Cubeful and cubeless equity

In our introduction to the doubling cube, we described, in terms of win percentages, why a player that’s trailing would be willing to accept a double. In a real backgammon game, both plays must consider gammons, which make the story more complicated (albeit way more interesting). For example, compared to a game with 70% win chances and no gammon chances, you would expect to win way more points in a game with 70% chances and another 70% gammon chances (that is, that every win is expected to be a gammon win). The first scenario is a typical D/T decision; the second is TG.

Instead of having to keep in mind all of the specifics of the various different situations, we use equity to describe the value of a position, which distills all of the relevant information about a position (i.e., your and your opponent’s win, loss, gammon and backgammon chances, and the cube position) into a single quantity. It corresponds to the value of your current position, where positive equity means you’re winning, negative means you’re losing, an equity of 1 means you’ve won the game, and an equity of 2 means you’ve won a gammon. The equation to calculate the cubeless equity, where we pretend there is no cube and the only way to win a game is to play it out until the end (i.e., neither player can ever cash out), is simply:

Equity = (Win % – Loss %) + 2 × (Gammon win %- Gammon loss %) + 3 × (Backgammon win % – Backgammon loss %)

In the language of cubeless equity, the “25% take point,” where the trailer breaks even when either taking or passing the cube, corresponds to an equity of -0.5: if your opponent chooses to double you when the game is valued is at -0.5, you have the choice between passing (equity = -1) or continuing the game at double the stakes (equity = -0.5 x 2 = -1).

If we want the equity to include the cube position (that is, centered vs. belonging to a player), and therefore also including the possibility of a player being doubled out, we need to use slightly more complicated equations for equity, called the Janowski formulas. These formulas consider, among other things, the fact that only the cube owner is capable of doubling out their opponent, and so does not need to play the game out to 100%.

Let’s consider a more concrete position to discuss these principles in more detail:

Cubeless equity: 0.430

Decision	Equity
No double	0.391
Double / Take	0.654
Double / Pass	1.000

In the position above, you have been in a stand-off with Gary, with both of you waiting for doubles to advance your anchors past each other and begin bearing off your checkers. In fact, Gary is a bit ahead in the race, but he has started crunching his board and was finally forced to break his anchor by rolling a 61 and playing a forced 18/12 (7/14 from our perspective). You now consider doubling, because a hit will nearly guarantee a win, and give you some small gammon chances. The analysis says this position is a D/T. How can we determine both sides of this cube decision from the equities listed in the table?

First, we note that our cubeless equity is 0.430, which means Gary’s equity is -0.430. If doubled, his equity will go to -0.860, which is still more than passing (-1.000). Therefore, if offered the cube, he should take. He isn’t thrilled to be taking, but doing so will maximize his equity. But should you double?

For our side of the decision, we must look at the cubeful equities to gain a clearer picture for the different outcomes. The equity for either doubling outcome — Gary taking (0.654) or passing (1.000) — is larger than the equity if we don’t double (0.391). Therefore, we should double. Gary’s decision can also be described in the language of cubeful equities — he would rather we win fewer points, so he should take (0.654) instead of pass (1.000).

Further reading

• A highly recommended classic article by Peter Bell that discusses the doubling window and equity.
• The original article by Rick Janowski deriving how to calculate cubeful equities.
• An article by Lasse Madsen with more detail explaining these formulas and how the bots use them to calculate cubeful equities.

Next lesson: Market losers