Take-Points in Money Games

by Rick Janowski

Guidance on doubling strategy in backgammon is provided by the following two theoretical models:

1. Dead-Cube Model – the classical model which makes no allowance for cube ownership.

2. Live-Cube Model – the continuous model which assumes maximum possible cube ownership value.

The former generally overestimates take-points and underestimates doubling-points (25% and 50% respectively, assuming no gammons). Conversely, the latter model underestimates takepoints and overestimates doubling-points (20% and 80% respectively assuming no gammons). When considered together, however, they provide an envelope in which correct cube action

decisions are to be found.

Dead-Cube Model

The owner of the cube is not afforded any additional benefits by it – he can neither double out his opponent nor raise the stakes at an opportune time. Effectively, the game is played out to its conclusion cubeless (but at the stake raised by the previous double).

Consequently, takepoints can be readily established from the risk-reward ratio. Assume a double occurs in a game where, if played to conclusion, both players will win a mixture of single-games, gammons and backgammons. The effects of gammons and backgammons can be dealt with by introducing the following two variables for the player making the cube action decision (in this case, the player doubled):

 
 

W = Average cubeless value of games ultimately won

L = Average cubeless value of games ultimately lost

Consequently, a take would risk 2L - 1 points to gain 2W +1 points. The minimum cubeless probability for a correct take (TP) is therefore:

TP =
(2L - 1)
(2W + 2L)
=
(L - 0.5)
(W + L)
…equation (1)

This formula is also applicable when the data considered represents effective game winning chances.

Live-Cube Model

The owner of the cube is guaranteed to use the cube with optimal efficiency if he redoubles, at which point his opponent will have an optional pass/take. All subsequent redoubles by either of the two players are similarly optimal. There are, in fact, an infinite number of different possible live cube models identifiable by the following two variable factors:

1. The number of possible subsequent optimal redoubles. This can vary between unity and infinity. The infinite model is a good approximation to any of the finite models – all odd-numbered finite models give slightly higher cube-ownership values, whilst the even-numbered models give slightly lower ones. The discrepancy reduces progressively towards infinity. The relationship can be imagined as a dampened-sinusoidal curve with the infinite model as its axis. The man on the six-point versus man on the six-point position is an example of the single-subsequent redouble live model (take-point = 18.75%). In fact, this live-cube model is the only one that exists in practice.

2. The change in gammon (and backgammon) rates throughout the life of the game. In most real backgammon positions, a player’s rate of winning gammons will decrease when his opponent redoubles. A typical example is when a shot is hit in an ace-point game, which subsequently gives the opponent little, if any, gammon risk. The same general reduction in gammon rate will normally occur in the live cube models, as the greater the number of subsequent optimal redoubles, the higher the chance that one or both players will, at some point, take men off. The rate of gammon loss could be linear (e.g., % loss per opponent’s redouble), or otherwise.

Assuming an infinite possible number of subsequent optimal redoubles, and a constant gammon rate (W and L are constant) for the sake of simplicity, the following formula was obtained, after some detailed mathematical analysis:

TP =
(L - 0.5)
(W + L + 0.5)
…equation (2)

Amazingly, the equation has a simple form. But what about the reduction in gammon rate, so far ignored? I investigated several different reducing models hoping to find that the above formula would still provide a reasonable estimate. What I found was much better; the formula is correct regardless of the gammon reduction rate considered, provided the W and L values used are average as opposed to initial ones! I wondered about this surprising result for some  time and developed the following argument to support it:

What is the difference, in terms of risk and reward, between the live and dead-cube models? There are additional benefits from holding the cube which add to the basic dead-cube reward (2W + 1). What are they and when do they occur? They occur on the point of redoubling when the redoubler’s equity jumps from 1·0 ppg (dead-cube) to 2·0 ppg (owning a 2-cube), a bonus of 1.0 ppg. (This is not strictly true, as the dead-cube equity is a little higher than 1·0 ppg, but this effect is balanced out by the equity jump occurring in more games than the cubeless take-point.) Consequently, if we add this bonus to the reward used in equation (1) for the dead-cube model, we arrive at equation (2) for the live cube model. As this argument is independent of any considerations of reducing gammon-rates, they would, indeed, appear to be irrelevant.

General Cube Model

Equations (1) and (2) above represent the take-point envelope in which correct take-points are to be found (the one known exception being the man on the six-point versus man on the six-point position). In any given position, the true take-point could be assessed by interpolating between the dead and live values, based on some intermediate value of cube-life, calculated, estimated, or just plain guessed at. The general form of these equations, given below again

for clarity, allows a more elegant solution:

TP dead =
(L - 0.5)
(W + L)
…equation (1)

#

TP live =
(L - 0.5)
(W + L + 0.5)
…equation (2)

Notice that the only difference is in the equations’ denominators, with the live value having the additional bonus from cube-ownership, explained before. As this bonus represents the expected equity jump, it is proportional to the degree of cube-life of the position (and inversely proportional to its long-term volatility). Intermediate models can, therefore, be represented by a cube-life index, x, which varies between 0·0 (dead cube, maximum volatility) and 1·0 (live-cube, zero volatility). The general form of equations (1) and (2) above is, therefore:

TP general =
(L - 0.5)
(W + L + 0.5x)
…equation (3)

Clearly the value of x varies from position to position, and will commonly be different for both sides. Some of the important factors that determine its value include:

 

1. The distance from the target – the further away from the optimal doubling point you are, the less likely you are to hit the bull’s-eye.

2. The size of the target – the size of the doubling window governs the size of the bull’s-eye.

3. The relative movement between the shooter and the target – the volatility of the position governs the likelihood of hitting the bull’s-eye, or even finding it, for that matter.

Finding accurate values for x is a difficult, almost impossible, task. However, we can make estimates of typical values for typical situations. In my opinion, for the majority of typical positions, x will commonly be between about 1/2 and 3/4 , with 2/3 being a normal value.

Cube Action Tables

To provide guidance on cube action, and to enable the reader to inspect the general results, the following tables are included:

Tables 1a, 1b, 1c – Cubeless take-points (for varying values of W and L) for x values of 0·0 (dead), 1·0 (live), and 2/3 (normal).

Tables 2a, 2b, 2c – Cubeless take-equities (for varying values of W and L) for x values of 0·0 (dead), 1·0 (live), and 2/3 (normal).

Cubeless take-equities (E take ) are calculated from the following general formula:

E take =
TP(W + L) - L
…equation (4)

 

Cubeless Take-Point Tables

Table 1a

Dead (x = 0.0)

Average cubeless win value W
1·00
1·25
1·50
1·75
2·00

Average

cubeless

loss

value

L

1·00
25·0%
22·2%
20·0%
18·2%
16·7%
1·25
33·3%
30·0%
27·3%
25·0%
23·1%
1·50
40·0%
36·4%
33·3%
30·8%
28·6%
1·75
45·5%
41·7%
38·5%
35·7%
33·3%
2·00
50·0%
46·2%
42·9%
40·0%
37·5%

#

Table 1b

Live (x = 1.0)

Average cubeless win value W
1·00
1·25
1·50
1·75
2·00

Average

cubeless

loss

value

L

1·00
20·0%
18·2%
16·7%
15·4%
14·3%
1·25
27·3%
25·0%
23·1%
21·4%
20·0%
1·50
33·3%
30·8%
28·6%
26·7%
25·0%
1·75
38·5%
35·7%
33·3%
31·3%
29·4%
2·00
42·9%
40·0%
37·5%
35·3%
33·3%

#

Table 1c

Normal (x = 2/3)

Average cubeless win value W
1·00
1·25
1·50
1·75
2·00

Average

cubeless

loss

value

L

1·00
21·4%
19·4%
17·6%
16·2%
15·0%
1·25
29·0%
26·5%
24·3%
22·5%
20·9%
1·50
35·3%
32·4%
30·0%
27·9%
26·1%
1·75
40·5%
37·5%
34·9%
32·6%
30·6%
2·00
45·0%
41·9%
39·1%
36·7%
34·6%

 

Cubeless Take-Equity Tables

Table 2a

Dead (x = 0.0)

Average cubeless win value W
1·00
1·25
1·50
1·75
2·00

Average

cubeless

loss

value

L

1·00
-0.500
-0.500
-0.500
-0.500
-0.500
1·25
-0.500
-0.500
-0.500
-0.500
-0.500
1·50
-0.500
-0.500
-0.500
-0.500
-0.500
1·75
-0.500
-0.500
-0.500
-0.500
-0.500
2·00
-0.500
-0.500
-0.500
-0.500
-0.500

#

Table 2b

Live (x = 1.0)

Average cubeless win value W
1·00
1·25
1·50
1·75
2·00

Average

cubeless

loss

value

L

1·00
-0·600
-0·591
-0·583
-0·577
-0·571
1·25
-0·636
-0·625
-0·615
-0·607
-0·600
1·50
-0·667
-0·654
-0·643
-0·633
-0·625
1·75
-0·692
-0·679
-0·667
-0·656
-0·647
2·00
-0·714
-0·700
-0·688
-0·676
-0·667

#

Table 2c

Normal (x = 2/3)

Average cubeless win value W
1·00
1·25
1·50
1·75
2·00

Average

cubeless

loss

value

L

1·00
-0·571
-0·565
-0·559
-0·554
-0·550
1·25
-0·597
-0·588
-0·581
-0·575
-0·570
1·50
-0·618
-0·608
-0·600
-0·593
-0·587
1·75
-0·635
-0·625
-0·616
-0·609
-0·602
2·00
-0·650
-0·640
-0·630
-0·622
-0·615


Example

Consider the following position, from the 12th game of the semi-finals match between Nack Ballard and Mike Senkiewicz at the Reno Masters in 1986. Senkiewicz, trailing 9-20 in this 23-point match, gave an initial double, which Ballard passed. Bill Robertie, analysing this match in his book Reno Quiz, evaluates the pass as correct at this match score. What would the correct cube action be in a money game?

# INSERT POSITION 01 #
Should White Take?

Using Robertie’s cubeless rollout figures:

Black wins single-game:
47%
Black wins gammon:
17%
Black wins backgammon:
1%
Black’s total wins:
65%
White wins single-game
31%
White wins gammon:
4%
White’s total wins:
35%
 
Black’s cubeless equity:
0·450 ppg

Considering White’s cube action,

L =
(47 + 17 x 2 + 1 x 3)
(47 + 17 + 1)
= 1.292  and    W =
(31 + 4 x 2)
(31 + 4)
= 1.114

1. Dead-Cube (x = 0·0)
From equations (1) and (4):

TP dead =
(1.292 - 0.5)
(1.292 + 1.114)
= 0.392 and    E take =0.3292 x (1.292 + 1.114) - 1.292 = -0.500 clearly

2. Live-Cube (x = 1·0)
From equations (2) and (4):


TP live =
(1.292 - 0.5)
(1.292 + 1.114 + 0.5)
= 0.2727 and    E take =0.2725 x (1.292 + 1.114) - 1.292 = -0.636

3. Normal-Cube (x = 2/ 3)
From equations (3) and (4):

TP 2/3 =
(1.292 - 0.5)
(1.292 + 1.114 + 0.333)
= 0.2892 and    E take =0.2892 x (1.292 + 1.114) - 1.292 = -0.596

In the actual position, White, with 35% winning chances, can take for money, regardless of the cube model considered.

Other Cube Action Decisions

So far, only take-points have been considered. There are many other doubling decisions to consider – when to redouble, when to beaver, etc. Correct cube-action can be established by comparing the resultant equities from the alternative cube positions – owned (Eo), unavailable (Eu), and centred (Ec):

Eo = Cv [p(W + L + 0.5x) - L]
…equation (5)
   
Eu = Cv [p(W + L + 0.5x) - L - 0.5x]
…equation (6)

#

Ec =
4 Cv
4 - x
[p(W + L + 0.5x) - L - 0.25x]
…equation (7)

where Cv = cube-value (i.e., the stake-level)
p = cubeless winning probability

Note that equation (7) is not applicable if the Jacoby Rule is in operation.


From manipulation of the above equations, the following table of formulae, covering the full range of cube-actions in money games, has been derived. Notice two particularly interesting features from this table:

 


1. In the live-cube model, when gammons and backgammons are active, it is never correct to double, as positions strong enough to double are also too good to double! This is understandable because the complete lack of volatility protects the double-out.

2. Assuming the Jacoby Rule is not in operation, then initial double-points are always lower than redouble-points. When the cube is dead or live, they coincide, but diverge for intermediate values of cube-life. Maximum divergence occurs when x is about 0·57, and typically ranges between 2.00% (W = 2, L = 2) and 3.75% (W = 1, L = 1).

 

Cube Action Formulae

 

where . . .

W = Average cubeless value of games ultimately won
L = Average cubeless value of games ultimately lost
x = Cube life index (0·0 for dead cube, 1·0 for live cube)
k1 = Jacoby factor (no beavers)
k2 = Jacoby factor (with beavers)

Appendix 1: Miscellaneous Equity Relationships


The various equities for the different cube positions may be expressed in terms of the cubelife index (x), cubeless probability of winning (p), and cubeless equity (E) as follows:

Cubeless Equity
E = p(W + L) - L
Cube-owned Equity Eo = Cv [E + 0.5x p]
Cube-unavailable Equity Eu = Cv [E - 0.5x(1 - p)]
Cube-centred Equity Ec =

4Cv

(4 - x)

[E + 0.5x (p - 0.5)]

The cube-centred equity may also be expressed in terms of the cube-owned and cubeunavailable equities (with their respective Cv values set at unity) as follows:

Ec =

4

4 - x

(Eo - 0.25x) =

4

4 - x

(Eu + 0.25x) =

2

4 - x

(Eo + Eu)


Note that the cube-centred equity formulae given above are not applicable if the Jacoby Rule is in operation. The cube-owned and cube-unavailable equities corresponding to the various cube-action points are shown by the following table:

 

Note that the above equities are independent of W and L apart for the initial double equities with the Jacoby Rule in operation. Also note that the cube-unavailable equity required for a redouble is the cube-life index (x) multiplied by the stake of the redoubled cube (Cv ). Using 2/3 as a normal value for x, the required equities after doubling to 2 are 0·667 and 0·500, for redoubles and initial doubles (no Jacoby) respectively. These values are fairly consistent with typical limiting values obtained from hand rollouts (generally minimum redoubles are between 0·6 and 0·7 ppg, and between 0·4 and 0·6 ppg for initial doubles). Consequently, 2/3 would appear to be a good estimate of the cube-life index.

Appendix 2: Refined General Model


A more rigorous analysis may be performed by considering different cube-life indices for both sides, which is what normally happens in practice. Let x1 and x2 be the cube-life indices for the player making the cube-action decision, and his opponent, respectively. Following a similar analysis as before, the equities from the alternative cube positions, owned (Eo), unavailable (Eu ), and centred (Ec), were derived: