How Experts Analyze Backgammon Games
First of all, I don't claim to be an expert. But I've read enough high-level backgammon material to know at least how experts approach backgammon problems. I chose a relatively simple one, that came from an actual match.
I was playing a match to 11 on the Microsoft Gaming Zone against a player named Relu, who was at the time the top-ranked player there. I was leading 3-2, when the following position came up. I was "o" and I had doubled my opponent to 2.
13 14 15 16 17 18 19 20 21 22 23 24 | | | x x x | | | | x x x | | | | x x x | | | | x | | | | | | | | | | |x| | | |x| | | | | | | | | | | | | o o | | o | | o o o o o x | | x o o | | o o o o o x | 12 11 10 9 8 7 6 5 4 3 2 1
As you can see, x is in a lot of trouble. He's likely to get gammoned unless he can hit a shot, either by entering both checkers from the bar and staying back, or by hitting from the bar itself. Even then, he still has to contain o's checkers. A lot will depend on him entering his checkers and getting some around to close his 5 and even 3 points (the 20 and 22 points on the board diagram), and he'd also like to hit sooner rather than later, before o gets too many checkers off. x's position isn't hopeless. If things go well, he can still win this game, but he needs things to go very well.
Well, things went pretty badly for x. He took a while to enter his two checkers, and o rolled a couple of sets of doubles, including a very important one just before this position came up:
13 14 15 16 17 18 19 20 21 22 23 24 | x | | x x x | | | | x x x | | | | x x x | | | | x | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | x | | o | | x x | | o x | 12 11 10 9 8 7 6 5 4 3 2 1
x now rolled a 6-4. Clearly, he has only one key decision, whether to run from the 1-point and save the backgammon or not. Should he?
Now, I've chosen this position because I think the answer is pretty clear and should be found by any decent player, but I thought I'd give the complete analysis.
First, what if he runs? He loses 4 points. That's easy.
What if he stays? I'm going to analyze this in terms of the next two rolls, one for each side, a total of 1296 possibilities.
First, I have 26 rolls that score an immediate backgammon. Of the 36 dice rolls, 25 don't contain a 1, plus one of those is double-1's. So I have 26 x 36 (it doesn't matter what his next roll is) immediate backgammons, or 936.
What about when I leave a blot on the 1-point? Well, he can do three things:
- He can hit me (11 rolls, any with an ace).
- He can enter without hitting but also leaving a blot in my board (4 rolls, 3-2 and 4-2), in which case I win a backgammon.
- He can enter and clear my board without hitting (21 rolls) in which case I win a gammon.
Cases 2 and 3 are easy. In case 2, I win a backgammon. This means 10 of my rolls times 4 of his, for 40 more backgammons. Case 3 is 10 of my rolls times 21 of his, for 210 gammons.
Now, we've actually "solved" 1186 of the 1296 possible outcomes already. What are left are the 110 times that I leave a shot and he hits me.
Well, a lot of things can happen. Remember, I can still win a backgammon! It's not likely, but suppose just for fun that I enter with 5-4, letting me hit him. He has to come in from the bar AND move the checker that he hit me with out. If I roll double 4's or 5's next, I will be off before he can get going. But let's say that's just a 1% chance, one of the 110 times.
If he hits me, he'll be trying to move his checkers around and hit me and send me back while I'm trying to go forward. He'll also be trying to close his board by making first his 5, then 3, then 1-point. He has only 3 points now, so it's going to be hard to keep me out for long. Let's say that he manages to close his board 25% of the time (28 times out of 110). In this case, his winning chances are about 10%, so he'll win 3 games and lose 25.
When he doesn't close his board, the question is whether he can hit me enough times to get his checkers around and get one off. Let's say that he can do that 60% of the time. Remember, all I need is to enter and get one nice set of doubles before he can make points to block me. Let's say that there are two positions coming up where he will have two shots at me, two numbers that can hit me. Let's say he can hit me at some point with a 5 or a 2. That hits with any 5, or 2, or 4-1, or 1-1, a total of 23 rolls. His chances of hitting a 23/36 shot twice in a row are 23/36 times 23/36, or about 40%. If I can ever slip past him, I'll get a gammon. So containing me 60% of the time isn't an unreasonable estimate.
So of the 81 rolls where he enters, hits, doesn't lose an unlucky backgammon, and doesn't close me out, we get 49 single losses and 32 gammons. Now let's add it all up:
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Backgammons: 936 + 40 + 1 = 977, total of 5862 points (the cube is at 2, remember). |
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Gammons: 210 + 32 = 242, total of 968 points. |
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Single losses: 25 + 49 = 74, total of 148 points. |
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Wins: 3, gain of 6 points. |
Adding it all up, we get a net loss of 5.38 points a game, compared to a loss of 4 if he runs. But you knew before we even started that running was the right play, I'm sure. (If you think that Relu, my opponent, really earned his ranking honestly, well, maybe this example of his play will change your mind.)