When backgammon
tournaments are held to determine an overall winner, the usual style
of competition is match play. Competitors are paired off, and each
pair plays a series of games to decide which player progresses to
the next round of the tournament. This series of games is called
a match.
Matches are played to a predetermined specified number of points.
The first player to accumulate the required number of points wins
the match. Points are awarded: one for a single game, two for a
gammon, and three for a backgammon. The doubling cube is used, so
the number of points a player receives is multiplied by the value
of the cube. Automatic doubles, beavers, and the Jacoby rule are
not used in match play.
We should emphasize that there is no bonus for winning more than
the required number of points in a match. The players' sole goal
is to win the match, and the final score is immaterial. This can
lead to some apparent inequities.
Consider a three-point match which one player leads 2 to 1. The
player with 2 points would seem to have a significant advantage
until you realize the strategy available to the player with 1 point.
That player will simply double at his first opportunity in the next
game. The player with 2 points is then forced to either (1) accept
the double, in which case that game will decide the match, or (2)
refuse the double, giving up a point, so that the following game
decides the match.
This situation has lead to the adoption of a rule called the Crawford
rule. The Crawford rule states that if one player reaches a score
one point short of the match, neither player may offer a double
in the immediately following game. This one game without doubling
is called the Crawford game. After the Crawford game, if the match
has not yet been decided, the doubling cube becomes available again.
Normalized Scoring
It is convenient when discussing doubling strategy to refer to match
scores in terms of the number of additional points each player needs,
rather than the number of points they have won so far. For example,
when one player leads a 7-point match 5 to 4, we would say that
the score is 2-away, 3-away, meaning that the player with 5 points
is 2 points away from victory and the player with 4 points is 3
away from victory. This is called a normalized score.
The advantage of normalized scoring, is that it is not necessary
to know the length of a match to be able to determine either player's
chances of winning. It allows us to talk about whole classes of
scores for which the same doubling strategy applies.
Normalized scores are sometimes written with minus signs to distinguish
them from scores expressed in terms of points won. For example,
the score is -7,-7 at the start of a seven-point match or when each
player has won 4 points in an 11-point match.
Post-Crawford Strategy
In the Crawford game itself, doubling strategy is very simple: neither
player is allowed to double, so there are no decisions to be made!
But after the Crawford game, the cube is in play again.
The trailing player should double in any post-Crawford game at
his earliest opportunity. After all, he has nothing to lose; if
he loses the game, he will lose the match anyway. By doubling early,
the trailing player guarantees that each game is worth two points
(maybe four if it is a gammon). This early double by the trailing
player in a post-Crawford game is called a mandatory double.
What about the leading player? When he is offered a mandatory double,
should he accept?
If the trailing player has an odd number of points to go (e.g.,
he is at 3-away or 5-away), then the leader should definitely accept
the mandatory double. Refusing would reduce the number of games
the trailing player needs to win the match.
If the trailing player has an even number of points to go (e.g.,
he is at 2-away or 4-away), then the leader has a choice. Suppose
the match score is 1-away, 2-away. When the trailing player offers
a mandatory double, the leader may either accept the double and
let the current game decide the match, or he may refuse and let
the next game decide the match. His best strategy, therefore, is
to accept if he is ahead in the current game and refuse if he is
behind. This option of being able to refuse a double when your opponent
is an even number of points from victory is called a free drop.
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