I don't have the link, but there was a website that said that if you were 25% or more to win the game, you should accept the doubling cube (when playing for money) if it is offered to you. Is this a common figure used in the backgammon community? It seems to be discounting quite a bit, but I'd like to see Bill and others' take on it.

There's a little bit more to it than that (you also have to consider how often you'll be gammonned/backgammonned), but it's essentially correct.

Say you're exactly 25% to win. The math works out like this: if you decline the double offer, you lose one unit 100% of the time. Your expected value calculation is thus:
-1 * 1.00 = -1

If you accept the offer and the game plays out to the end, you will lose 2 units 75% of the time, but win 2 units 25% of the time. The expected value calculation is then
-2 * 0.75 + +2 * 0.25 = -1

Thus, based on the expected value, you're neutral between accepting the double and passing when you're at exactly 25% to win. If your chances are any better, you'd prefer to take the double over passing.

The calculations above also don't take into consideration that there is an advantage in owning the cube, making taking the cube more attractive in close situations.

Thanks whitewolf. My problem was that I was assuming the 25% came from the fact that holding the cube is an advantage, but I get it now.

In practice, do players offer the doubling cube whenever they are significantly better than 50% to win (excluding the complexities of gammon and backgammon) or does the shift of power (the doubling cube) prevent them from doing this until they are even better off?

I'm not an expert, but as a recreational player the way I think about the cube is to find a spot to double when you have a decent advantage that with a good roll can become a dominant one. Such as a mid to late game position with relatively even pip counts where your opponent has left you a shot and you have 4 or 5 points covered in your board. that if you hit will give you a significant advantage.

Theoretically, you would like to offer the cube at exactly 75% winning chances. This is, however, quite hard to do because factoring in gammons, backgammons and the advantage of holding the cube is hard to do for most positions.

There are some common situations in practical play that usually lead to a cube offer. Let's have a look at them (please forgive me for the absence of diagrams... I'm writing this from scratch and with no BG tool at hand).

• You have managed to escape your last checker and now you have a race without contact, i.e. no hits are possible anymore. As a rule of thump you offer an initial double if you lead the pipcount by 8% and you offer a redouble if you lead the pip count by 9%. This is a situation, where you have a solid advantage.
• You double to avoid a market losing roll. Imagine you have a closed home board wheras the opponent would have the 5-point and 6-point of his board open. So imagine that you have two checkers on your midpoint and the opponent has one checker left on your bar point, so you hit with any 6 (11 rolls), 51 (2 rolls), 42 (2 rolls), 33 (1 roll) and 22 (1 roll) for a total of 17 out of 36 rolls that win the game immediately by hitting. You also very close to winning if you roll the high doubles 55 and 44 (66 is already accounted for). So we have 19 of 36 rolls winning, whichs is slightly above 50%. If you throw these rolls, your opponent won't take a cube anymore, so you're said to have lost your market in this case. So you double now, because, you're opponent will still be in the game if you don't roll the winning roll. In case you don't hit but manage to run one checker to safety, your opponent should not be able to redouble right back (in case he can, I have chosen a bad example to demostrate the point).
• The action double: A lot of hits are possible that result in a huge swing in your favour. If you don't hit, you're in trouble. Otherwise you're crushing the other guy. An example could be if you had a 6-prime from your 2-point to your barpoint and you're on the bar with one checker and you have two direct shots to blots in your opponents home board (e.g. he holds 4 points and has a blot on each of the still open points) and the opponent has one blot on his midpoint. If you hit, he's stone dead and most likely gammon whereas you yourself are not as likely to lose gammon because it's only one checker he could close-out.
• The double to end the game. This has actually a lot to do with the advantage of cube ownership. So imagine both players have only three checkers left on the 1-point during the bearoff and you're to roll. If you double, he can't take because he can only win if he rolls a double the next roll (6/36 or 1/6 or 16% chance to do that). So if you have access to the cube, you win 100% of the time. If your opponent has the cube, you can't double, and thus you lose if you don't roll a double and then your opponent does (1/6 * 1/6 = 1/36 or 2,8%). By doubling, you remove the tiny chance that you still lose the game, e.g. increase your winning chances from 97,2% to 100%.

This last post explained most of the common doubling situations very well. I'll just add that a position can also be too good to double. Here's how that works.

Let's say you own the doubling cube at 2, the game is turning in your favor, but you don't think your game is good enough to double yet. Now a great sequence occurs: you roll a good number, and your opponent rolls badly. You assess that your opponent would now pass a double if you offered one, but the combinations of high gammon chances (where you would win 4 points instead of just 2) and low losing chances makes it more profitable to play on for the gammon while holding onto the cube.

Mr Robertie,

I never thought I would have a chance to say this, but: What do you believe I learned that from?



I'm honoured.


PS:

I messed up the numbers in the last example. The chance to lose the game is to NOT roll a double, and then the opponent rolls a double. So correct is (5/6 * 1/6) = 5/36 = 13,8% instead of 2,8%.

To give an example in hopes that I'm understanding this: I've been playing around with GNU BG and I'm seeing an endgame spot where I think this applies. I have the cube in a bearoff position of 6543321. My opponent has 554432. GNU BG suggests that I'm a 63.2% favorite here, and in isolation it would seem that I have a profitable redouble. But the situation you describe applies twice: first that I could have a more profitable autorefused double later denying my opponent equity from good dice, and second that my opponent could catch up and redouble in a situation where I would have to refuse, thus denying me my equity from good dice. The combination of those two is what makes it unprofitable to double in this situation. Do I have that right?

You have to take into account that in the situation I described, the opponent would make a huge mistake by taking because he only has 13% winning chances. He also doesn't have any benefit from taking the cube because if he does roll his double he has won and if he does not he has lost. There is no way he could return the cube.

In your situation, your opponent would have a clear take based on winning percentages alone. In addition to that, you would give him exclusive access to the cube, which means that you have to win the game by the dice alone. You can't cube him out anymore in case you reach a situation like I described, leaving him with 13,8% in this spot.

In case you roll an ugly sequence, for example 61 followed by 41, he might be able to cube you out because he already has one checker more borne off. I would say that since you are talking about a redouble, cube ownership is too much of an asset in this position because it gives you the chance to recover from an ugly sequence.

If the cube was in the middle it might be a reasonable double (and take, of course) because you would lose your market if you throw something strong.

Sort of off-topic, but I think the best way to play backgammon is with the Jacoby rule, where you don't get into situations where it is too good to double. This rule doesn't count gammons unless the cube has been turned. Of course, this rule does not apply to match play.

Sort of off-topic again but for our poker friends I have some nice analogies:

Value betting = doubling, opponent has odds to take

Protection = doubling, opponent has to pass

implied odds = cube ownership

slow playing = opponent can't take if doubled, he can catch up on the next roll but is still not able to take

LAG style = opting for hitting loose if a safe move is also reasonable

bad beat = you play for a gammon, getting hit during bear-off, you dance 10 times straight on a 3-point board, you get redoubled way too early but you cannot beaver, you take and dance another 4 times, opponent leaves a shot and you miss. You are happy that you have borne off a checker so you can't lose a gammon.

Note: wasn't sure if this warranted its own post

So what if you're playing against someone who will accept the doubling cube, unless he is near-certain that he will lose. Should you wait until you are a significant favorite to offer, since this reduces the threat of him giving the cube back to you?

If the opponent takes to much, you can let your advantage increase a bit more before doubling. Optimally he eats the double when he's already in danger of getting gammoned (-:

Yes, ideally you should wait until he'll close to having a tough choice (in his mind) between accepting and declining. This is done both to reduce the threat of a redouble, as well as to maximize the value you gain by doubling.

The mathematical explanation runs like this: Say you're up against a player who will accept a double offer even when he's 90% to lose (say with 0% chance at gammon). If you offer the cube when you're 75% to win, your EV in the game goes from 0.5 units to 1.0 units, a gain of 0.5 units. However, if you wait until you're 90% to win before offering, your EV in the game jumps from 0.8 to 1.6, an increase of 0.8. So you gain more by waiting, and you avoid the situations where you would have doubled at a more normal range, but your game ends up going south and you wish you hadn't doubled.

The first two responders to this question were right on the money, but there's another factor to consider. When your loose opponent sees that you're not doubling positions that are clear doubles, he'll usually figure out that you're taking advantage of him, and quickly tighten up. I've seen this happen in chouettes and heads up many times. This kind of player is really just looking for action, and you'll win more money if you supply the action he craves. You can tighten up just a tad in non-volatile positions, but if there are gammons in the air, double normally.

You'll make your big money in two situations: (1) you suddenly go from a slight favorite to an 85% with a great shot, and your double gets taken anyway, or (2) you double, he takes and gets back to even, and redoubles you. His doubles will be as loose or looser than his takes, and because of that you won't have to worry about making serious errors when he doubles. That's worth a lot in the long run.

But don't get too cute. Just play, and he'll give his money away

Okay, so if my opponent will pass if I offer, my losing chances are low, and my gammon chances are high, then I reach a point where my gammons make the expectation of holding the cube higher than the expectation of offering it. So, these situations come up at times when my opponent wants
to forfeit the game because of my prospects for gammoning him?

Is there anyone who provide me a link to an example where this is the case. I think it's important for me to understand the cube in theory, but I think I'm ready for something a little more pragmatic.

Bill and others, thanks. I'm learning a lot from this forum.

A simple example is a case where you own the cube at 2, but have just closed out a couple of your opponent's checkers and are bringing your last few men home. The rest of your opponent's men are in his inner board.

Clearly, if you double he will pass (his winning chances are much less than 5%) and you will win 2 points.

If you play on, you will win a gammon (4-points) about 40% of the time, and get unlucky and lose a game once in a great while. The rest of the time he will save a gammon and you still win your 2 points. So not doubling is better.