Backgammon match equity table
A match equity table gives each player's chance of winning a backgammon match from any score, written as how many points each side still needs. It is how a match player turns a score into a cube decision: a double is right or wrong by what it does to these chances, not by the points on the cube. This is the table the HedgeHog engine prices every match decision with, the Rockwell-Kazaross table, with its post-Crawford values.
Your chances at a score
Your chance of winning the match: 33.1%
- Doubled to 2, you need 37.2% winning chances to take.
- Doubling to 2, your opponent needs 19.8% to take.
Take points here ignore gammons and treat the cube as dead once it is taken, which is how they are usually quoted. The engine's own take point counts gammons and what owning the cube is worth, so an analysis can show a different number at the same score.
The table
Find the points you need down the side and the points your opponent needs across the top. At 4-away, 2-away you win the match 33.1% of the time, and your opponent 66.9%: the two cells mirror each other across the diagonal. A player 1 away faces the Crawford game, so the first column is the trailer's chances in it.
| Away | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1 | 50.0 | 67.7 | 75.1 | 81.4 | 84.2 | 88.7 | 90.7 | 93.2 | 94.4 | 95.9 | 96.6 | 97.6 | 98.0 | 98.5 | 98.8 |
| 2 | 32.3 | 50.0 | 59.9 | 66.9 | 74.3 | 79.9 | 84.2 | 87.5 | 90.2 | 92.3 | 93.9 | 95.2 | 96.2 | 97.1 | 97.7 |
| 3 | 24.9 | 40.1 | 50.0 | 57.1 | 64.8 | 71.2 | 76.3 | 80.5 | 84.0 | 87.1 | 89.4 | 91.5 | 93.1 | 94.4 | 95.5 |
| 4 | 18.6 | 33.1 | 42.9 | 50.0 | 57.7 | 64.3 | 70.0 | 74.6 | 78.8 | 82.4 | 85.4 | 87.9 | 90.0 | 91.8 | 93.3 |
| 5 | 15.8 | 25.7 | 35.2 | 42.3 | 50.0 | 56.7 | 62.7 | 67.8 | 72.6 | 76.7 | 80.3 | 83.4 | 86.0 | 88.3 | 90.2 |
| 6 | 11.3 | 20.1 | 28.8 | 35.7 | 43.3 | 50.0 | 56.3 | 61.7 | 66.8 | 71.3 | 75.3 | 78.9 | 82.0 | 84.7 | 87.0 |
| 7 | 9.3 | 15.8 | 23.7 | 30.0 | 37.3 | 43.7 | 50.0 | 55.5 | 60.9 | 65.6 | 70.0 | 73.9 | 77.4 | 80.5 | 83.3 |
| 8 | 6.8 | 12.5 | 19.5 | 25.4 | 32.2 | 38.3 | 44.5 | 50.0 | 55.4 | 60.4 | 65.0 | 69.1 | 72.9 | 76.4 | 79.4 |
| 9 | 5.6 | 9.8 | 16.0 | 21.2 | 27.4 | 33.2 | 39.1 | 44.6 | 50.0 | 55.0 | 59.8 | 64.1 | 68.2 | 71.9 | 75.3 |
| 10 | 4.1 | 7.7 | 12.9 | 17.6 | 23.3 | 28.7 | 34.4 | 39.6 | 45.0 | 50.0 | 54.9 | 59.3 | 63.6 | 67.5 | 71.1 |
| 11 | 3.4 | 6.1 | 10.6 | 14.6 | 19.7 | 24.7 | 30.0 | 35.0 | 40.2 | 45.1 | 50.0 | 54.6 | 58.9 | 63.0 | 66.8 |
| 12 | 2.4 | 4.8 | 8.5 | 12.1 | 16.6 | 21.1 | 26.1 | 30.9 | 35.9 | 40.7 | 45.4 | 50.0 | 54.4 | 58.6 | 62.5 |
| 13 | 2.0 | 3.8 | 6.9 | 10.0 | 14.0 | 18.0 | 22.6 | 27.1 | 31.8 | 36.4 | 41.1 | 45.6 | 50.0 | 54.2 | 58.3 |
| 14 | 1.5 | 2.9 | 5.6 | 8.2 | 11.7 | 15.3 | 19.5 | 23.6 | 28.1 | 32.5 | 37.0 | 41.4 | 45.8 | 50.0 | 54.1 |
| 15 | 1.2 | 2.3 | 4.5 | 6.7 | 9.8 | 13.0 | 16.7 | 20.6 | 24.7 | 28.9 | 33.2 | 37.5 | 41.7 | 45.9 | 50.0 |
After the Crawford game
Once the Crawford game is over the cube is live again, and the trailer doubles at the first chance: the leader's match is on the line either way. That makes the trailer's chances higher than the Crawford column above. At 2-away the trailer wins 48.8%, not 50%: passing the trailer's double leaves both players 1 away, a coin flip, so the leader can pass for free any game that starts badly for them.
| Trailer away | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Trailer wins | 50.0 | 48.8 | 32.3 | 31.0 | 19.0 | 18.1 | 11.6 | 10.9 | 7.0 | 6.5 | 4.2 | 3.9 | 2.5 | 2.3 | 1.5 |
Using it
In a money game a take needs 25% winning chances, whatever the stake. In a match it depends on the score. Leading 2-away, 4-away, you can take a first double with only 19.8%: once the cube is on 2 your next win ends the match. Trailing at the same score you need 37.2%, because a pass still leaves you in the match and a lost 2-cube does not. The calculator above does that arithmetic for any score. What match equity and a take point mean is in the glossary, and how the app shows a cube decision at a match score is in The cube.
Analyse a match and HedgeHog prices every cube decision in it from this table, gammons and cube ownership included, and marks the ones you got wrong.