Bearing Off

[crazycowboy]

Say you have 2 checkers on the 6 point, 2 checkers on the 5 point, 2 checkers on the 3 point, 2 checkers on the 2 point and zero checkers on the 1 point. You roll a 5 - 1. What is the best strategy for moving the checkers?

I think I read once that the best thing to do in this case is to use the 1 to move from the 6 point to the 5 point.

Thanks for any input.

[adamosad]

I think the 5 is easy (5-out) then you must think about the 1. 6-5 move might be good but personally I would play 2-1 move. I do not know what gnu is suggesting for this situation but I would play that way because I have more chances to get 2 checkers out with my next roll...

[Hardy_whv]

Well, the general principle in the bearoff (without contact) is:

If you are a favourite to win the game, you should work on a better distribution. This will improve your chances bearing off 2 checkers with most rolls. In your position the move 5/off 2/1 is improving the distribution as it only leaves a gap on the 4-point.

If however you are the underdog, you should search for moves, that let you bear off more checkers if you roll some jokers (= doubles). You have to optimize for the miracle, that might happen. For the given position, 6/off is a bit better, letting you bear off 3 checkers with a 55 (instead of only 2 if you move 5/0ff 2/1).

So the answer to your question depends on the opponents position.

But, take it easy. In this case, the difference between the two moves is marginal.


Hardy    

[lewscannon]

If you are losing, you should drop immediately

[adamosad]

hmmm Hardy gives a better answer  

If you think that you are better than your opponent you should play 5-off 2/1, if you are the underdog you should play 6/off and if you think that you are both equal it depends on your expectations. Actually, if you are risk lover you must choose 6/off and if you are risk averse you must choose 5-off 2/1 (In the case that you are both equal)

[snowjake]

I've always believed in getting a piece off any time you have the chance. Passing off the chance to remove a piece to position your other pieces better is, IMO, a wasted roll.  You might never get that number you moved a piece onto. I've always felt the best you can do is break even in the event ou happen to move a piecewhere a number will help you but passing a chance to get a piece off is jst wrong.

Does anyone else see it this way?

[Hardy_whv]

I've always believed in getting a piece off any time you have the chance. Passing off the chance to remove a piece to position your other pieces better is, IMO, a wasted roll.

Well, in more than 99% of all cases it IS correct to bear off a checker if you can do so. But there are some positions, where bearing off is an error and you better smooth the position. Have a look at this position (DMP, so no cube):



Here it is much better to use the 2 to move 4/2. Otherwise you have a gap you will never be able to fill again. And any 2 will not allow you to bear off a checker.

Winning chances after the two possible moves are:

6/off 4/2: 71,20 %
6/off 2/off: 70,09%

So by playing 2/off you give away more than 1% of match winning chances! A huge blunder.


Hardy  

[snowjake]

Hardee

Im sure you either worked it out or saw the math someplace but it just doesn't make sense to me.

Forget about the 6, that's obvious.

But what have you accomplished by moving the 4 to 2? You've made your next 2 a bear off instead of merely moving the 4 to 2 you avoided. In the meantime, what if you never even roll a 2?

I'm sorry, I just don't see it.  

[snowjake]

Hardy

I've been thinking about this one. You've got me mystified

Give me a series of rolls where you'd be off the board before me by going 4/2 rather than 2/off.

[gogul]

To move 4 - 2 here costs you no additional move and you will avoid the risk of rolling a useless 2 or even bader a double 2 and very probably smoothly bearoff in the next five moves and:

If you are a favourite to win the game, you should work on a better distribution. This will improve your chances bearing off 2 checkers with most rolls

[diane]

To move 4 - 2 here costs you no additional move and you will avoid the risk of rolling a useless 2 or even bader a double 2 and very probably smoothly bearoff in the next five moves and:

If you are a favourite to win the game, you should work on a better distribution. This will improve your chances bearing off 2 checkers with most rolls

How is the 22 worse?  If you move the 6 off, 2 off, then roll 22, the one from the 4 goes off - and 2 from 3 to 1, which is EXACTLY the same result as if you had moved 6 off, 4-2 in the previous move?

[Hardy_whv]

Hardee
Im sure you either worked it out or saw the math someplace but it just doesn't make sense to me. ...

After 2/off you still have 9 checkers to bear off, after 4/2 you have 10 left.

If you don't roll doubles you will probably bear off in 5 rolls in both cases, 9 or 10 make no difference there.

If you roll one 2 in those 5 rolls, after moving 2/off you are still able to bear off in 5 rolls, but what if you roll two times a 2? Then you already need 6 rolls to bear off, while after move 4/2 you are still able to bear off in 5 rolls. One WHOLE ROLL! Thats a difference, isn't it?

What are the odds to roll a 2 two times within 5 rolls? Around 16%. So in 16% of all cases you will need one whole roll more after 6/off 2/off compared to 6/off 4/2 (The numbers are a bit rought, as I am neglecting the cases where you roll 3 or 4 times a 2. But the principle is nevertheless correct.)

Hardee
I'm sorry, I just don't see it.

In Backgammon a lot of calculations have to be done. Actually things are VERY complex in backgammon sometimes, so I strongly recommend to get backgammon software helping you for analyzing problems. Many problems can't be approched as easy as the one we dicussed here. So go and get GNU Backgammon. It will help you analyzing problems and also analyzing the matches you played. You will be surprised how many errors one can make, even if the situation looks easy  

Hardy  

[Hardy_whv]

Perhaps I should add two remarks, I didn't state clearly enough above:

1. Counting the resulting moves required to bear off is an important thing! In the discussion above its either 9 or 10, which makes no difference in terms of rolls required to bear off. And with 4/2 you are likely to miss no single roll in the bear off process (only if you roll more than two times a 2).

2. The generals principle as stated in the very beginning of this discussion is still valid: If you are an underdog to win, you are more likely to be correct to bear off as many as you can. Lets modify the position discussed a bit on the opponents side, so he is a favourite now to win:



Now your match winning chances are:

6/off 2/off: 27,84%
6/off 4/2: 27,64%

Now it's slightly better to bear off.

Yes, I know, backgammon is confusing sometimes. But I can't help it  


Hardy  

[snowjake]

It still doesn't make any sense to me. I don't care whether you're ahead or behind. I can't think of one valid instance where not bearing off can pay dividends in the long run.

Draw up any board you want  where we're in a bear off situation. Show me any series of rolls and tell me how you will bear off passing a chance to bear off. I will counter with bearing off in either as many or fewer.

I just don't see how not bearing off can be a faster path to winning regardless of the situation. Bearin off will never be slower It might not always be quicker but it will never be slower.

[Hardy_whv]

I can't think of one valid instance where not bearing off can pay dividends in the long run.

Draw up any board you want  where we're in a bear off situation. Show me any series of rolls and tell me how you will bear off passing a chance to bear off. I will counter with bearing off in either as many or fewer.

snowjake, you are a hopeless case. I have drawn a board where NOT bearing off is better than bearing off. I did explain why and I gave a precise descritpion of rolls, where you are better of having played 4/2 instead of 2/off.

In the given example you probably require 5 rolls to bear off. If just two of those rolls contain a 2 (16% probability), you require 6 rolls instead if you played 2/off instead of 4/2. When playing 4/2, you still require 5 rolls to bear off, even if two of your 5 rolls contains a 2. Just count the rolls required.

In the give example you don't loose anything by playing 4/2. You gain instead. Get your board, try it, count the moves.

And again: Get the recommended backgammon software. And use it.


Hardy  

[snowjake]

I don't know what you've showed me. You've thrown some percentages at me but have not backed them up with anything substansial.

Humor me, with the example you've drawn up, or any other that you choose show me each individual roll along with the corresponding moves you could make rather than bearing off the max men and give me the opportunity to show you that I can (or can't bbear off in as few or fewer moves.

[Hardy_whv]

... show me each individual roll along with the corresponding moves you could make ...

Sorry snowjake, I wont be able to satisfy your request. You want me to guide you to every individual roll until you are convinced that playing 4/2 is better than bearing off??? That's impossible!

With 9 or 10 black checkers left, the match will be over in about 10 moves (5 moves of WHITE, 5 moves of BLACK), or a bit less if one of the player rolls some doubles. For any roll there are 21 possible different outcomes of the dice (11, 12, 13, 14, 15, 16, 22, 23, 24, 25, 26, 33. 34. 35, 36, 44, 45, 46, 55, 56, 66). If we ignore, that some moves can be played differently, that are about 21 ^ 10 different moves to go through (=16.679.880.978.201 moves). If the game is over after 8 rolls it's still 21 ^ 8 (=37.822.859.361 moves). That's a bit too much - even for me.

How do you work it out? In the early days of backgammon (70s and 80s) you conducted manual rollouts. I.e. playing this position to conclusion for many hundred times. Each time you wrote down the result. At the end you had a rough estimate of the percentages. A very boring procedure. Since the middle of the 90s there is modern, reliable backgammon software available which can do the rollouts for you. You ask the software to conduct the rollouts, the software will tell you which move results in which percentage of wins and losses. With the help of that software you can identfy which move is the best. Thats what I did. And I added the general principles that apply for the bearoff. Either you believe me (and backgammon software), or stay with your preferences.

I wish you much joy in working through that position yourself.


Hardy  

[snowjake]

NO, Hardy, Im not asking you to show me every one of the possible individual rolls possible for 10 rolls of the dice. That's 21o different possibilities even before we factor in sequence.

I am asking you to show me just one particular set of 10 throws of the dice where not bearing off the 4 will pay dividends in the end. Idon't think it's possible and you're not able to show me just one example.

I know BG is a game of percentages and assuming that the players know strategies the guy who knows the small percentages better will win more often than not.

But you keep saying that it's better not to bear off at times yet you aren't able to produce a single time that it is better.

You can throw words at me but you can't back them up. If you can I will admit that you got the better of me.

You won't.

[Hardy_whv]

I am asking you to show me just one particular set of 10 throws of the dice where not bearing off the 4 will pay dividends in the end. I don't think it's possible and you're not able to show me just one example.

Snowjake, this is not the way it works. Sorry.

It doesn't help to show one particular sequence of rolls where one move is better than the other. That doesn't prove anything. You have to compare it to all other possible combinations.

What I can tell you is:

If during the bearoff you roll two times a 2, you will finish your bearoff one whole move earlier when you had played 4/2 than if you had played 2/off. If that helped you to win, depends of the rolls of your opponent and the rest of your rolls. Can you follow me in that reasoning?

If I can't persuade you, my patience inventory is exhausted. Try to google for similar discussions (keywords: "backgammon bearoff smooth"). Perhaps others can persuade you.

Hardy  

[nabla]

Hardy, I am sorry to open this controversial thread again, but that is a very interesting example you gave, nevertheless like snowjake I can't find the sequence of rolls which make it better to play 4/2 than to bear off.
A pair of 2s is not the example snowjake asked for, it rather makes both moves equivalent :
1) In the diagramed position hardy plays 4/2, snowjake plays 2/off
2) With the next 2, hardy will play 2/off, snowjake will play 4/2
Conclusion : already after the first 2, hardy and snowjake have exactly the same position on the board !

I am confident in the computer's numbers, so there must be a mysterious sequence of rolls which makes hardy's play better, but I have not been able to find it. What is it ? Maybe 11 followed by furthers 1s ? I must set up a board at home to see it !

[diane]

This was my conclusion in this situation too - the 'dreaded' next 22 roll being the prime example - it makes absolutely no difference!!  However, I do take the point that there are situations where not bearing off is better - but someone once told me Kit Woolsey uses greedy bearoffs, since the difference is too minimal to matter.  I am running with that...    

[Hardy_whv]

A pair of 2s is not the example snowjake asked for, it rather makes both moves equivalent ...

I am confident in the computer's numbers, so there must be a mysterious sequence of rolls ...

I've never been talking about a double 2 (representing moving a 2 FOUR times), but two single 2s. Try that. After smoothing, you are able to bear off two checkers with two 2s. After bearing off with 2/0 you can not bear off a single checker with a 2. So after rolling a 2 two times (not talking about double 2s), you are better off one whole roll after smothing. Just count the remaining checkers for both cases.

Don't look for any other "mysterious" sequences, it's the 2s that count here.

Btw, theres a nice article about smooting play at Gammon Village by Walter Trice. Perhaps he has more convincing power than I do: http://www.gammonvillage.com/backgammon/ne...resourceid=4279 (only for GV members).

Hardy ;-)

[Hardy_whv]

... the 'dreaded' next 22 ...

Well, as said in my last post, I never talked about a 22, but about two single 2s.

Trice mentions 259 positions, in which it is correct to smooth instead of bearinf off. Those positions usually look a bit weird, as they consist of stacks and gaps (or possible future gaps). So as long as your position looks "normal", well distributed, greedy bearoff is okay. But if your positions looks a bit stacked, you might consider to not use greedy.

Btw, the 259 positions consist of
- 88 positions, where it's correct to smooth with a 2,
- 121 posision, where smoothing with a 3 is correct and
- 50 positions, where smoothing is correct with a 4.

It's never correct to smooth with a 1 or a 5.

... but someone once told me Kit Woolsey uses greedy bearoffs, since the difference is too minimal to matter.  I am running with that... 

Well, as most of the positions, where smoothing is better than bearing off, look a bit "weird", with stacks and gaps, I guess, that even Kit Woolsey will become alert and switch greedy off, if he faces such a position.

Hardy  

[nabla]

I've never been talking about a double 2 (representing moving a 2 FOUR times), but two single 2s. Try that. After smoothing, you are able to bear off two checkers with two 2s. After bearing off with 2/0 you can not bear off a single checker with a 2. So after rolling a 2 two times (not talking about double 2s), you are better off one whole roll after smothing. Just count the remaining checkers for both cases.

Noooo, I never talked about a double 2 either, a pair of 2s is not a double 2  

Why don't you try a second roll of say 52 ?
snowjake : plays 4/2 3/off (but certainly not 4/off 3/1??)
hardy : plays 3/off 2/off

snowjake and hardy are now in the exact same position.
If there is a second 2 at some point after that, both will probably play it in the same way (2/off) !
What difference can it make whether one plays 2/off at move 1 and 4/2 at move 2 (snowjake) or 4/2 at move 1 and 2/off at move 2 (hardy) ? You need to make snowjake miss twice in order to make him worse than you having missed deliberately in the first move.

[Hardy_whv]

I have to offer my excuses. My explanation was simply not right. At least not in the way given. I should have tried it over the board, not purely in my head.  

Still the fact is true, that smoothing is better here than bearing off. Try the following sequence for example:

62, 65, 52, 52.

After smoothing, your position is 301000 (three checkers in the one-point, one on the 3-point). After bearing off it is 500000 (five on the one-point). So after smoothing you are likely to bear off in 2 rolls (except after rolling 21), after bearing off you will require 3 more rolls.

So my feeling is, that the effect only works, after the remaining checkers on the higher points have been taken off.


Hardy .... promising to check his explanations better next time    

[nabla]

I have to offer my excuses. My explanation was simply not right. At least not in the way given. I should have tried it over the board, not purely in my head.   

Still the fact is true, that smoothing is better here than bearing off. Try the following sequence for example:

62, 65, 52, 52.

After smoothing, your position is 301000 (three checkers in the one-point, one on the 3-point). After bearing off it is 500000 (five on the one-point). So after smoothing you are likely to bear off in 2 rolls (except after rolling 21), after bearing off you will require 3 more rolls.

So my feeling is, that the effect only works, after the remaining checkers on the higher points have been taken off.


Hardy .... promising to check his explanations better next time 

Thank you, this sequence is much better, I should have found it ! First a big roll making the 4 disappear, then there are indeed two misses with the twos. Very nice example, which just needed a better explanation    

[blitzxz]

Intresting discustion. To summarize Hardy's point I would say that at least 4 condition has to be present when it is not correct to bear-off all the possible checkers.

1. Uneven number of checkers. (after you bear-off everything possible)

2. All the checkers close to home and no gaps. (so that you are almost certain to bear-off two checkers every roll.)

3. Lead in race.

4. Bad distribution if you bear-off.

All in all these situations are rare when you play correctly and even then the difference isn't big.