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# Doubling Cube

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Nov 16, 1993, 7:24:35 PM11/16/93
to
I am an avid Acey - Duecy player. I recently found out the game I always
called Backgammon is Acey - Deucy. I am learning to play "the proper
way", but some thing bothers me. How does one use the doubling cube,
when are the appropriate times to do so, and what is the actual purpose
of the cube. Thanx in advance for any and all comments. OOOHHHH also is
there a acey - deucy line like FIBS.

### Michael J Zehr

Nov 18, 1993, 9:52:24 AM11/18/93
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How to use the cube?

The cube lets one play offer to double the stakes of the game.
Initially either player can offer to double, but after the initial
double the player who most recently accepted a double "owns" the cube
and only that player may offer a double.

After one player doubles, the other may accept the double, and play
continues at the doubled stakes, or the double can be declined, in which
case the declining player resigns the game at the current stakes.

When to use the cube?

This is a very complicated subject. Entire books have been written
about the topic. Here's my attempt at an introduction though.

[First, a brief glossary. Equity is the expected winnings from a game.
If a player has a 65% chance of winning the game (worth one point) and a
35% chance of losing the game (worth negative one points), the equity is
.65 - .35 or .30. The other player's equity would be -.30.]

Suppose it's the last roll of the game and you have the opportunity to
double. If your equity is positive (or your chance of winning greater
than 50%) you should double. But should your opponent accept?

Well, the opponent can drop for an equity of -1, or accept for an equity
of twice the previous equity. So if before the double the equity of the
person being doubled was greater than -.5, s/he should accept. This
represents a 25% chance of winning.

This is the key to roughly determining whether you should take or drop a
cube. Look at the 36 possible rolls by the doubler, consider how many
of those 36 variations you think you can win, and accept if there are 9
or more of them.

So, why don't people normally double as soon as they get a slight edge
in the game? Because what I said above is only true on the LASST roll
of the game. Remember that by doubling you give up the opportunity to
double later, thus you give up a bit of value. Here's an example:

1 2 3 4
O
O

X
X
1 2 3 4
X on roll, cube action?

X wins outright 26/36 rolls (any roll that doesn't have a 1 plus 11) and
has a big double.

Should O take? O wins when X misses and O rolls one of 26 rolls, so
it's 10/36 * 26/30 or 20%. Looks like a drop, right? But if O takes, O
has a chance of redoubling, and in fact should redouble if X doesn't win
immediately. This gives O enough equity to accept the cube. (I'll let
you work out the math yourself.)

In this case, the value of being able to redouble is worth around 5%
equity to O, changing the situation from a big drop to a close take.

In general you can count on cube ownership being worth around 5%,
possibly more.

Some other things to keep in mind about doubling.

A "market loser" is a roll (more generally a roll by you and a roll by
your opponent) after which he will drop a double (provided on this turn
your opponent will take the double). If you have no market losers, then
it cannot be correct to double. Even after your best sequence your
opponent will still take, so there's no point in doubling.

(It's hard to have zero market losers. Most positions would have a
variation in which one side rolls a double, hitting and making an inner
board point, and the other side fans, leading to a double/drop
position.)

In a match situation, the value of each game to the players varies with
the score, and this changes the basic 25% drop value. The decision then
depends on what's known as match equities. See Kit Woolsey's "How to
Play Tournament Backgammon" for a good introduction to match play.
(It's available from Gammon Press -- see the FAQ for more information.)

What is the purpose of the cube?
"purpose" sounds philosophical. It's affect on the game is to make
things more interesting, speed play, add a new level of complexity, etc.

As I said, entire books are devoted to cube handling. It's probably the
most important aspect of the game. The gain or loss from perfect or
poor cube handling is far greater than gains or losses from checker
play. I hope this provides enough information to get you sstarted and
give you an idea on what else to study.

-michael j zehr

### Michael A Urban

Nov 18, 1993, 7:56:37 PM11/18/93
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Thanks Michael Zehr for your excellent synopsis on doubling!
I particularly liked your example. I have found these two
positions to be quite illustrative as well.

#1

1 2 3 4 5 6
O
O
X owns the cube, on roll, cube action?

X X
1 2 3 4 5 6

#2

1 2 3 4 5 6
O O

X owns the cube, on roll, cube action?

X X
1 2 3 4 5 6

In position #1 X has 19/36 rolls that win and 17/36 that lose.
To maximize the equity, X doubles and O accepts. To get,

(19-17)/36 * 2 = 0.11 equity.

Interestingly O's situation in position #2 is worse than #1, yet in
this case X should NOT double. Plainly put, by doubling X
relinquishes control of the cube, allows O to double back on those
17/36 rolls that X fails to bear off, X drops and so the equity is the
same as above, 0.11. If X does not double immediately, O must roll,
and X picks up wins on the 7/36 that O fails to bear off. This
amounts to more than twice the equity of when X doubles immediately.
The math:

19/36 - 17/36 * (29-7)/36 = 0.24 equity.

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