Problem of the Week #100: March 13


Cash game, White owns the cube. Black on roll.





Black to play 5-1.



Note: All ‘cash game’ problems assume the Jacoby Rule is in effect. That is, you can’t win a gammon unless the cube has been turned.

Note: I’ll be away next week, so I’ll post the solution to this problem around March 27.

If I break now he has 24 shots and getting hit is a disaster. If I waiit a turn and he rolls a 6 then something will look better for me. It looks like there's a clear gain in waiting a so 6/1 6/5.

This looks like a classic pay now, or pay later type problem. If we break the midpoint now we give 24 shots with any ace or four plus 22, 33 and 63. If we play the other way, and break the six point (with 6/5 6/1) we leave no shots this roll.

BUT, after we play the safe way we are in danger of rolling some bad numbers next throw. 61 62 52 51 41 31 (ten shots) all leave single direct shots and 54 and 64 leave double shots! also, 44 leaves us with a hopeless position. Even after we throw one of these numbers and survive, there is the potential for more awkwardness on the following roll.

I think that we should chose to pay now and play 13/8 3/2.

Our position is weakening and our opponents is getting stronger. Typically this is the critical criteria in these type of positions. If our position is weakening while our opponents is getting stronger we should tend to pay now, and gamble early. If our position is not so brittle or our opponents board is getting close to crunching we should wait. Here, we should take this opportunity to gamble and give up 24 shots while our opponent has only a four point board.

Disagree.

I say play safe 6-5 6-1 and our opponents position will not get stronger next roll. Just the opposite, he will likely have to make some concession in the outfield.

One move I would not make is 13/8 3/2. Chances are that white will need to either crunch up his board, give me a direct shot, or move off the midpoint on his next roll, so no way I'm leaving two direct shots on this roll. This narrows my choice down to between 6/5 6/1 and 8/7 8/3. The 6/5 6/1 choice is safest for a single roll, however this would make it harder for me to get my guys in safe on subsequent rolls. So, I'm leaning towards 8/7 8/3 where he's only got an 11/36 chance to hit me on his next roll, and if he misses me I'll most likely have an easier time playing my next roll(s).

Yeah this reminds me of one of those problems from new ideas in backgammon (know in my circle as ..."the book that retired me from backgammon" as I got almost none of them right....f ing jellyfish).. where the ugly play is just going to be better somehow.

The one thing I remember about that pay now vs pay later section in that book, was that it is almost always right to pay later....sometimes you dont ever have to pay, and if u pay now, you might have to pay again anyway lol


So suck it up, strip the 6 and pray.

Grunch.

I'm gonna be a nit on that one, I go with the safe play, I don't want to be on the bar against a 4-pt board.

When ahead, run and play safe? Yes for me this time.

6/1 6/5

Edit (still in grunch mode): I just noticed that White had the cube. The play I chose might eventually tend toward an efficient cube for White. Is it better to provoke things now? Not sure.

He he, you're not stripping it, you're dismantling it!

can't see anything other than 6/5 6/1 here

Black can only play a five from three places in Problem 100, his six point, eight point or midpoint.

• 6/1, 6/5 – Although this play is completely safe for now, it surrenders the best asset Black has, his strong home board. If he makes this move, on his next turn Black will face the same dangers he faces now, but with only four home board points instead of five.
  
  
• 8/3, 8/7 – This is the safest move available to Black that will allow him to keep his home board. After this play, White will have 11 shots to hit, but when he does hit, White will have to sacrifice his anchor, the point he most wants to keep. If White hits and leaves a blot, Black may have a few return shots from the bar. If Black makes this play and gets away with it, he will still face the likelihood of future expose when he clears his midpoint.
  
  
• 13/8, 3/2 – This play begins to clear Black’s most difficult point. If it works, Black will become a heavy favorite, but first he must survive a double-direct shot, plus indirect nines. After this move, White will have 24 hitting numbers. If he is hit, Black will probably have no return shots from the bar.

Before selecting his move, Black should consider White’s position. If it is likely to deteriorate, Black may wish to defer any risk-taking. So…

White will hit whenever he can, but in case he cannot, White will likely play two checkers down from the midpoint. This would leave Black essentially where he is now, with White holding Black’s four and nine points.

When White rolls a one, he will not be able to clear his midpoint. But even then, he can often play safely. If White rolls 11, 12, or 13, he will move inside without damaging his board. Only if he rolls 14, 15 or 16 will he be forced to consider leaving a blot on his midpoint. What all this means is that White has a lot of playability, and that his position will usually improve before it begins to deteriorate.

From these observations, an ordinary analysis goes something like this:

1. White’s position is getting stronger, not weaker.
   
   
2. Black, therefore, should take a risk now, and hold onto his home board. Deferring the issue will only make things worse. This means Black must choose between breaking his eight point and breaking his bar point. Breaking the six point is not an option.
   
   
3. The nominally safer play, breaking the eight point, contains as much risk as the alternative. This is because Black will likely leave another shot when he breaks his midpoint later. The combined risk of a single shot now and another shot later is about the same as the 24 shots Black offers when he breaks the midpoint now.
   
   
4. Although the overall risk incurred by breaking the midpoint is about the same as that entailed in breaking the eight point, the benefits are not commensurate. Clearing the midpoint is clearly superior. If Black survives White’s shot, he will be nearly home free.
   
   
5. Ergo, Black should break the midpoint, playing 13/8, 3/2.

What this analysis ignores, however, is the possibility that Black might somehow hit White, and in the follow up, close him out and win a gammon. Consider what could happen if Black breaks the eight point, and White is forced to hit by abandoning his anchor. Right away, Black would have 33 to hit the blot left behind (on the Black four point), plus whatever combinations (1x and 3x) that would let Black enter and hit outside. That’s 5 potential shots immediately, and there could be others later. Compare this with the gammon-winning chances Black might have if he breaks the midpoint: virtually none!

Interesting as all this might be, it is not enough to dissuade me from accepting the orthodox analysis.

My solution: 13/8, 3/2


For the Record
I am so often wrong that I like to post my record in these messages. It's kind of a truth-in-advertising thing. Grunch: I have been answering these problems without the use of a bot, and before checking the excellent solutions of others, since Problem 28. My record at this writing is 53%.

I don't want to pay now because I don't think that his position is going to get stronger in the next roll. It's going to get weaker before it gets stronger (maybe 2-3 rolls before it gets better for him). His home board is not going to improve, and his outfield control is going to fade as one of those anchors is almost certainly going to go. Plus, there's a chance he's going to leave a shot for us. Leaving a double direct shot right now just seems unnecessarily reckless. It's not like you can even cube him out of the game if he misses. You've got to play this one all the way to the end.

Pay later, lose the 6 point. It's not doing all that much for you, anyway. It's not like your plan is to hit something in the outfield and try to keep him on the bar for a few rolls (a hit is more like a distraction to make it harder for him to hit you as you scramble your checkers home). And it's not doing much against his advanced anchor since you're not trying to keep him hemmed in.

Ditch the six point imo. 6/1 6/5

That's some excellent analysis as usual Taper_Mike. The only place I would disagree with you is that on white's next roll he is likely to have to move off of the midpoint, which will improve black's position by eliminating the double direct shot. This would be enough for me to hold off leaving that direct shot on this turn.

6/5 6/1 seems like the obvious play here, that gets my vote. If my opponent was soft I might consider 13/8 3/2.

How many rollouts will we need to get the "right" answer?

Grunch.

After our affair with Mr. KT, we Germans learned: working honestly with your source. So once again, Mr Woolsey said:

“1. If you pay now and get away with it, will the position be a near claim or will there still a lot of work to do? If you have a near claim that argues for paying now, but if you still aren’t home free, there is a greater incentive to pay later.

2. Is your opponent’s position improving or is it likely to get worse in the next few rolls? If your opponent’s board is now at maximum strength, this argues for paying later since it will be very costly to be hit. On the other hand if your opponent has not yet built up his board but is in the process of doing so, it may be correct to pay now since the cost of being hit won’t be as great as it will be later.

3. Is your own position deteriorating quickly, or do you still have some flexibility? If you have sufficient flexibility that argues for paying later. You will have time to hold out for a while, and maybe roll the doubles necessary to clear the difficult point. On the other hand if your position is stripped, now may be the time to pay before things get worse.

Generally at least two of these conditions have to point forwards paying now-otherwise it is usually correct to pay later.”

I think, this problem is also a difficult decision, where the tradeoffs for paying now and paying later are not so obvious.

1. With a centered Cube, we could cash, if we get away with paying now. But with the Cube on the other side, we have to play the 25 yards to the end to score a touchdown. And with the Cube there, we have to play even some extra yards. 1:0 for paying later.
2. The opponent’s board is not at maximum strength now. But there are also no builders in the zone, to make the roof better. For several moves, it will retain as a life after death house. At first sight, it will improve, but if we look closer, not. 2:0 for paying later.
3. We may consider our house as an asset, worthy to hold. But let us consider following scenario. Hitted, playing with a 5 point house and playing with a 4 point house. Playing with 5 points is better, but being hit with a 4 point house is far from death here and I wouldn’t hesitate to take a recube. Second, if we break the 6 point, a little flexibility is even gained. Tied.

So if we sum up, clear victory for paying later. At least to something completely different. You have also to look, how your decision will influence the gammon rate. If the pay now decision is not indicated according to our criteria, but wins way off more gammons, it may be correct to play against the criteria. After all, equity rules.

6/1 6/5 for me.

That will depends on the variance left in your rollout. For XG, for example, if the margin between the candidates is bigger than the added confidence intervals, you will have likely enough trials. From my experience, minimum is 1296 trials 3ply with variance reduction (predefined).

In the diagram below, I have played 62 for White after clearing Black's 6 point. Can Black (as X) reasonably expect to bring his checkers home and leave fewer than 24 shots from this position? It won't be easy!

Having said that, I've learned to respect the group mind on this site. When I post my solution, and then discover that all the heavies are on the other side, it usually means I'm wrong!

The real question is whether this is any worse than breaking on the previous roll. If it's just as ugly the next time around (and not uglier), but you give yourself the chance to roll doubles to safely get home, then what do you lose by waiting a roll to pay?

Is this not uglier? Black has cracked! His 5:4 advantage in home board points is now even.

My first attempt at analysis...hope I don't make fool out of myself!

Black is going to get hit eventually no matter what move he makes. Since he has a 5 point board he should maximize his chances of getting a return shot after being hit. And why equalize his board with whites by playing 6/1 6/5? Black needs the midpoint for return shots so I vote 8/7 8/3.

I'm going with 6/1, 6/5. It is ugly but we can hope to wait this out or get lucky. White may have to break one of his outfield points next roll. With black still having the checkers on the 13 point, he wants to keep the 8 point as long as possible.

What value is your home board to you at this point in the game? You're so far ahead that your game plan is just to scramble home. It buys you some safety trying to break your 8 point, but not a huge amount.

Compare this to the value of white's home board. His plan is to hit something and keep black on the bar a little while so he can catch up in the race. If he were to crack, he's in huge trouble because he cannot execute his desired game plan.

So the fact that we're now tied on the number of home board points doesn't really bother me.

White's position is actually even less flexible than I realized- Other than 55 filling his 3, and that's even double-edged, is there any roll he would voluntarily play if he could just pass the dice back? He has to break an anchor or screw up his home board, so I don't see how to justify leaving the 24-shot now. Well, I guess 11 for him isn't a big loss after 16/14 (2), but it's still worse.

What happens after Black is hit? If he fans, can he accept a double? If Black breaks the midpoint now, the odds are almost 30% that exactly that sequence will occur:

P(White hits) * P(Black fans) = (24/36) * (16/36) = 30%

Of course, Black could also find himself on the bar facing a cube even if he breaks the six point now.

I'm playing 8 - to - 2
duplicating a few rolls and keeping the point that will hopefully force white to move or bust it's board