Match Equity: Tables & Calculations

What Is Match Equity?

Match equity is the probability of winning a match from a given score, assuming both players are of equal ability. It is the foundation for all doubling decisions in match play.

Using Match Equity for Doubling Decisions

When you are playing a backgammon match, whether or not to double, and whether or not to take, depends on many factors:

• The odds of winning the game if you don't double
• The odds of losing the game if you don't double
• The odds of winning a gammon or backgammon
• Whether your opponent will take or drop if you double
• Your odds of winning the match if you win or lose 1, 2, 4, or more points

Estimating the odds of all but the last item is guesswork - and the more experience you have, the better your guesses will be. Experts can often estimate their winning odds within one or two percentage points.

The last item, however, is not something expert players estimate - it is something they know. They know the odds of winning or losing at every score because they have memorized the match equity tables and take points for every given score.

The Take/Drop Decision

Assume you are in a position where you believe you will win the game 30% of the time. Your opponent has given you the cube. You should drop if your odds of winning the match after dropping are greater than your odds of winning by taking. You should take if dropping gives worse match-winning odds than taking. You can't know which is better unless you know your take point.

A Practical Example

Consider a match to 5, Black leads 3-1, cube in the middle, Black on roll. Black is winning and might even win a gammon, so maybe he should play on. But if he doesn't double and wins just one point, he doesn't win the match. A tough decision.

Using Snowie, we find that Black should not double. And if doubled, White should take. The reason: White can redouble on the next roll, and whoever wins plays for the match. White has a better chance of winning the match by taking this cube and redoubling than by dropping.

If White drops, they are only 18% to win the match. But White wins this game 23% of the time - so taking gains 5% over dropping.

Approximation Formulas

You don't always need a computer. Three practical formulas can estimate match equity in your head. In each formula, D is the difference in scores (leader's distance minus trailer's distance), and T is the trailer's distance from winning.

Janowski's Formula

Developed by Rick Janowski:

Leader's equity = 50 + (85 × D) / (T + 6)

Example: 3-away vs. 8-away. D = 5, T = 8. Equity = 50 + (85 × 5) / (8 + 6) = 50 + 425/14 = 80%.

The challenge: mental arithmetic with numbers like 425/14 can be difficult over the board.

Turner's Formula

Easier to calculate mentally:

Leader's equity = 50 + (24/T + 3) × D

Example: 3-away vs. 8-away. D = 5, T = 8. Equity = 50 + (24/8 + 3) × 5 = 50 + (3 + 3) × 5 = 80%.

Works in whole numbers at 3, 4, 6, 8, and 12-away. Accurate within 1% for T up to 12.

Neil's Numbers

Developed by Neil Kazaross - learn a short lookup table instead of calculating:

Example: 3-away vs. 8-away. D = 5, multiplier = 6. Equity = (5 × 6) + 50 = 80%.

Important Notes

1. None of these formulas work when L = 1 (the Crawford game). You'll just have to learn those equities separately.
2. Don't mix formulas within one calculation.
3. All three underestimate slightly when L = 2 and T is between 5-9. Consider adding 2%.
4. For T of 13 or more, all formulas lose accuracy.

Take Point Calculation

The take point formula tells you the minimum winning probability needed to accept a double:

Take point = risk / (risk + gain)

Where risk is the equity lost by taking and losing (compared to dropping), and gain is the equity gained by taking and winning (compared to dropping).

Worked Example

You're losing 13-11 in a 15-point match. Your opponent doubles to 2. What should you do?

• If you drop: score is 14-11, your equity is 17%
• If you take and win: score is 13-13, your equity is 50%
• If you take and lose: you lose the match, equity is 0%

Risk = 17% (what you lose by taking vs. dropping). Gain = 33% (what you gain by taking and winning vs. dropping).

Take point = 17 / (17 + 33) = 34%

Quick count reveals your opponent bears off unless rolling a 1, so you win 11 out of 36 games = 31%. Since 31% < 34%, you should drop at this match score (even though you'd take in a money game at 25%).

It Takes Time

Does all this sound complicated? It did to me, for many years, and it took a lot of study, patience, memorization of tables, and coaching before I fully understood it - and I still have problems with these calculations from time to time. But I do not feel lonely, as everyone has some difficulty with this.

Until you become a top player, you do not have to calculate these numbers exactly - you can estimate, as we all do to some extent. But you do need to understand the concept in order to estimate well.

Equity formulas by Rick Janowski, Neil Kazaross, and the author. Match equity table by Kit Woolsey. Practical examples using Snowie.