Match Equity in Backgammon: Take Points and Cube Action

The Match Equity Table (MET)

The Match Equity Table (MET) is the central object of competitive backgammon mathematics. It maps every possible match score to the leader's probability of winning the match — and from those probabilities flow take points, double points, gammon prices, and post-Crawford strategy. Every serious player works with an MET, every modern bot uses one internally, and the modern reference is the Rockwell-Kazaross MET, compiled by Tom Rockwell and Neil Kazaross from large-scale rollout data and published in the GNU Backgammon distribution.

This page reproduces the pre-Crawford Rockwell-Kazaross MET, derives the formal mathematics of take points and gammon prices from it, presents the revised Janowski formula for those who prefer a closed-form approximation, and previews Neil's Numbers — the mental heuristics most strong tournament players use over the board. The dedicated Neil's Numbers page covers those heuristics in full.

Definitions and Conventions

The MET is indexed by the away-score of each player. A player who needs $a$ more points to win the match is at "$a$-away." Smaller $a$ means closer to winning. The MET cell $M(a, b)$ gives the match-winning chance (MWC) of the player who is $a$-away, given that the opponent is $b$-away, evaluated at the start of a new game (no current-game state).

Two structural identities follow from definition:

$M(a, a) = 50\% \quad \text{for all } a \geq 1$

$M(a, b) + M(b, a) = 100\% \quad \text{for all } a, b \geq 1$

The diagonal is 50% by symmetry: an equal-away score is by definition a balanced match. The second identity is a conservation law: someone wins the match, with probability 1.

The MET is conventionally evaluated pre-Crawford — that is, at scores where neither player has yet reached the Crawford game $(N-1)$. The 1-away column captures Crawford-then-post-Crawford sequences, and is computed by composing the Crawford-game probability (effectively a single-cube game) with subsequent post-Crawford games (where the trailer doubles immediately).

The Rockwell-Kazaross Pre-Crawford MET (Core 9×9)

The table below gives the match-winning chance (MWC), expressed as a percentage, for all combinations of away-scores from 1 to 9 — the range covering every score combination in matches up to 9 points (the most commonly played tournament length). The cell at row $a$, column $b$ is $M(a, b)$ — the MWC of the player who is $a$-away when the opponent is $b$-away.

The full Rockwell-Kazaross MET extends to 25×25 in its canonical published form. The 9×9 core reproduced here covers the bulk of the strategically relevant cube decisions; the full table is distributed with GNU Backgammon as met/rockwell-kazaross.xml and is the authoritative reference for cell-level precision. Compiled in 2010 by Tom Rockwell and Neil Kazaross: every score of a 15-point match was rolled out 38,880 times with GNU Backgammon 2-ply, and the table was then extrapolated to 25 points.

Source-of-truth note. The values above are the published Rockwell-Kazaross values rounded to one decimal; cells below the diagonal are 100% minus their mirror cell. For exact rollout-grade values consult the Rockwell-Kazaross file distributed with GNU Backgammon; serious cube-action analysis should be performed against the live engine.

How to read the table. A 5-point match starts at 5-away vs 5-away → $M(5, 5) = 50\%$. A player who wins the first single game (1 point) goes to 4-away vs 5-away → leader's MWC is $M(4, 5) = 57.7\%$. A player who wins the first game with a gammon at cube value 2 (so 4 points won) skips to 1-away vs 5-away → MWC is $M(1, 5) = 84.2\%$.

Cube Equity Derivations from the MET

The match equity table is the source of every cube decision in match play. The derivations below produce take points, double points, and gammon prices as direct functions of the MET cells.

The take point

Suppose the current cube is at value $C$, and Player A doubles to $2C$. From the receiver's perspective, after the double the score will resolve as follows:

Take the receiver's perspective: the receiver is $r$-away and the doubler is $o$-away.

• If receiver takes and wins: the receiver scores $2C$ points and moves to $(r - 2C, o)$ — receiver's MWC becomes $M(r - 2C, o)$.
• If receiver takes and loses: the doubler scores $2C$ points and the receiver is at $(r, o - 2C)$ — receiver's MWC becomes $M(r, o - 2C)$.
• If receiver drops: receiver loses $C$ points; new score is $(r, o - C)$; receiver's MWC becomes $M(r, o - C)$.

Setting expected match equity from taking equal to match equity from dropping, and letting $p$ be the receiver's winning probability in the current game:

$p \cdot M(r - 2C, o) + (1-p) \cdot M(r, o - 2C) = M(r, o - C)$

Solving for $p$ gives the take point:

$p_{\text{take}} = \frac{M(r, o - C) - M(r, o - 2C)}{M(r - 2C, o) - M(r, o - 2C)}$

For money play, all $M$ values reduce to plain $\pm 1$ wins and the formula collapses to the well-known $p_{\text{take}} = 25\%$ result. In match play, the take point varies dramatically with score — values from 8% to 35% are not unusual, and at certain post-Crawford scores the take point is meaningfully reduced by the asymmetry of the MET.

Gammon price

The gammon price at a given score is the marginal match equity gained from converting a plain win into a gammon win, normalised to the equity differential between winning and losing a single game. Formally, if a plain win at cube $C$ takes the player to $(r - C, o)$, a gammon win at cube $C$ takes them to $(r - 2C, o)$, and a plain loss takes them to $(r, o - C)$:

$\text{Gammon Price} = \frac{M(r - 2C, o) - M(r - C, o)}{M(r - C, o) - M(r, o - C)}$

The numerator is the equity gained by upgrading a single win to a gammon win. The denominator is the equity swing between winning and losing the single game — the natural scale against which the upgrade is normalised.

The gammon price varies with the score: 0.50 in money play, 0 at double match point, low for a leader whose plain win already ends the match, and high for a trailer whose gammon would.

For a 7-point match at 7-away vs 7-away, with a 1-cube:

$\text{Gammon Price (initial)} = \frac{M(5, 7) - M(6, 7)}{M(6, 7) - M(7, 6)} = \frac{62.0 - 56.0}{56.0 - 44.0} = \frac{6.0}{12.0} = 0.50$

(Note: $M(7, 6) = 100 - M(6, 7) = 100 - 56.0 = 44.0$ by the conservation identity.)

A gammon price of approximately 0.50 is typical for symmetric mid-match scores: 0.50 is exactly the money-game gammon price, where a gammon is worth twice a plain win. At other scores it moves far from that: for a leader who needs only the cube value to win, extra points are worthless (gammon price 0), while for a trailer whose gammon would land the match it climbs. With the trailer 4-away post-Crawford and the cube on 2, the gammon price approaches 1.00 — the gammon wins the match outright.

The double point

The double point for the doubler is the equity at which the value of turning the cube exceeds the value of holding it. Unlike the take point (a clean function of MET cells), the double point depends on cube efficiency — the likelihood that the position will reach a better cube turn later — and on gammon prices. Modern engines compute it by full rollout. For typical positions the double point is around 65–70% of game-winning equity, but the variation across positions and scores is wide.

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The Janowski Formulas

Memorising every cell of a 25×25 MET is impractical for over-the-board calculation. Rick Janowski published closed-form approximations of match equity and revised them in 2013 (Match Equity Formula Reviewed and Revised); the revised formulas below track the Rockwell-Kazaross MET closely across most scores.

Critically, two distinct formulas are needed — one for normal pre-Crawford scores and one for Crawford / post-Crawford scores. The cube-suppression in the Crawford game distorts the equity surface enough that a single closed-form approximation cannot accurately fit both regimes.

Pre-Crawford formula

For normal pre-Crawford scores where both players have away-scores ≥ 2 and the cube is in play:

$M = 0.5 + 0.87 \times \frac{D}{T + 6}$

where:

• $D$ = the leader's lead in points (trailer's away-score minus leader's away-score).
• $T$ = the trailer's away-score — the number of points the trailer still needs to win the match.

Crawford-specific formula

For Crawford-game scores (one player at 1-away), where the cube is removed:

$M = 0.525 + 0.57 \times \frac{D}{T + 2}$

with the same variable definitions. The shifted constants reflect the equity-surface distortion produced by the cube's suspension in the Crawford game.

Worked example (pre-Crawford)

Score: leader 4-away, trailer 8-away (pre-Crawford). $D = 8 - 4 = 4$. $T = 8$ (the trailer's away-score).

$M = 0.5 + 0.87 \times \frac{4}{8 + 6} = 0.5 + 0.87 \times \frac{4}{14} \approx 0.5 + 0.249 = 0.749$

Predicted leader's MWC: ≈74.9%.

Table cell $M(4, 8)$ from the 9×9 above: 74.6%.

The closed-form approximation matches the rollout-derived MET cell to within 0.3 percentage points — well inside any over-the-board cube decision tolerance. The pre-Crawford Janowski formula is a useful approximation across the bulk of the symmetric MET surface.

Common mistake to avoid

A frequent error in informal play is to use $T$ = leader's away-score instead of $T$ = trailer's away-score. The formula is calibrated with $T$ as the trailer's away — using leader's away in the example above (with $T = 4$ instead of $T = 8$) would yield an inflated and incorrect prediction.

In practice, strong players use the Janowski approximations for quick over-the-board mid-match equity estimation, and rely on memorised MET cells (Neil's Numbers, see below) for the score situations they actually encounter most often.

Neil's Numbers (Preview)

Because exact MET lookup is impractical at the board, Neil Kazaross developed a mental heuristic system that gives MWC values to within 1–2% for the most-common away-score combinations. The core entries are reproduced here; the dedicated Neil's Numbers page covers the full set with worked examples.

The basic table:

The three bolded entries — 5a/1a = 15%, 4a/2a = 33%, 2a/3a = 60% — are the canonical "starting set" cited by Kazaross in his published lectures on match equity heuristics. Memorising these (and a few derived rules for cube-turned scores) handles most of the meaningful match-equity decisions in tournament play.

Crawford and Post-Crawford in the MET

The Crawford game is captured in the MET as a special case in the 1-away column. When the leader is 1-away and the trailer is $b$-away ($b \geq 2$), the cube is removed for the Crawford game, so the trailer must win a single 1-point game (~50% probability for skill-balanced players). If they win, the new score is leader 1-away, trailer $(b - 1)$-away — and the post-Crawford rule kicks in.

Post-Crawford strategy: the trailer doubles to 2 on the first roll. If the leader drops, the score becomes 1-away vs $(b-1)$-away; if the leader takes and loses, 1-away vs $(b-2)$-away. The leader's take point $W$ follows from comparing the two:

$W \geq \frac{M(1, b - 1) - M(1, b - 2)}{1 - M(1, b - 2)}$

At $b = 2$ this gives 50%: a take only pays when the leader is the favourite in the game. At even away-scores the trailer gains nothing from the dropped point — $M(1, b-1) \approx M(1, b)$ — so the drop is free: the free drop windows at specific scores. The full mechanics are on the Crawford Rule page.

Modern Practice: Bot-Generated METs

The Rockwell-Kazaross MET was compiled by rollout. Modern variants now exist that have been re-tuned against XG2 and Wildbg rollouts, with small corrections (typically <0.5% per cell) reflecting improvements in engine evaluation. The major contemporary alternatives are:

• Snowie MET — older, slightly different at high-asymmetry cells.
• Kazaross-XG2 MET — Neil Kazaross's re-tuning against XG2 rollouts (2010s).
• Wildbg MET — open-source 2020s re-tuning, publicly auditable.

In tournament play, the choice between MET versions changes individual cube actions by at most a few percent and is not material to most decisions. The Rockwell-Kazaross table reproduced above remains the most widely cited reference.

Implementation on GamesGrid

GamesGrid's analysis runs on GNU Backgammon at full strength, and its match-play evaluations use the match equity table distributed with that engine. After a match, every cube decision is graded for the score it was taken at — so the player can see whether their cube action was correct for the score, not merely against a generic money-game baseline.

See Also

• Mathematics hub
• Neil's Numbers
• Crawford Rule
• Jacoby Rule — money-play counterpart.
• PR & ELO — how the MET feeds into Performance Rating.
• Glossary — formal definitions for MET, gammon price, take point, Janowski formula.
• How to Play Backgammon — the beginner's guide: the rules in plain language, in fifteen minutes.