Ignore the cube in the picture. Suppose this position occurs in a cubeless game, perhaps in a match with score 1-away to 1-away. Black is on the roll but has not yet tossed the dice. The game will be decided as a bearoff race, the first to remove all fifteen checkers will win.

In the posts on this blog, several reference cube positions have appeared. This position, however, is a benchmark position, but not an example of the more restrictive category of reference positions. What is remarkable about this benchmark position? Black and White are equally likely to win this game — fifty-fifty to a high accuracy. Black is on the roll but White has a five pip lead in the bearoff race. And, of course, there are no gammons.

As a benchmark position the situation is a dead heat, which in itself is noteworthy. Playing either White or Black is equally as good. Future winning prospects are a dead heat. And yet this benchmark position will offer some insights into the doubling cube in both cash games and short matches.

At the start of this classic bearoff race, White is ahead by a half roll but Black has the dice tinkling in his dice cup. Assume the position is from a three-point match at double match point (DMP). The cube is dead. Gammon losses don’t matter. Just win the game. At all costs.

Let’s get specific. White and Black’s situation is a hidden gem. Black trails the count by five pips but fortunately Black is on the roll. Yet, with a dead cube, this position is absolutely a dead heat. Black wins or White wins. No gammons to be had. Fifty-fifty. Exactly. Well within a penny. Equity up for grabs is 0.0003 after 41472 rollout cases.

Now transport this position around the local backgammon circuit — cash game here and weekly matches there — and the meaning of the doubling cube is teased out.

First plunk this position into a cash game situation. Doubling cube, centered, live.

Merely introduce the doubling cube centered between the two players and the equity is no longer zero, as was the case at DMP. Although Black is on the roll, White actual has the positive equity — just under two cents on the dollar. Equity for Black is -0.0171 (or 17mp, seventeen milli-points) after 41472 rollout cases.

The difference in pips is unchanged at five, the position is the same, and no gammon win is possible, and yet Black becomes an indisputable underdog. All by the creation of the doubling cube in the rules of backgammon.

What gives?

The bearoff of all checkers is not the same for White as it is for Black. When Black wins the bearoff, Black enjoys an extra dice roll. When White wins the bearoff, White must accomplish this feat using the same number of dice rolls as Black.

In addition, the cube’s latent power to alter the definition of the finish line and cash the game operates differently for White and Black. White, starting with a five pip lead, may reach the take point earlier when White has good luck and Black has bad luck. But if both White and Black have equivalent luck during the bearoff, Black can often forgo Black’s very last bearoff roll and claim the game with the cube. For White to accomplish this eleventh hour capitulation, White needs to get lucky or have Black get quite unlucky, likely both. It is Black who begins on the roll. But, of course, White has a shorter distance to travel. An asymmetry in opportunity, yet not necessarily unfair.

Now consider the situation where Black owns the cube in a cash game. See position below:

Give Black ownership of the cube and Black becomes a favourite. In equity Black has improved by nearly sixteen cents on the dollar, partly from the right to claim the game whenever White’s situation deteriorates below the take point, and Black halts the game by a recube. That bullying, plus twice the stake when White takes an earlier recube, both combine to positive 15.7 cents in equity for Black, according to 41472 rollouts cases.

Now consider the situation where White owns the cube in a cash game. See position below:

When the cube is on the other side, White owns the cube. The race still favours White by five pips. Black is still on roll. Then White has positive 17.2 cents in equity, not Black.

The swing is ferocious between cube ownership. Black to White spans over a third of a dollar. To be exact: from equity 0.0003 dead cube to (unowned, centered, owned) is ( -0.1722, -0.0171, +0.1565 ) for Black equity in cash games after 41472 rollout cases.

Although the introduction of the doubling cube into this balanced position is itself relatively small — a couple percent when centered — any future double has equity invested in cube ownership. Note: None of these situations is a cube now in this dead heat position.

Return to the centered cube position.

Now consider various scores in a three-point match.

This position post Crawford acts like the original dead heat of the cubeless double match point. A dead heat to win the match.

Before the Crawford game, such as in the first game of the match, a centered cube with Black on the roll has White’s match equity positive at 0.0248 (negative for Black), and 25mp is similar to the centered cash game 17mp but slightly better for White. Thus the score 3-away 3-away (Game #1 of the match) is similar to a cash game during a bearoff race, but a shade better for the leader in pip count. (Caveat: most tournament players know the definition of equity between cash game and match is not quite identical.)

Suppose White already doubled Black earlier during the first game of a three-point match. Black’s equity in a later parry redouble situation will heavily favour Black, with positive +0.1881 match equity, due to cube ownership. By contrast, if Black doubled White earlier in the game, then Black’s match equity at the start of this balanced bearoff race is -0.2063 (White a big favourite). Advice? Don’t be frivolous with racing doubles at the score of 3-away 3-away. Again, notice no one actually doubles anyone yet in this balanced bearoff position. The equity advantage is from cube ownership (or access) and possible future cube actions.

Suppose in a three-point match Black draws first blood in Game #1 and leads the match by a score of one-zero in Game #2 of the short match. Imagine this position occurs in Game #2. With a centered cube in the second game, and Black leading the match, White’s equity here grows to +0.0491 (negative for Black).

Suppose instead in Game #2 White leads with the score one-zero. Black trails in the match and is thinking about a catchup cube. Now Black is already a small favourite to win this game; the match equity is +0.0152 for Black (negative for White).

Let’s return finally to this bearoff position with a centered cube in a cash game. One study technique observes rollouts between bot versus bot and concludes that the doubling cube usually gets turned. Therefore, plenty of market losers will soon appear during the bearoff. These market losers are typical: somebody rolls low dice then someone rolls high doubles like 66s, 55s, 44s, and so on. Or someone rolls high doubles then somebody rolls low dice. Asymmetry in rolls, yet closer to home.

Conclusions? Any thoughts? Maybe leave a brief reply.