This article originally appeared in the May 2000 issue of GammOnLine. Thank you to Kit Woolsey for his kind permission to reproduce it here.

## Post-Crawford Doubling

By Matt Reklaitis
As most experienced players already know, proper use of the doubling cube involves utilizing the concept of market losing sequences. It is not incorrect to roll if there aren't any market losing sequences—sequences after which your opponent will not have a take on your next opportunity to offer the cube. This rule of logic brings about some interesting paradoxes when applied to tournament play.

The most common of these is the proper cube strategy at the 2-away, 2-away score. Assuming perfect play, it is an easy exercise to show that it not incorrect to double at the first opportunity—even if no market losing sequences exist. So what's the proper strategy? Double immediately or wait for at least one market loser? Well, it's not incorrect to do either. (I.e., this problem has more than one correct answer). Many players are quick to point out that the assumption of perfect play is not valid in practice, and so they adopt doubling strategies which present their opponents with the opportunity to make errors. (Of course, these players risk making errors themselves.) Enough articles have previously been written concerning the 2-away, 2-away score that it does not need to be addressed further here. Instead, we are going to look at some lesser discussed situations which you wouldn't want to have to face unprepared.

Consider this situation:

### Blue leads White, 1-away, 3-away, post-crawford

Blue: 4-1: 24/23, 13/9

Ok, its post crawford, obviously White, who is trailing in the match, should automatically double. Or should he? Just as with the 2-away, 2-away score, if you assume perfect play it is not incorrect to double immediately—and almost all of us have been trained to do this. Since Blue doesn't have a free drop available, the take is trivial. But what about waiting for a market losing sequence? If there aren't any market losers, it is not incorrect to roll. At this score, it doesn't really matter much to White if he wins 1 point or 2, so by rolling he may be able to present Blue with the opportunity to make an error later on. The value of the cube to White is when he scores a gammon to win the match. So, does White have any market losing sequences? Lets say that White likes the idea of suckering his opponent and rolls:

White: 4-4: 24/16*, 6/2*(2)

Blue: Dances

 151180 ``` ``` White 25 point matchBlue 4

White: Doubles

Who's the fool now? Did White make an incredible error by rolling? Or does Blue still have a take and will he be blundering by passing into (essentially) double match point. Let's see..

```ACTION                       SCORE (Blue,WHITE)       MATCH EQUITY

Blue Drops                   1-away, 2-away            50% (ignoring free drop equity)
Blue Takes and Wins          Blue wins                 100%
Blue Takes and Loses Single  1-away, 1-away            50%
Blue Takes and Loses Gammon  White wins                0%
```
There are 3 variables to consider:
W, the amount Blue wins
L, the amount Blue loses single
G, the amount Blue loses a gammon

By definition, W + L + G = 1

Blue's match equity if he takes is the amount he wins (W) plus 50% of the amount he loses single (L):

Blue can take if this equity is greater than the 50% equity he would
have by dropping:

E-take > .5
W + .5*(1 − W − G) > .5
W + .5 − .5*W − .5*G > .5
.5W > .5G
W > G

So Blue can take if he wins more than he gets gammoned. Back to the above position...can Blue take? JF level 7 evaluates the position as follows:

Blue wins: 33%
White wins: 67% (41% single, 26% gammons)

Since Blue wins more than he gets gammoned, it's still a clear take. Even with this favorable sequence, White has not lost his market. And many good players would drop this position, justifying White's earlier decision to roll.

Continuing the exercise, consider this situation:

### Blue leads White, 1-away, 5-away, post-crawford

White: 4-2: 8/4, 6/4

Blue: 4-3: 24/20 13/10

White decides to wait and rolls his best:

White: 3-3: 8/5*(2), 6/3(2)

Blue: Dances

 149165 ``` ``` White 27 point matchBlue 6

Now White doubles. Does Blue have a take?

This time I will give JF's level 7 evaluation first:

Blue wins: 25%
White wins: 75% (40% single, 35% gammons)

From the analysis above, we know that this would be a drop for Blue if the score was 1-away, 3-away post crawford. What about the current score:

```ACTION                        SCORE (Blue,WHITE)        MATCH EQUITY

Blue Drops                   1-away, 4-away            70% (ignoring free drop equity)
Blue Takes and Wins          Blue wins                 100%
Blue Takes and Loses Single  1-away, 3-away            70%
Blue Takes and Loses Gammon  1-away, 1-away            50%
```
Using the same variables as before, we can calculate Blue's match equity if he takes:
```E-take  =  W + .5*G + .7*L
=  W + .5*G + .7*(1 - G - W)
= .3*W - .2*G + .7
```
Blue can take if this equity is greater than the 70% equity he would have by dropping:
```E-take > .7
.3*W - .2*G + .7 > .7
W > 2/3 * G
```
At this score, Blue can take if he wins at least a fraction of the amount he gets gammoned! Plugging in JF's numbers we see that Blue still has a slim take, since his 25% wins is more than the 23% factored gammons (35% actual gammons * 2/3 = 23% factored gammons). Again, White has not lost his market and gains if Blue drops.

The formula above can be generalized: In all odd-away, post crawford positions, the leader can take trailers cube if he wins at least a fraction F of the amount he gets gammoned. F is easy to calculate:

F = (D − TG) / 100 − D,

where D is the match equity if leader drops and TG is the match equity if leader takes and gets gammoned. (Example: At 1-away, 5-away post-crawford, D = 70% and TG = 50%, so F = 2/3 )

```POST CRAWFORD  SCORE                 GAMMON FRACTION (F)

-1, -3                               1
-1, -5                               2/3
-1, -7                               2/3
-1, -9                               4/5
-1, -11                              2/3
-1, -13                              1/2
```
But buyer beware! Remember, the example positions were chosen very carefully. Doubling immediately in these situations is not incorrect. If instead you decide to adopt the post-crawford cube waiting strategy, you really need to be aware of your market losing sequences. Otherwise, the small gains you make from this strategy will quickly be wiped out when you roll that joker only to see your opponent dance and then correctly pass your sucker cube.