The Crawford Rule

The Crawford Rule is the single most consequential adaptation of the doubling cube for match play. Introduced by John R. Crawford in the late 1940s and codified in his and Oswald Jacoby's The Backgammon Book (1970), it governs the use of the cube in the critical late stage of a match — specifically, in the game in which one player first reaches a score one point short of the match target. The rule is universally adopted in international tournament backgammon.

This page covers the formal definition, the equity argument for why the rule exists, and the standard game theory of post-Crawford play — including the trailing player's near-mandatory aggressive doubling and the "free drop" windows where the leader has an unusual cube-receiving option.

1. The Formal Definition

In a match to $N$ points, the Crawford game is the first game in which either player reaches a score of $N - 1$ . In that single game:

The doubling cube is removed from play.
Neither player may offer a double.
The game is settled at its natural value: 1 point for a single win, 2 for a gammon, 3 for a backgammon.

In every subsequent game of the same match — the post-Crawford games — the cube returns to standard match-play rules. Either player may offer a double on their turn before rolling, subject to ownership.

The rule is invoked exactly once per match. Once the Crawford game ends, the rule does not re-apply, even if the score later returns to a configuration in which one player is again one point from winning.

2. Why the Rule Exists

The Crawford Rule was introduced to repair a specific game-theoretic defect in unmodified match play. Without it, a player trailing in a match would have a guaranteed-correct strategy that effectively eliminates the meaning of the cube at the critical moment.

Consider a 7-point match. Suppose the leader has reached 6-away/0 (i.e., the score is 0-6, with the leader needing 1 more point and the trailer needing 7). Without the Crawford Rule:

The trailer must win every remaining game to win the match. Cube efficiency for the trailer is infinitely high — every point won at value $n$ is worth as much as every point won at value $2n$ in terms of match progress (the trailer still needs to win the next game regardless).
The trailer should therefore double immediately, on the first legal opportunity of every game, before rolling.
The leader has a clear take/drop decision based purely on their winning probability from the post-double position. Standard cube actions, market gain, and timing nuance vanish.

In effect, every game from that score until match end becomes a sequence of automatic 2-point games. The cube has been collapsed to a no-information signal. The rule was specifically designed to remove this degeneracy and preserve the cube's normal informational and strategic content in the most critical game of the match.

By suspending the cube for one game — the Crawford game — the rule eliminates the trailer's automatic-double exploit at precisely the moment it would matter most: the game in which the leader can clinch.

3. The Equity Argument

The mathematical case for the rule rests on the asymmetry of equity at the score $N-1$ vs. $k$ -away, where $k > 1$ .

Let $M(a, b)$ denote the match equity of the player whose away-score is $a$ when the opponent is $b$ -away. From a Match Equity Table such as the Rockwell-Kazaross MET:

$M(1, 1) = 50\%, \quad M(1, 2) \approx 70\%, \quad M(1, 3) \approx 75\%, \quad M(1, 4) \approx 81\%$

The leader at 1-away gains relatively little match equity from winning at 2× value rather than 1× — but the trailer at $k$ -away with $k$ large gains disproportionately from doubling. By taking the cube out of play in the Crawford game itself, the rule prevents the trailer from extracting that disproportionate value at the worst possible moment for the cube's informational role.

4. Post-Crawford Play

Once the Crawford game ends, all subsequent games are post-Crawford games. The cube returns. The score is now (leader 1-away, trailer $k$ -away) for some $k \geq 2$ at the start of each post-Crawford game.

The trailing player's near-universal cube strategy in post-Crawford games is: double at the first legal opportunity, before rolling, on the very first turn.

The reason is the same asymmetry the Crawford Rule was designed to suppress in the Crawford game itself: the trailer needs to win every remaining game, so winning at value 2 is no worse than winning at value 1 for the trailer's match progress, but is exactly twice as valuable for the leader. Specifically:

The trailer trails by $k - 1$ points. To win, they need a sequence of $k - 1$ wins (or fewer gammons/backgammons). Each game they win is a step toward closing the gap. A win at cube value 1 cannot close the gap unless they then also win the next game.
A win at cube value 2 may close the gap entirely (if at $k = 2$ , i.e., 2-away vs 1-away) or reduce it by 2 points instead of 1.

The trailer therefore doubles immediately. The leader's decision is then a pure take/drop based on winning chances and the resulting MET re-anchoring.

5. The Free Drop

There is one subtlety in post-Crawford play that creates an unusual strategic option for the leader: the free drop.

At certain post-Crawford scores, dropping the trailer's first-turn double costs the leader zero match equity, and may even gain a tiny amount. The mechanics:

Suppose the score is leader 1-away, trailer 2-away (i.e., trailer needs 2 more points to win, leader needs 1).
The trailer doubles to 2 on the first turn.
If the leader drops, they lose 1 point. New score: leader 1-away, trailer 1-away. This is a 50% MET position (the "double match point" or DMP).
If the leader takes, they play a 2-point game from approximately even equity. Winning the 2-point game wins the match. Losing the 2-point game means the leader has lost 2 points → score is leader 1-away, trailer 0 (wins match).

In some positions the drop loses essentially zero equity — the leader is approximately as well off dropping to DMP as taking and playing the volatility-heavy 2-cube game. This is the free drop window. Skilled match players exploit it by dropping the trailer's first-turn double at certain score configurations even when the take is mathematically borderline, particularly when the position itself has structural disadvantages.

The free-drop window depends on the specific score and on gammon prices. Modern computer analysis identifies the exact MET cells where the free drop is correct, incorrect, or break-even. Memorising these cells is part of standard tournament preparation. See the match equity page for the underlying table.

6. When to Double in Backgammon — Practical Summary

Combining the rules above produces a practical doubling guide:

Score situation	Trailing player's cube action	Leading player's cube action
Both players ≥ 2-away, no Crawford yet	Standard cube theory: double when efficient, take to 25% (with recube vig adjustments).
Either player reaches $N-1$ (Crawford game)	No doubling. Cube is removed for one game.
Post-Crawford, trailer ≥ 2-away	Double on first turn, every game.
Post-Crawford, leader receiving double	Take in most positions. Apply the free drop at specific scores (notably 2-away).
Post-Crawford, trailer reaches DMP (1-away/1-away)	No double — both players need exactly 1 point regardless.

This compresses the entirety of match-play cube action in the late stage into a small number of decisions, with the principal nuance reserved for the free drop and the gammon-price adjustments at specific scores.

7. Historical Note

The rule's name preserves John R. Crawford's contribution to mid-20th-century backgammon theory. Crawford was one of the strongest competitive players of the era, and his collaboration with Oswald Jacoby on The Backgammon Book (1970) was the most influential single text on cube strategy until Bill Robertie's Advanced Backgammon (1991). The rule has been part of the standard tournament ruleset since at least the 1967 World Championship organised in the Bahamas by Prince Alexis Obolensky.