Equity in Backgammon

In backgammon lessons you often hear things like “If I pass I am 43% to win” or “My equity here is over 0.6, but I still don’t think it’s a double.” These concepts of “equity” are important building blocks for an understanding of backgammon, yet they remain a mystery to many players.

We are going to explain the basic concepts of match equity and game equity in backgammon.

Match Equity

a) Terminology

Match equity is often referred to as called match-winning chances or MWC%.

In an earlier backgammoned article we explained that the odds of winning a 3-point match at a score of 2-0 Crawford is 25%. All match equity is defined in terms of “points away.” After all, it doesn’t matter if the score is 2-0 to 3 or 24-22 in a 25pt match. There is no standard notation for noting the score. We will use notation of –a/-b to mean “I need a points, and my opponent needs b.” “C” at the end will mean “This is the Crawford game” and “pC” will mean “The Crawford game has already passed.”

b) Gammon Frequency

The frequency of Gammons obviously affects match equity. There is some dispute among experts as to the frequency of gammons. Kit Woolsey and Hal Heinrich did a study of match equities around 1990, and determined that the frequency of gammons in a game played to conclusion without the doubling cube was about 20%. More recent work tends to support a higher gammon frequency. For the balance of this article, we will assume a 20% gammon frequency. At the end, we will discuss what happens if a higher gammon frequency is assumed.

Of course, you can say “But matches are played with the doubling cube!” Very true. In some of the examples we will give – Crawford and post-Crawford games – the cube action is known. In other games, the reluctance of players to lose a doubled gammon will often lead them to drop a double with win chances well over 25%.

c) Match equity and Gammons in backgammon

Consider the fairly simple score of –2/-1C. Let’s assume that the cubeless gammon frequency is 20%. The following outcomes are possible:

I win a gammon this game, and the match, 10% of the time.
ii) I win this game without a gammon 40% of the time, and win the next game 50% of the time

Adding these together, I will win the match 10% + (40% * 50%), or 30% of the time. So my match equity at –2/-1C is 30%.

d) Cube Leverage

Suppose that you lead 1-0 in a match to 3. If the doubling cube is never turned by either side, your match-winning chances are as follows:

10% of the time you will win a gammon and win the match
40% of the time you will win a single game, and be 75% to win the match
40% of the time you will lose a single game, and be 50% to win the match
10% of the time you will lose a gammon game, and be 30% to win the match

10% + (40% * 75%) + (40% * 50%) + (10% * 30%) is 63%. However, actual collected data shows that the leading side wins only about 60%. Why?

Remember in an earlier article, we showed that the trailer at this score can double much earlier than usual. We also didn’t discuss gammons in that article.

Let’s consider the win chances that each side needs to win the game with the cube. The leader needs to reach 75% win chances – if he doubles, gammons for his side are irrelevant. If we hold to our estimate of 20% gammons, when the trailing side reaches 65.8% wins, he will be winning 13.2% gammons. Winning 65.8% with 13.2% gammons equates to 50% match-winning chances, the same as if the opponent drops a double.

If we take a simple model and say that each player starts with 50% chances to win the game, the leader has to go from 50% to 75% to win with the cube. The trailer has to go only from 50% to 65.8%. If we round this for simplicity to 65%, we find that the trailer has to go only 15% while the leader has to go 25%. The leader should win only 3 games for every 5 won by the trailer!

Simply knowing match equities does not accomplish much. Match equities are building blocks for further concepts. We have already discussed simple examples, of doubling at scores of 1-0 in 3-point matches. In a later article, we will give more complex examples.

Position Equity

The equity of a position is its mathematical expectation. To give a simple example, if you are 60% to win and 40% to lose, with no gammons possible, your equity if 0.60 – 0.40, or 0.20 points.

What if there are gammons and backgammons? These are easily reflected. Consider a position with the following odds:

Backgammon win: 2%
Gammon win: 30%
Simple win: 38%
Simple loss: 20%
Gammon loss: 9%
Backgammon loss: 1%

The equity of this position is:

.02 * 3 + .30 * 2 + .38 * 1 – .20 * 1 – .09 * 2 – .01 * 3

Or 0.63 points.

When a computer evaluation of a position is given, it usually appears as follows:

Backgammon wins
Gammon and backgammon wins
Total wins
Total losses
Gammon and backgammon losses
Backgammon losses

The above position would therefore appear as:

2.0% 32.0% 70.0% 30.0% 10.0% 1.0%

You can get the same result by adding the numbers on each side:

(2 + 32 + 70) – ( 30 + 10 + 1) = 63
Why Equity Matters

Equity matters for a number of reasons.

One is that it governs cube decisions. In a money game, if you cannot use the cube later on, you should take if your equity is better than –0.50 points, and drop if it is worse than –0.50 points. Allowing for the cube, the threshold is around –0.55 to –0.56 points. But if you don’t know the equity, you can’t assess whether to take or pass.

A second is that equity is the way decisions are evaluated. If one checker play gives you equity of +0.25 points and another +0.23, the first play is the better one. The same is true for cube decisions. If your equity before doubling is +0.65 points, and after doubling (allowing for giving up ownership of the cube) it is +0.55 points, you have forfeited 0.10 points per game by doubling.

Cubeless vs. Cubeful equity

The example we gave above is an example of cubeless equity. But let’s consider cubeful equity.

Take this position:

backgammonboard0018

This position will be won by Black about 70% of the time. It is not quite a double. If the cube were not in play, Black’s equity would be 0.70 – 0.30, or 0.40 points per game.

However, because of the cube, Black’s equity is in fact about 0.60 points. Black is more likely to get use out of the cube than White. One way of looking at this position is that if Black’s winning chances get up to 78%, he will give White a borderline decision between taking and passing. He only needs to increase his pure win chances by 8%, to a cubeless equity of 0.56 points, to have a cubeful equity of a full point.

Obviously, your cubeful equity in a position:

Equals the cubeless equity if the cube is dead for any reason.
Will be equal to or greater than the cubeless equity if you own the cube.
Will be equal to or less than your cubeless equity if your opponent owns the cube.
Is usually greater than your cubeless equity when you are the favorite in the game, and less than your cubeless equity when you are the underdog in the game.