I was thinking a bit recently about "bearoff databases". These can be used near the end of many games, where both players are bearing off. The number of possible game states is actually fairly manageable in these cases, unlike the case of a generic board where the number of states is truly huge.

So in the case of bearing off, you can just enumerate the board layouts and the expected number of points in each.

Here's how it works: for a simple board, eg when both players have just one piece left, it's pretty easy to work out the expected number of points. Just run through all 24 possible rolls and check what happens in each one. In most of those cases the player with the dice will bear off and win a point. If not, you check what the expected number of points is for the opposing player post your move, and your expected number of points is just -1 times his.

You can then recursively figure out all the more complex board layouts: for a given layout, run through all possible rolls; for each roll check all possible moves; for each possible move check the opponent's expected number of points post-move, and choose your move to minimize your opponent's benefit. "Check the opponent's expected number of points" is the recursive bit - that will do a similar calculation, which will also recurse, and so on.

To make things easier you can store each board layout->expected number of points in a hash table so you don't have to keep repeating calculations. That's the database part of "bearoff database". Then once you have the database, you can quickly look up the expected number of points given a board layout.

That's a very brute force approach, and it only works if there aren't too many possible board layouts. Remember, the board layout is a combination of the player's setup and the opponent's setup: this is called a "two sided" bearoff database.

With a bit of math you can work out the total number of board layouts if you restrict your database to N points nearest the end and up to M checkers on those points:

Number of board layouts = ( (N+M)!/N!/M! - 1 )^2

That is, the number of ways to lay out just one player's side of the board is (N+M) choose N minus 1; and for each of those there are the same number of ways for the opponent's side to be laid out.

There are also "one sided" bearoff database. Here you enumerate just one player's board layouts, and for each you record the distribution of number of moves before all pieces are born off. If you have that data for both players in can approximate the expected number of points for either. I'm not that clear on how that approximation works, and haven't tried to investigate it much: the two sided database seems manageable and easier to understand.

Some examples for (number of points,number of checkers): (3,3): 361; (3,10): 81,225; (3,15): 664,225; (4,3): 1,156; (4,10): 1,000,000; (6,3): 6,889; (6,10): 64.1M; (6,15): 2.9B. On my Mac Pro in a single process I can calculate around 100k/minute. Storing them as an uncompressed text file uses around 1.8MB/100k entries.

Just for interest I note that the number of layouts for 24 points and 15 checkers is 6.3e20. This is an upper limit to the total number of valid backgammon boards, since it includes layouts where there are white and black checkers on the same point. It ignores layouts where either player has a checker on the bar, but I suspect those number much less than the overlapping cases. So the total number of backgammon boards is order 1e20 anyways.

So in the case of bearing off, you can just enumerate the board layouts and the expected number of points in each.

Here's how it works: for a simple board, eg when both players have just one piece left, it's pretty easy to work out the expected number of points. Just run through all 24 possible rolls and check what happens in each one. In most of those cases the player with the dice will bear off and win a point. If not, you check what the expected number of points is for the opposing player post your move, and your expected number of points is just -1 times his.

You can then recursively figure out all the more complex board layouts: for a given layout, run through all possible rolls; for each roll check all possible moves; for each possible move check the opponent's expected number of points post-move, and choose your move to minimize your opponent's benefit. "Check the opponent's expected number of points" is the recursive bit - that will do a similar calculation, which will also recurse, and so on.

To make things easier you can store each board layout->expected number of points in a hash table so you don't have to keep repeating calculations. That's the database part of "bearoff database". Then once you have the database, you can quickly look up the expected number of points given a board layout.

That's a very brute force approach, and it only works if there aren't too many possible board layouts. Remember, the board layout is a combination of the player's setup and the opponent's setup: this is called a "two sided" bearoff database.

With a bit of math you can work out the total number of board layouts if you restrict your database to N points nearest the end and up to M checkers on those points:

Number of board layouts = ( (N+M)!/N!/M! - 1 )^2

That is, the number of ways to lay out just one player's side of the board is (N+M) choose N minus 1; and for each of those there are the same number of ways for the opponent's side to be laid out.

There are also "one sided" bearoff database. Here you enumerate just one player's board layouts, and for each you record the distribution of number of moves before all pieces are born off. If you have that data for both players in can approximate the expected number of points for either. I'm not that clear on how that approximation works, and haven't tried to investigate it much: the two sided database seems manageable and easier to understand.

Some examples for (number of points,number of checkers): (3,3): 361; (3,10): 81,225; (3,15): 664,225; (4,3): 1,156; (4,10): 1,000,000; (6,3): 6,889; (6,10): 64.1M; (6,15): 2.9B. On my Mac Pro in a single process I can calculate around 100k/minute. Storing them as an uncompressed text file uses around 1.8MB/100k entries.

Just for interest I note that the number of layouts for 24 points and 15 checkers is 6.3e20. This is an upper limit to the total number of valid backgammon boards, since it includes layouts where there are white and black checkers on the same point. It ignores layouts where either player has a checker on the bar, but I suspect those number much less than the overlapping cases. So the total number of backgammon boards is order 1e20 anyways.

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