The five point match

Why this?

This is walk-through of all scores in a five point match. We examine how take points and cube dynamic varies across scores, with sample positions to illustrate differences.

Positions types

To show differences due to gammons, we consider five different types of positions:

Last roll: Truly last roll, not even automatic recubes
No gammons: Zero gammon but possibly cube leverage
A little gammon: Early middle games with anchors
Some gammons: No-anchor middle games
Many gammons: Blitzes and back games

What do the different gammon situations look like? Below are the examples we will use for examination here, along with money game rollouts.





Player Winning Chances:   75,00% (G:0,00% B:0,00%)
Opponent Winning Chances: 25,00% (G:0,00% B:0,00%)
Cubeless Equities: No Double=+0,500, Double=+1,000
Cubeful Equities:
 No double:     +0,500 (-0,500)
 Double/Take:   +1,000
 Double/Pass:   +1,000
Best Cube action: Double / Take





Player Winning Chances:   77,93% (G:0,00% B:0,00%)
Opponent Winning Chances: 22,07% (G:0,00% B:0,00%)
Cubeless Equities: No Double=+0,559, Double=+1,117
Cubeful Equities:
 No double:     +0,838 (-0,144)
 Double/Take:   +0,983
 Double/Pass:   +1,000 (+0,017)
Best Cube action: Double / Take





Player Winning Chances:   73,58% (G:14,79% B:0,70%)
Opponent Winning Chances: 26,42% (G:4,53% B:0,16%)
Cubeless Equities: No Double=+0,581, Double=+1,159
Cubeful Equities:
 No double:     +0,923 (-0,052)
 Double/Take:   +0,975
 Double/Pass:   +1,000 (+0,025)
Best Cube action: Double / Take





Player Winning Chances:   70,79% (G:23,06% B:1,12%)
Opponent Winning Chances: 29,21% (G:6,10% B:0,23%)
Cubeless Equities: No Double=+0,588, Double=+1,189
Cubeful Equities:
 No double:     +0,871 (-0,101)
 Double/Take:   +0,972
 Double/Pass:   +1,000 (+0,028)
Best Cube action: Double / Take





Player Winning Chances:   67,23% (G:35,78% B:0,95%)
Opponent Winning Chances: 32,77% (G:7,68% B:0,40%)
Cubeless Equities: No Double=+0,616, Double=+1,262
Cubeful Equities:
 No double:     +0,843 (-0,152)
 Double/Take:   +0,995
 Double/Pass:   +1,000 (+0,005)
Best Cube action: Double / Take

We’ll use these as examples to consider how the score influences take points in a five-point match.

Overview

First, a high level overview of the scores we see in five-point matches.

Match equity table

For convenience, here’s the default match equity table used by XG, for five-point matches, rounded to three decimal points:

      1     2     3     4     5
1 0.500 0.677 0.751 0.814 0.842
2 0.323 0.500 0.599 0.669 0.744
3 0.249 0.401 0.500 0.572 0.648
4 0.186 0.331 0.428 0.500 0.577
5 0.158 0.256 0.352 0.423 0.500

Take point plots

Plotting take points under different situations, might help to build an intuition of the cube dynamics in the five-point match. We first calculate take point assuming the player will never redouble (except for automatic redoubles when we have nothing to lose). This is to see the effect of the score and gammons, for now ignoring the cube.

Several patterns stand out; some well-know, some might not be so well-know. A few observations:

The absolutely lowest take point you can face, is when leading 2-away, 5-away in a straight race. You can take here with as little as 17% winning chances.
Not surprisingly, if you get doubled at 4-away, 2-away, the automatic redouble to 4 also generates a low take point: 19%.
Leading 2-away, 4-away, get you the third-lowest race take point: 20%.
However, leading 2-away, 3-away, makes for a slightly higher take point (26%) than for money, since winning two points is also very valuable for the opponent here.
Same thing at 4-away, 3-away; we have the very highest dead-cube race take point: 35%. (Note that this of course changes when we introduce the cube, since re-cubes are very potent here. But be careful if this is a last-roll situation).
The innocent 5-away 4-away is also dangerous: Take point 30%.
At the other end of the spectrum we see (not surprisingly) very high take points when the opponent need 4 and has a clear gammon threat. Of course we have the notorious 2-away 4-away, where you need tremendous compensation in terms of winning chances, if you get doubled with a good risk of losing a gammon. But 3-away, 4-away and 5-away, 4-away, are almost as precarious.
In general, introducing gammons pushed the take points up, a good deal more than for money (compare the purple dots to the purple dotted line), because winning a gammon with the cube at two generally is pretty decisive.

Let’s focus on the effect of introducing the cube. If we look at a straight, medium length race, like in our reference position above, access to the cube in a in a money game generally allows us to take with about 22% rather than the 25% we would need in a true last roll situation.

At match scores, that effect varies considerably. Look at the plot below:

Of course, at scores where a double/take (and possibly an automatic re-cube) leaves the cube dead, there’s no difference between the dead-cube take point and the take point including the cube.

At other scores, in general, the cube benefits the trailer the most: At 4-away, 3-away, for example, cube access makes a massive difference, lowering the take point from 35% to 26%.

In contrast, at 3-away, 5-away, if the leader redoubles, the cube comes back at 8, making his lead meaningless, so being able to redouble is not so helpful. Take point goes from 21% cubeless to 19% with cube leverage. (Both lower than the money take point of 22%, however.)

Pre-crawford

5-away scores

5-away, 5-away

We give this score a pretty detailed treatment, then apply the same methodology to the rest of the pre-Crawford scores. The intention is to examine the anatomy of all score: How are they different from money in general?

Estimated live cube take points at 5-away, 5-away
Position type	Take point, match	Take point, money	Difference
Last roll	0.239	0.250	-0.011
Race	0.201	0.214	-0.014
A little gammon	0.262	0.261	0.001
Some gammons	0.297	0.291	0.006
Many gammons	0.344	0.335	0.010

This table compares take point for the five example positions for money and at the score 5-away, 5-away. When the difference between the match and the money take point is negative, it means the position is easier to take at the match score. When the difference is positive, we have less of a take at the score.

Let’s spot check with a few rollouts at the score: Our sample race position and the blitz position:





Money game, cubeful equities:
 No double:     +0,838 (-0,144)
 Double/Take:   +0,983
 Double/Pass:   +1,000 (+0,017)
 Best Cube action: Double / Take
5-away, 5-away, cubeful equities:
 No double:     +0,817 (-0,093)
 Double/Take:   +0,910
 Double/Pass:   +1,000 (+0,090)
 Best Cube action: Double / Take

The rollout confirms that this race is indeed a bigger take at the score, as we would expect from the slightly lower take point.





Money game, cubeful equities:
 No double:     +0,843 (-0,152)
 Double/Take:   +0,995
 Double/Pass:   +1,000 (+0,005)
Best Cube action: Double / Take
5-away, 5-away, cubeful equities:
 No double:     +0,856 (-0,144)
 Double/Take:   +1,096 (+0,096)
 Double/Pass:   +1,000
Best Cube action: Double / Pass

Again, the rollout shows that the blitz position, which was a marginal take for money, is a pass at the score, due to the slightly higher take point.

4-away, 5-away

Estimated live cube take points at 4-away, 5-away
Position type	Take point, match	Take point, money	Difference
Last roll	0.227	0.250	-0.023
Race	0.188	0.214	-0.026
A little gammon	0.265	0.261	0.004
Some gammons	0.305	0.291	0.014
Many gammons	0.359	0.335	0.025

3-away, 5-away

Estimated live cube take points at 3-away, 5-away
Position type	Take point, match	Take point, money	Difference
Last roll	0.209	0.250	-0.041
Race	0.187	0.214	-0.027
A little gammon	0.275	0.261	0.014
Some gammons	0.320	0.291	0.029
Many gammons	0.379	0.335	0.044

2-away, 5-away

Estimated live cube take points at 2-away, 5-away
Position type	Take point, match	Take point, money	Difference
Last roll	0.173	0.250	-0.077
Race	0.173	0.214	-0.041
A little gammon	0.279	0.261	0.018
Some gammons	0.332	0.291	0.041
Many gammons	0.400	0.335	0.066

5-away, 4-away

Estimated live cube take points at 5-away, 4-away
Position type	Take point, match	Take point, money	Difference
Last roll	0.304	0.250	0.054
Race	0.243	0.214	0.029
A little gammon	0.301	0.261	0.039
Some gammons	0.333	0.291	0.042
Many gammons	0.380	0.335	0.045

5-away, 3-away

Estimated live cube take points at 5-away, 3-away
Position type	Take point, match	Take point, money	Difference
Last roll	0.287	0.250	0.037
Race	0.223	0.214	0.009
A little gammon	0.250	0.261	-0.011
Some gammons	0.270	0.291	-0.021
Many gammons	0.301	0.335	-0.034

5-away, 2-away

Estimated live cube take points at 5-away, 2-away
Position type	Take point, match	Take point, money	Difference
Last roll	0.395	0.250	0.145
Race	0.234	0.214	0.019
A little gammon	0.216	0.261	-0.045
Some gammons	0.212	0.291	-0.079
Many gammons	0.210	0.335	-0.125

4-away scores

4-away, 4-away

Estimated live cube take points at 4-away, 4-away
Position type	Take point, match	Take point, money	Difference
Last roll	0.288	0.250	0.038
Race	0.224	0.214	0.010
A little gammon	0.299	0.261	0.038
Some gammons	0.338	0.291	0.047
Many gammons	0.392	0.335	0.057

3-away, 4-away

Estimated live cube take points at 3-away, 4-away
Position type	Take point, match	Take point, money	Difference
Last roll	0.240	0.250	-0.010
Race	0.211	0.214	-0.004
A little gammon	0.312	0.261	0.051
Some gammons	0.361	0.291	0.070
Many gammons	0.426	0.335	0.091

2-away, 4-away

Estimated live cube take points at 2-away, 4-away
Position type	Take point, match	Take point, money	Difference
Last roll	0.199	0.250	-0.051
Race	0.199	0.214	-0.015
A little gammon	0.333	0.261	0.072
Some gammons	0.396	0.291	0.105
Many gammons	0.477	0.335	0.142

4-away, 3-away

Estimated live cube take points at 4-away, 3-away
Position type	Take point, match	Take point, money	Difference
Last roll	0.352	0.250	0.102
Race	0.258	0.214	0.044
A little gammon	0.279	0.261	0.018
Some gammons	0.297	0.291	0.006
Many gammons	0.325	0.335	-0.010

3-away scores

3-away, 3-away

Estimated live cube take points at 3-away, 3-away
Position type	Take point, match	Take point, money	Difference
Last roll	0.302	0.250	0.052
Race	0.252	0.214	0.037
A little gammon	0.296	0.261	0.035
Some gammons	0.322	0.291	0.031
Many gammons	0.361	0.335	0.026

2-away, 3-away

Estimated live cube take points at 2-away, 3-away
Position type	Take point, match	Take point, money	Difference
Last roll	0.262	0.250	0.012
Race	0.262	0.214	0.048
A little gammon	0.326	0.261	0.065
Some gammons	0.361	0.291	0.070
Many gammons	0.411	0.335	0.076

3-away, 2-away

Estimated live cube take points at 3-away, 2-away
Position type	Take point, match	Take point, money	Difference
Last roll	0.368	0.250	0.118
Race	0.249	0.214	0.035
A little gammon	0.249	0.261	-0.012
Some gammons	0.249	0.291	-0.042
Many gammons	0.249	0.335	-0.085