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Problem of the Week #118: Solution
09-10-2011
, 12:38 PM
Problem of the Week #118: Solution
Black on roll.
Black to play 4-3 in these situations:
(a) Cash game, center cube.
(b) Tournament match, center cube, Black trails 6-12 to 15.
Note: All ‘cash game’ problems assume the Jacoby Rule is in effect. That is, you can’t win a gammon unless the cube has been turned.
High-level backgammon is played in two ways. In clubs, people normally play cash games, either heads-up or in chouettes. In these contests, each game is independent of any games that might have come before or after. In tournaments, however, players play matches to a certain number of points. In a 15-point match, for example, the first player to accumulate 15 points is the winner. In matches, optimal checker and cube play is often affected by the score in the match.
In general, the leader in a match becomes more defensive. He wants to protect the lead he’s created, and protecting a lead takes two forms:
(1) Reducing the likelihood that the cube will reach a high level. If the cube gets big enough, the entire match may swing on the result of the current game, rendering the leader’s previous advantage null and void. Clearly, if the leader’s chances in the current game are big enough (whatever that means in the situation) then the leader will cheerfully let the game decide the match. But mostly, he’s a little more reluctant to offer a speculative cube, and a little more reluctant to take.
(2) Avoiding gammonish positions. The leader is obviously even more unhappy to be gammoned than he would be in a money game. So positions where gammons are likely need to be avoided.
The trailer’s interests, of course, are directly opposed to the leader’s. So he is willing to double a little earlier than normal, and perhaps take cubes more aggressively. He will seek out positions that are more likely to lead to a gammon, since he has more to gain from a big swing, and less to lose.
That’s the theory, of course, and it’s easy to understand. The real artistry lies in deciding just when you need to adjust your money play to take advantage of the match score, and when you don’t. Can you judge when the time is right, and when it isn’t?
Take a look at Position 118. The opening rolls of this game were 4-1 for Black (which he played 13/9 24/23) and then 2-1 for White (played 13/11 24/23). Now Black has a 4-3 to play, and he obviously has two reasonable choices: 24/20 23/20, making the defensive 20-point, or 9/5 8/5, making the offensive 5-point.
For money, making the 20-point is correct by a pretty clear margin. The reason is not that the 20-point is intrinsically stronger than the 5-point. It’s not. Both points are valuable. The real reason is that making the 5-point with the builders on the 8-point and the 9-point leave Black with a relatively poor position in front. His 8-point is now stripped, while he remains with a big stack on the 6-point. While he’s made a good point, future progress will be difficult. Making the 20-point, in contrast, preserves a good distribution of offensive builders.
When you translate a position from money play to match play, however, things change. Trailing 6-12 to 15, Black benefits greatly from winning a gammon instead of a single game, whereas the cost of losing a gammon instead of losing a single game is much less. As a result, Black seeks out plays that increase his own gammon chances. If such a play happens to result in more lost gammons as well, so what? He’s very likely to lose those matches anyway.
Playing 9/5 8/5 now suits his purposes perfectly. Anytime you make a point in your board, your gammon chances increase by a few percent compared to alternatives, and making the 5-point is especially strong since it’s both an attacking point and a priming point. Failing to make the defensive 20-point will cost Black some extra gammons, but those don’t matter nearly as much. Rollouts indicate that making the 5-point creates about 3% more gammon wins for Black while conceding about 2.5% more gammon losses. Given the relative value of gammons won versus gammons lost, that’s a good trade for Black.
Solution:
(a) For money, 23/20 24/20
(b) Trailing 6-12 to 15, 9/5 8/5
Black on roll.
Black to play 4-3 in these situations:
(a) Cash game, center cube.
(b) Tournament match, center cube, Black trails 6-12 to 15.
Note: All ‘cash game’ problems assume the Jacoby Rule is in effect. That is, you can’t win a gammon unless the cube has been turned.
High-level backgammon is played in two ways. In clubs, people normally play cash games, either heads-up or in chouettes. In these contests, each game is independent of any games that might have come before or after. In tournaments, however, players play matches to a certain number of points. In a 15-point match, for example, the first player to accumulate 15 points is the winner. In matches, optimal checker and cube play is often affected by the score in the match.
In general, the leader in a match becomes more defensive. He wants to protect the lead he’s created, and protecting a lead takes two forms:
(1) Reducing the likelihood that the cube will reach a high level. If the cube gets big enough, the entire match may swing on the result of the current game, rendering the leader’s previous advantage null and void. Clearly, if the leader’s chances in the current game are big enough (whatever that means in the situation) then the leader will cheerfully let the game decide the match. But mostly, he’s a little more reluctant to offer a speculative cube, and a little more reluctant to take.
(2) Avoiding gammonish positions. The leader is obviously even more unhappy to be gammoned than he would be in a money game. So positions where gammons are likely need to be avoided.
The trailer’s interests, of course, are directly opposed to the leader’s. So he is willing to double a little earlier than normal, and perhaps take cubes more aggressively. He will seek out positions that are more likely to lead to a gammon, since he has more to gain from a big swing, and less to lose.
That’s the theory, of course, and it’s easy to understand. The real artistry lies in deciding just when you need to adjust your money play to take advantage of the match score, and when you don’t. Can you judge when the time is right, and when it isn’t?
Take a look at Position 118. The opening rolls of this game were 4-1 for Black (which he played 13/9 24/23) and then 2-1 for White (played 13/11 24/23). Now Black has a 4-3 to play, and he obviously has two reasonable choices: 24/20 23/20, making the defensive 20-point, or 9/5 8/5, making the offensive 5-point.
For money, making the 20-point is correct by a pretty clear margin. The reason is not that the 20-point is intrinsically stronger than the 5-point. It’s not. Both points are valuable. The real reason is that making the 5-point with the builders on the 8-point and the 9-point leave Black with a relatively poor position in front. His 8-point is now stripped, while he remains with a big stack on the 6-point. While he’s made a good point, future progress will be difficult. Making the 20-point, in contrast, preserves a good distribution of offensive builders.
When you translate a position from money play to match play, however, things change. Trailing 6-12 to 15, Black benefits greatly from winning a gammon instead of a single game, whereas the cost of losing a gammon instead of losing a single game is much less. As a result, Black seeks out plays that increase his own gammon chances. If such a play happens to result in more lost gammons as well, so what? He’s very likely to lose those matches anyway.
Playing 9/5 8/5 now suits his purposes perfectly. Anytime you make a point in your board, your gammon chances increase by a few percent compared to alternatives, and making the 5-point is especially strong since it’s both an attacking point and a priming point. Failing to make the defensive 20-point will cost Black some extra gammons, but those don’t matter nearly as much. Rollouts indicate that making the 5-point creates about 3% more gammon wins for Black while conceding about 2.5% more gammon losses. Given the relative value of gammons won versus gammons lost, that’s a good trade for Black.
Solution:
(a) For money, 23/20 24/20
(b) Trailing 6-12 to 15, 9/5 8/5
