Is backgammon solved?
Checkers is solved and chess engines crush grandmasters. So has backgammon been solved? Not quite. Here is what computers know, what they still estimate, and what that means when you sit down to play.
The short answer
No. Standard backgammon has not been solved: nobody knows the perfect move and cube decision for every position. But computer programs such as eXtreme Gammon (XG) and GNU Backgammon play so well that they beat every human over a long series, and small pieces of the game, like the three-checker variant hypergammon and many bear-off endgames, are solved exactly.
What "solved" would mean
A game is solved when the best result from any position is known for certain. Tic-tac-toe is solved by hand. Checkers was solved by computer in 2007: perfect play from both sides is a draw. For backgammon, which has dice, "solved" means knowing the exact equity of every position: the average points you win or lose with perfect play, for every roll that could come, including the doubling cube.
Why it is so hard
- Too many positions. Counting just where the 30 checkers can stand gives about 18,528,584,051,601,162,496 legal positions, roughly 18.5 quintillion.
- Dice branch every turn. Each position must be considered for all 21 distinct rolls, and each roll usually allows many legal plays.
- The cube adds another layer. Who owns the cube and at what value changes the right play, and in match play so does the score.
- Games can loop. Checkers get hit and re-enter, so positions repeat. There is no simple "work backward from the end" order, as there is for a pure race.
The parts that are solved
- Hypergammon, the variant with three checkers each, was solved by Hugh Sconyers in the early 1990s. Its roughly 8 million checker positions have exact values.
- Bear-off endgames. Once all checkers are home and there is no more contact, engines use bearoff databases that store the exact winning chances. See bearing off for the human version.
- Short races are close to solved: with a good pip count and a few rules of thumb, humans can match the databases on most cube decisions.
How bots play so well
Modern bots do not look positions up. They use a neural network that has played itself millions of times and learned to estimate a position's winning, gammon and backgammon chances. To decide a move, the bot looks a couple of rolls ahead and picks the play with the best estimate. To settle a hard question, it runs a rollout: it plays the position out thousands of times with different dice and averages the results.
Those rollouts are where today's opening moves come from. Several plays that experts swore by in the 1980s turned out to be wrong once bots rolled them out.
A short history
| When | What happened |
|---|---|
| 1979 | Hans Berliner's BKG 9.8 beats world champion Luigi Villa 7–1 in a short exhibition match, the first program to beat a world champion at any board game, with some help from the dice. |
| Early 1990s | Hugh Sconyers solves hypergammon, the three-checker version of the game, computing the exact value of every position. |
| 1992 | Gerald Tesauro's TD-Gammon at IBM learns by playing itself with a neural network and reaches the level of strong human players, changing how experts play some openings. |
| 1990s–2000s | Neural-network bots such as Jellyfish, Snowie and GNU Backgammon become the standard way to analyse games. |
| Late 2000s onward | eXtreme Gammon (XG) becomes the reference engine for top players; its rollouts settle most opening and cube questions. |
For more of the game's past, see the history of backgammon.
What it means for your games
Because the game is unsolved, there is always more to learn, and because the dice decide single games, a beginner can still beat an expert on the night (see luck or skill). Bots are the best teachers ever made: play a game, then ask what they would have done.
Backgammon Lounge's computer opponent uses a neural network too, with three levels and move hints. Play a game against it and use the hints to see where your plays differ.
Questions
Is backgammon solved?
No. Nobody has computed the perfect move and cube decision for every position of standard backgammon. There are about 18.5 quintillion legal checker positions, and the dice and doubling cube multiply the work. Smaller versions, such as three-checker hypergammon, have been solved exactly.
Can a computer beat the best backgammon players?
Yes, over a long series. Bots like eXtreme Gammon and GNU Backgammon play better than any human on average. In a single game anyone can win with good dice, so even the best bot loses many individual games.
What is the best backgammon engine?
eXtreme Gammon (XG) is the engine most top players use for analysis. GNU Backgammon (gnubg) is free and open source and nearly as strong. Both are neural networks that estimate the value of a position rather than looking it up.
Which parts of backgammon are solved?
Endgames with few checkers left. Bearoff databases store the exact value of positions where all checkers are in the home boards, and engines look these up instead of estimating. Hypergammon, with three checkers each, is solved completely.
Will backgammon ever be solved?
Not by brute force soon. The number of positions is far larger than chess endgame tables that took decades to build, and every position has 21 different rolls to consider. Bots are already close enough to perfect that the gap matters only to the very best players.
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