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.
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):
Consequently, a take would risk 2L - 1 points to gain 2W +1 points. The minimum cubeless probability for a correct take (TP) is therefore:
This formula is also applicable when the data considered represents effective game winning chances.
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:
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:
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:
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:
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:
Cubeless Take-Point Tables
Cubeless Take-Equity Tables
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 #
Using Robertie’s cubeless rollout figures:
Considering White’s cube action,
1. Dead-Cube (x = 0·0)
2. Live-Cube (x = 1·0)
3. Normal-Cube (x = 2/
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):
where Cv = cube-value (i.e., the stake-level)
Note that equation (7) is not applicable if the Jacoby Rule is in operation.
Cube Action Formulae
Appendix 1: Miscellaneous Equity Relationships
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:
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