:

>James Eibisch (jeib...@revolver.demon.co.uk) wrote:

>: I understand that, when 2-away, 2-away, one ought to double as soon as

>: one has an advantage. If this is to be taken literally, should the cube

>: always be turned on the 3rd or 4th move?

>

>In fact you don't even have to have any advantage. You might just have

>say 40 % and you should double anyhow.

>

>The important thing here is whether you have a market looser sequence.

>

>If you do - you should double.

>If you don't - you can wait.

Sometimes I think I understand and agree with this doubling strategy and at

other times I am not so sure about it. Kit Woolsey's conclusion on page 11

of _How to Play Tournament BG_ is "If you are playing a perfect opponent, it

is always correct to double at this match score if there is any way you can

lose your market on the next exchange."

Iit is the notion of a "perfect opponent" that gets me a little perplexed.

In this context do I rightly take this expression to mean "a player who

follows this policy"? If I were playing a BG program which had this policy

built-in, I see that I should follow the Woolsey rule....

But I take it that if I am playing an opponent who has never considered this

match score closely, and who therefore may double much more

conservatively, OR a player who has (genuinely?) expressed his disagreement

with the strategy, it doesn't seem to me that I am well-served by following

Woolsey's rule by doubling in positions where my market-losing sequences are

remote, especially if I am actually an underdog in the game.

So does that make me an "imperfect" player, and so should my opponents take

the Woolsey rule with a grain of salt when -2/-2 against me? --and are they

then imperfect players, too?

In practice, reading Woolsey's argument on this score has made me

hyper-aware of the danger of losing my market at -2/-2, and so I do double

with quite slender provocation at that score --but my trigger-finger is much

itchier against a strong opponent who has considered this rule and is likely

to follow its spirit than against a less experienced player who is

morelikely to lose his own market over the next few rolls due to a lack of

understanding of this match score situation.

A general policy that occurs to me is that, if I am less likely to lose my

market on the next parlay than my opponent is to lose _his_ market on his

next turn, taking into consideration both the current position and my

opponent's skills, then I should refrain from doubling. Is this sensible?

I have a vague recollection on this topic a year ago that was very involved

in "game theory" questions ---I don't mean to stir up those questions.

Please make follow-up posting relevant to practical tournament play if

possible.

Albert

--

"When it was proclaimed that the Library contained all books,the

first impression was one of extravagant happiness. All men felt

themselves to be the masters of an intact and secret treasure.

-Jorge Luis Borges, "The Library of Babel"

Mar 2, 1995, 10:17:51â€¯PM3/2/95

to

Darse Billings <da...@cs.ualberta.ca> wrote:

>...

>Absolutely. If you roll 66, for example, your opponent could have a

>legitimate drop next turn. [...]

>...

>Absolutely. If you roll 66, for example, your opponent could have a

I've been thinking about what happens when you get 66s for your first roll

(obviously 2nd roll in game) and make both bar points. Frequently, it

screws up your timing, allows your opponent to build a board behind you,

and then hit you and nail you to the bar.

I've even had folks double me right after I rolled the sixes. I wonder

if there's an analysis of this situation?

--

=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=

Don A. Hanlen Computer Science

dha...@beta.tricity.wsu.edu Washington State University

(509) 588-4528 Tri-Cities

Mar 3, 1995, 10:40:31â€¯AM3/3/95

to

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

: Albert's explanation is a very accurate summary of what I do in

: Albert's explanation is a very accurate summary of what I do in

: Kit

Im a really good backgammon player but I want you to send me some technics to become a better one. Thanks

Mar 6, 1995, 12:37:13â€¯AM3/6/95

to

In article <D4uH5...@serval.net.wsu.edu>, dha...@beta.tricity.wsu.edu

(Don A. Hanlen) wrote:

(Don A. Hanlen) wrote:

> Darse Billings <da...@cs.ualberta.ca> wrote:

> >...

> >Absolutely. If you roll 66, for example, your opponent could have a

> >legitimate drop next turn. [...]

>

> I've been thinking about what happens when you get 66s for your first roll

> (obviously 2nd roll in game) and make both bar points. Frequently, it

> screws up your timing, allows your opponent to build a board behind you,

> and then hit you and nail you to the bar.

>

> I've even had folks double me right after I rolled the sixes. I wonder

> if there's an analysis of this situation?

6-6 is an excellent opener unless your opponent started off by making the

bar point. It give you a big jump in the race, secures your bar point.

and makes an advanced anchor! As you note, however, it does leave you

somewhat "stiff," what with the stack on your 6-point and the scarcity of

spares and buiders to play with.

If my opponent doubled me after my 6's, I think I would beaver it, unless,

perhaps, he had rolled 4-4 ---maybe even then.

More often, I have seen opponents double after *their own* 6-6 opening,

without appreciating the drawbacks that you imply in your note.

Definitely a bad idea.

I like rolling the 6-6 opener, but my next roll often determines just

_how_ good it turns out to be.

Albert

Mar 23, 1995, 4:19:06â€¯PM3/23/95

to

How can a duffer from the provinces quibble with Kit?

Feebly, and by pretending that I just need more explanation.

OK: I guess I still don't get it, although I'm trying.

Feebly, and by pretending that I just need more explanation.

OK: I guess I still don't get it, although I'm trying.

[ I don't know who wrote this: ]

}

} : A common misconception is that if you have any market-losing sequence

} : early in the game at that score, you should double. That's not exactly

} : true, because often such a sequence will leave you too good to double.

} : That is, if such a sequence comes up, you can afford to play on, hoping

} : for an undoubled gammon, and trying to balance the chance of that against

} : the chance of losing your market for the cube altogether.

In article <kwoolseyD...@netcom.com>,

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

}

} Not true, Marty. If you know what you are doing, you would have doubled

} before you got to such a situation where playing for an undoubled gammon

} is correct. I'll say it again -- if there is any possible market losing

-} sequence, then it is theoretically correct to double. And, it is never

} wrong to double at your earliest opportunity. The only mistake with the

} cube one can make at this score is to risk losing one's market by failing

} to double.

}

} Kit

I think you are saying that if there is any chance your opponent will

drop later but would take now, you give up equity to the extent the

opponent later improves his equity by dropping.

Now it seems to me that, conversely, situations

could develop where *you* might later improve your own equity by dropping,

and playing two more games. (I think the official reply is "your opponent

should have doubled already, so you won't get that improvement ever".)

If the cube goes to 2 in a -2;-2 game, one of the two players is going

to lose the match. That player will wish the cube hadn't been turned.

The losing player will now be sorry, if it was he who turned the cube

in the first place.

Imagine being on roll in some volatile midgame. A couple of your possible

numbers would be so horrible that they would guarantee you will lose.

A couple other numbers would be so good that you are likely to win gammon.

The majority of the rolls just continue the battle without noticable

change in the equity. What will your own equity be one full turn later ?

In the games you are going to lose, your equity is infinitely better if

the cube is still on one. In the games you are likely to gammon, your

equity is again better without the cube, although your opponent will

likely turn it for you and you will take. Apparently, then, the struggle

games must have lots of equity swing.

What if, in all the struggle games, the only market loser is a

1/36 * 1/36 sequence in which the opponent will lose but not be gammoned ?

I give up equity by not doubling this now. Couldn't this forgone equity

be smaller than the equity I throw away in the "horrible" & "gammon"

scenarios ? Kit seems to be saying it doesn't matter, I should double

if there is *any* market loser.

Is this some kind of zero-sum deal, where if it's not right for me to

double, it's automatically right for my opponent to double ?

-- Spider

Mar 23, 1995, 5:38:20â€¯PM3/23/95

to

In article <hb7SlCps...@garnet.msen.com> l...@mail.msen.com writes:

>How can a duffer from the provinces quibble with Kit?

>Feebly, and by pretending that I just need more explanation.

>OK: I guess I still don't get it, although I'm trying.

>

>How can a duffer from the provinces quibble with Kit?

>Feebly, and by pretending that I just need more explanation.

>OK: I guess I still don't get it, although I'm trying.

>

>}Marty Storer wrote:

>} : A common misconception is that if you have any market-losing sequence

>} : early in the game at that score, you should double. That's not exactly

>} : true, because often such a sequence will leave you too good to double.

>} : A common misconception is that if you have any market-losing sequence

>} : early in the game at that score, you should double. That's not exactly

>} : true, because often such a sequence will leave you too good to double.

>kwoo...@netcom.com (Kit Woolsey) wrote:

>} Not true, Marty. If you know what you are doing, you would have doubled

>} before you got to such a situation where playing for an undoubled gammon

>} is correct. I'll say it again -- if there is any possible market losing

>} sequence, then it is theoretically correct to double. And, it is never

>} wrong to double at your earliest opportunity. The only mistake with the

>} cube one can make at this score is to risk losing one's market by failing

>} to double.

>

>Imagine being on roll in some volatile midgame. A couple of your possible

>numbers would be so horrible that they would guarantee you will lose.

>A couple other numbers would be so good that you are likely to win gammon.

>The majority of the rolls just continue the battle without noticable

>change in the equity. What will your own equity be one full turn later ?

>In the games you are going to lose, your equity is infinitely better if

>the cube is still on one. In the games you are likely to gammon, your

>equity is again better without the cube, although your opponent will

>likely turn it for you and you will take. Apparently, then, the struggle

>games must have lots of equity swing.

>

>What if, in all the struggle games, the only market loser is a

>1/36 * 1/36 sequence in which the opponent will lose but not be gammoned ?

>I give up equity by not doubling this now. Couldn't this forgone equity

>be smaller than the equity I throw away in the "horrible" & "gammon"

>scenarios ? Kit seems to be saying it doesn't matter, I should double

>if there is *any* market loser.

>

>Is this some kind of zero-sum deal, where if it's not right for me to

>double, it's automatically right for my opponent to double ?

Well, yes and no Spider. I've gone through some swings back and forth

on this one, and so having recently understood it myself, perhaps I can

explain it better.

There are two kinds of sequences: market losers, a roll by you plus a

roll by your opponent such that afterwards if you double your opponent

will drop; and what I'll call market crashers -- a roll by you such

that after you roll your opponent will double and you must drop.

The rest of the rolls leave you playing on.

Your potential equity loss by doubling now is the sum of the equity loss

for all your market crashers -- rolls on which you wish you could drop

on your opponent's turn, getting an equity of .30 (the match equity for

-1:-2), but are now stuck in the game for the match with a win percent

less than .30.

Your potential equity loss by not doubling now is the sum of the equity

loss for all yoru market losers -- 2 rolls after which your opponent

will drop giving you an equity of .70 rather than a match equity of a

higher win percent.

If the potential equity loss by doubling is greater than the potential

equity loss by not doubling, then you shouldn't double. For example:

1 2 3 4 5 6

X

X

O

O

1 2 3 4 5 6

O on roll, score -2:-2, cube action?

O has some market losers: 33, 44, 55, 66. All other rolls are market

crashers.

But if you work it out it's pretty clear O should *not* double. Is

there a position for X last turn for which it is incorrect to double and

yet will reach this position? Yes -- if you assume X had 4 checkers on

his 6pt and rolled a 66. It would have been wrong for X to double, even

in view of various market losing sequences (namely 66 followed by

anything other than 33, 44, 55, 66).

So... if there are both market losers and market crashers, then it

might not be correct to double. If there are market losers and no

market crashers, then the double is mandatory because if you don't

double you risk losing equity.

Here's the key that makes this line of reasoning moot: It's generally

accepted that there are no market crashers on the first roll of the

game. Since market losers are two roll sequnces and market crashers are

one roll sequences, then market crashers cannot appear without there

having been market losers the roll before. Since there are no market

crashers on the first roll, someone will at some point reach a situation

with market losers and no market crashers. And at that point, it is

incorrect to not double.

Ahhh, but what about gammons you ask? Let's suppose you reach a

situation in which your'e too good to double but yet haven't had any

market losers. (For example, you win 65% gammons, your opponent wins

35% normal games.) So if one side cubes, the other side takes and the

match equity at that point is 35%. Now the match equity for the leading

side is either above or below 65%. If below, the leading side should

double and the side behind should take. The leader increased equity to

65%, the trailer took for 35% instead of dropping for 30%. If the

undoubled equity is above 65%, then the *trailer* should double, raising

his equity from below 35% to 35%. So if one side is too good to double,

yet doesn't have any market losers, then the other side should double!

[I'm glossing over the fact that the equity is changing between the time

when the leader has the choice to double and the time when the trailer

has a chance to double. But I think the line of reasoning still holds

together.]

Okay, so that takes care of the theory side of things. [As an aside, my

racing analyses show that 66, 12 is going to be a market loser for

either one side or the other, hence the game can't go to the end without

a market loser appearing.]

But now practice rears it's ugly head.... (Remember that in theory

there's no difference between theory and practice, but in practice there

is. *grin*)

It's quite possible that your opponent will not spot a 1/1296 market

loser when it first appears. If he then rolls the first half of it,

you're in a situation in which 1/36 or your rolls are market crashers,

and you might possibly have market losers as well. But those market

losers might be more rare than the market crasher, leading to the

situation in which you have market losers but your market crashers

outweight them so you don't double.

So, *IF* your opponent first makes a mistake in not doubling when he

should, it is not true that you should double with any market losers.

Hence with perfect play the cube will be turned and the match will end

in that game, but with imperfect play on one roll, it is possible that

perfect play for the rest of the game will still leave the cube at one.

So both Marty and Kit are correct. As long as both sides play

perfectly, then you should double at your first market loser, no matter

how remote. But you might reach a situation in which you're too good to

double, and conceivably you could have reached that without having

market losers (win rate under 70%, but virtually all wins gammons). In

that case, you don't double, but your opponent should!

Hope this helps.

-michael j zehr

Mar 29, 1995, 3:00:00â€¯AM3/29/95

to

Michael J Zehr <ta...@ATHENA.MIT.EDU> writes:

>So both Marty and Kit are correct. As long as both sides play

>perfectly, then you should double at your first market loser, no matter

>how remote. But you might reach a situation in which you're too good to

>double, and conceivably you could have reached that without having

>market losers (win rate under 70%, but virtually all wins gammons). In

>that case, you don't double, but your opponent should!

Let's not be so polite that we mislead readers. Kit's>So both Marty and Kit are correct. As long as both sides play

>perfectly, then you should double at your first market loser, no matter

>how remote. But you might reach a situation in which you're too good to

>double, and conceivably you could have reached that without having

>market losers (win rate under 70%, but virtually all wins gammons). In

>that case, you don't double, but your opponent should!

statement "I'll say it again -- if there is any possible

market losing sequence, then it is theoretically correct

not exactly true" is right but his "because often such a

sequence will leave you too good to double" is the wrong

reason.

You can arrive at a position that is not a double, even

though you can lose your market, provided that your

opponent has previously made a mistake so that you can

immediately roll something that loses HIS market. In this

situation the fact that some of your market-losing

sequences leave you too good MAY be relevant, but probably

is not a major factor.

Suppose you can't roll a market crasher but can lose your

market, and 100% of your market losers leave you too good.

You should double. That is because your chance of winning

the match can't be greater than your cubeless probability

of winning the game. The reason for THAT is that your

opponent can ensure that if you play optimally all his wins

will come at the 2 level. He can do this by doubling

next roll no matter what, when per assumption (no market

crashers) you should take.

Since you can lose your market even if you play perfectly

(but your opponent doesn't) sometimes you will wind up being

too good. When this happens you will always be sorry you

didn't double, if there is any chance whatever that you will

have to cash later. (But at your last chance to double your

opponent in, you rated to be sorrier if you did than if you

didn't.) *Too good* is a misnomer here. *Stuck having to play

for the gammon* describes it better.

Earlier, you wrote:

>Your potential equity loss by doubling now is the sum of the equity loss

>for all your market crashers -- rolls on which you wish you could drop

>on your opponent's turn, getting an equity of .30 (the match equity for

>-1:-2), but are now stuck in the game for the match with a win percent

>less than .30.

>Your potential equity loss by not doubling now is the sum of the equity

>loss for all yoru market losers -- 2 rolls after which your opponent

>will drop giving you an equity of .70 rather than a match equity of a

>higher win percent.

you can ignore all the sequences that do not lose either

your own or your opponent's market. I believe this is

not the case. There may be sequences such that next roll

it would be wrong to double. If you double now you will

have lost equity if one of those sequences happens, so

you have to take them into account in deciding whether

to double. (Again, this can only happen after someone has

made a mistake.)

Walter Trice

trice on FIBS

walt...@delphi.com

Apr 12, 1995, 3:00:00â€¯AM4/12/95

to

Spider, l...@mail.msen.com, (that's me!) writes:

[ ... most of my article deleted here ... ]

} >How can a duffer from the provinces quibble with Kit?

} >Feebly, and by pretending that I just need more explanation.

} >OK: I guess I still don't get it, although I'm trying.

/.../

} >Imagine being on roll in some volatile midgame. A couple of your possible

} >numbers would be so horrible that they would guarantee you will lose.

} >A couple other numbers would be so good that you are likely to win gammon.

} >The majority of the rolls just continue the battle without noticable

} >change in the equity. What will your own equity be one full turn later ?

/.../

ta...@ATHENA.MIT.EDU (Michael J Zehr) wrote:

}

} Well, yes and no Spider. I've gone through some swings back and forth

} on this one, and so having recently understood it myself, perhaps I can

} explain it better.

}

}

} There are two kinds of sequences: market losers, a roll by you plus a

} roll by your opponent such that afterwards if you double your opponent

} will drop; and what I'll call market crashers -- a roll by you such

} that after you roll your opponent will double and you must drop.

/... excellent explanation by michael snipped here .../

}

} So... if there are both market losers and market crashers, then it

} might not be correct to double. If there are market losers and no

} market crashers, then the double is mandatory because if you don't

} double you risk losing equity.

}

}

} Here's the key that makes this line of reasoning moot: It's generally

} accepted that there are no market crashers on the first roll of the

} game. Since market losers are two roll sequnces and market crashers are

} one roll sequences, then market crashers cannot appear without there

} having been market losers the roll before. Since there are no market

} crashers on the first roll, someone will at some point reach a situation

} with market losers and no market crashers. And at that point, it is

} incorrect to not double.

I'm not convinced that market crashers are necessarily one roll sequences.

Your inductive proof here seems to rely on the idea that the human

player's look-ahead is two ply. This, coupled with the one-roll (1/2 turn)

lag in doubling opportunity, provides the crux of the argument: "market

crashers cannot appear without there having been market losers the roll

before."

But can't I turn this induction on its head by claiming my look-ahead

is three ply? If I can see a three roll market crasher, and a two roll

market loser, should I double? Not necessarily, say I. We can continue

this line of reasoning back to the first roll of the game, too.

/... more excellent Zehr exposition snipped here .../

}

} But now practice rears it's ugly head.... (Remember that in theory

} there's no difference between theory and practice, but in practice there

} is. *grin*)

}

} It's quite possible that your opponent will not spot a 1/1296 market

} loser when it first appears. If he then rolls the first half of it,

} you're in a situation in which 1/36 or your rolls are market crashers,

} and you might possibly have market losers as well. But those market

} losers might be more rare than the market crasher, leading to the

} situation in which you have market losers but your market crashers

} outweight them so you don't double.

}

} So, *IF* your opponent first makes a mistake in not doubling when he

} should, it is not true that you should double with any market losers.

} Hence with perfect play the cube will be turned and the match will end

} in that game, but with imperfect play on one roll, it is possible that

} perfect play for the rest of the game will still leave the cube at one.

}

This is kind of what I am trying to say, except perhaps I am not so bold

as to use the word "mistake". If the opponent refrains from doubling

and brings us to your situation where I now have both market crashers

and market losers, is it necessarily a "mistake"? What if his superior

look-ahead spotted my market losers, which are market crashers to him?

} So both Marty and Kit are correct. As long as both sides play

} perfectly, then you should double at your first market loser, no matter

} how remote. But you might reach a situation in which you're too good to

} double, and conceivably you could have reached that without having

} market losers (win rate under 70%, but virtually all wins gammons). In

} that case, you don't double, but your opponent should!

}

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