| Backgammon Variants |
Every game has luck in it. Games that don’t have luck in them are no fun to play.
Even tic-tac-toe has luck in it, at least it does when you are new to the game. Children just learning will try different moves somewhat at random until they get a sense of what they are doing. Once they figure out a successful strategy, the luck disappears and tic-tac-toe ceases to be fun. You can’t have fun without luck being part of the game.
This applies even to games that don’t appear to have any luck, so called “pure skill” games, such as chess or golf. These games have luck too; it is just that the luck in these games is harder to see. (Bill Robertie has a nice explanation of how luck manifests itself in chess.)
That’s what I like about backgammon. Backgammon has luck and skill, but backgammon doesn’t shy away from being a game of chance. When you see the dice, you know there is luck involved.
In hypergammon, we can even measure the luck, because we know the exact equity of every position. The “luck” of a roll is the difference in your equity before you roll and after you roll.
Largest Joker
A joker is a particularly lucky roll, the sort of roll that can turn a game around. For some reason, people tend to notice and remember their opponent’s jokers more than their own. You might even say, “that’s the biggest joker I’ve ever seen!” If you were playing hypergammon, wasn’t the biggest joker you’ve ever seen if you have seen this position:
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| Black rolls 6-6 |
Before black rolled, his position was pretty bleak. He might even have been gammoned. Black’s equity was a lowly −1.24241.
Then black rolled sixes. Glorious sixes! In one roll black gets all his checkers off and wins the game. He even wins a gammon! Black’s equity soars from −1.24241 to +2.0, a difference of 3.24241.
That is the biggest equity swing possible for a single roll in cubeless hypergammon.
Largest Anti-Joker
An anti-joker is the opposite of a joker. It’s an unusually unlucky roll. What is the biggest anti-joker in hypergammon? Look at this position:
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| Black rolls 5-2 |
Before he rolled, black was looking pretty good; he had a triple shot from the bar at white’s three blots. Only 5 rolls out of 36 miss those blots (6-5, 5-5, and 5-2). Black’s equity was +.82902.
Of course black rolls one of his missing numbers, and the worst one at that. Black is forced to enter and bypass all three white blots. He is now likely to lose a gammon. Black’s equity plunges to a sickening −1.76175.
The difference between +.82902 and −1.76175 is 2.59076. This is the worst roll you can throw in cubeless hypergammon.
The dice giveth, and the dice taketh away.
Biggest Checker Play Error
A whopper in backgammon is big error, usually defined as an error 0.1 or more of lost equity. You can’t afford to make too many whoppers and expect to fair well in the long run. In cubeless play especially, where equities are not multiplied by the cube, whoppers are quite serious errors. Nevertheless, everybody makes a whopper once in a while, sometimes worse.
What is the biggest error you can make in cubeless backgammon? Bob Koca figured it out. Look at the following position:
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| Black rolls 1-1 |
It takes only a little thought to realize the correct play. Black should use three of his 1’s to hit whites’s blots, 24/23*/22*/21*, then use his final 1 to cover the blot on his three point, 4/3.
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| After the right play |
At this point, black stands a good chance of winning a gammon. His equity is +1.73588.
But what if black instead played 4/1, 3/2?
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| After the wrong play |
That is a seriously bad way to play this roll. It leaves white a triple shot at black’s lone checker on 24 and white stands a good chance of winning a gammon or a backgammon. Black’s equity plunges to −2.06012. Yikes!
The difference is 3.79600 in lost equity — a 38-whopper — the biggest error you can make in cubeless hypergammon.
Gammon Rate
When you win a game in backgammon, sometimes you also win a gammon (a.k.a., a “double game”). The number of gammons you win divided by the number of games you win is called your gammon rate.
Each person has a slightly different gammon rate because gammon rate depends partly on your style of play. If you play aggressively, you will win gammons at a higher rate and lose them at a higher rate too.
When bots play backgammon against each other, their gammon rate is usually around 27 or 28 percent. That is to say, about 27 or 28 percent of all games played without a cube end in a gammon or a backgammon. Most good players have a gammon rate in this area too.
What do you think the gammon rate is for hypergammon? Do you think it is higher or lower than in backgammon? I’ll give the answer next time.
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