On the planet of Gallifrey a time

lord’s pension is a cash sum that

is proportional to the number of

Daleks that he has battered during

his working lifetime. This cash sum will be

paid to him, or to his estate, on his retirement

date whether or not he is alive on that date. The

task of projecting a time lord’s final Dalek total

(and therefore his pension) is an interesting

challenge to time lord actuaries, and one that

this article sets out to explore.

First I should explain the demographics of the

time lord population. All time lords start waging

war on Daleks at the same age and all retire at

the same age, so that all who live long enough

have the same working lifetime of 50 Gallifreyan

years (about 300 of our years). What

makes the problem more interesting though is

the time lords’ ability to regenerate.

For the benefit of anyone that has never seen

Doctor Who, I should explain that time lords

effectively have ten lives. Every time they are

close to death (up to a maximum of nine

times), they can regenerate into a new body,

effectively starting a new life. Nobody ever

needs to regenerate before their coming of age

the dangers that necessitate regeneration are

all due to chasing Daleks. The new body always

looks different to the last one, which is a real

pain for the BBC as it has to bring in a new actor

each time

Starting with the basics

Of course different time lords will have different

appetites for battering Daleks, and differing

Dalek-battering abilities. So our pension projections

should ideally base the future rate of Dalek

battering for a particular time lord on his past

record. One possible starting point is the basic

Dalek-rate calculation, given by the formula:

projected Daleks = Daleks to date × 50

/working Gallifreyan years used

For many years this was the standard formula

for pension projections, but many commentators

pointed out two key faults with the formula:

? The formula does not take into account the

well recognised tendency for Dalek battering

to accelerate as time lords approach retirement,

assuming instead that the Dalek-rate

stays fixed.

? The formula does not take into account the

number of regenerations that a time lord has

undergone to date. The fewer the number of

regenerations a time lord has in hand, the

fewer chances he is likely to take during his

adventures, and the lower his likely Dalekrate

(not to mention the increased probability

that he will use up all ten of his lives without

using up all his available working years).

Dalek-battering resources

At this point, a couple of innovative statisticians

announced their presence with an alternative

methodology that changed the face of time lord

pension projections forever. Their idea was to

introduce the concept of Dalek-battering

resources. At every point in his working life a

time lord has a number of outstanding working

years (y) and remaining regenerations (r). His

Dalek-battering resources are a function F(y,r)

that reflects how much more opportunity he

has in his career to batter Daleks. It is defined

in such a way that F(50,10) = 100% (so that

everybody starts their career at 100%). Also

F(0,r) = 0 for all r, and F(y,0) = 0 for all y; if a time

lord reaches retirement or uses up all ten lives,

his battering resources are zero. The new projection

formula is now:

projected Daleks = Daleks to date

/ (1 F(y,r))

I imagine that the method of construction of

the two-dimensional table F(y,r) (see table) was

as follows:

1 All historical Dalek batterings and time lord

regenerations were divided up into 500 cells,

corresponding to the number of regenerations,

r, that the corresponding time lord had

used up to that point (09) and the curtate

number of Gallifreyan years, y, that the time

lord had been working up to that point (049).

2 Regeneration rates q(y,r) were calculated for

each cell by dividing regenerations by the

corresponding exposed to risk and then

graduating the resulting rates.

At this point a time lord’s outstanding

working years and regenerations have been

modelled as a two-dimensional Markov

chain, where the process can move from (y,r)

at one point in time to either (y 1,r 1)

with probability q(y,r), or to (y 1,r) with

probability 1 q(y,r) (see figure 1).

3 Next, a battering rate b(y,r) was calculated for

each cell by dividing Dalek batterings by

exposure and then graduating.

4 The next step was to construct a table of

B(y,r), being the expected number of future

Dalek batterings for a time lord y years from

retirement and with r remaining regenerations.

The starting point was at the boundaries

where B(y,0) = 0 for all y and B(0,r) = 0

for all r. The rest of the table was filled in by

working backwards using: B(y,r) =

b(y,r) + q(y,r)B(y 1,r 1) + (1 q(y,r))B(y 1,r)

(see figure 1 again).

5 Finally the resulting table would have been

normalised using F(y,r) = B(y,r)/B(50,10) so

that F(50,10) = 100%.

And there you have it. Problem solved. Please

tell me that wasn’t difficult.

Betting on success

As well as its use in pension projections, the projection

formula can also be used in the world of

competitive Dalek battering. Imagine that the

Doctor and the Master have entered into a bet

about who will be the most successful Dalek

batterer over their careers. The Master is sent

out to batter first and scores 300 Daleks (using

up five lives) in his 50 Gallifreyan years. The

Doctor is on 150 for two after 30 Gallifreyan

years, at which point the BBC cancels his show

for reasons beyond his control and he does not

have the opportunity to use his remaining

20 Gallifreyan years.

Given that the bet was based on who was the

‘most successful’ rather than on who batters the

largest number of Daleks, the time lord actuaries

will need to be called in to determine the

winner. Using the old-fashioned Dalek-rate calculation,

the Master would win as his Dalek rate

of 300/50 = 6 exceeds the Doctor’s rate of

150/30 = 5, but this does not allow for how the

Doctor’s Dalek rate was expected to increase

over the remaining 20 Gallifreyan years, especially

with eight regenerations in hand. In order

to get a fair comparison between the time lords,

we bring in the new tables and note (from the

figures in the F(y,r) table) that the Doctor was

allowed to use only 1 F(20,8) = 47.8% of his

resources, so his 150 could be extrapolated forwards

to 150/0.478 = 314. A comfortable victory

for the Doctor.

On the other hand, what if the Doctor’s show

is unexpectedly relaunched? The bet with the

Master is on again. The problem is that he has

been off air for ten years and has only ten years

left to batter Daleks. How many Daleks should

he be aiming to batter if he is to be able to win

his bet? We know that the Doctor has already

used 47.8% of his original resources. He now

has F(10,8) = 30.8% of his original resources left,

so in total will be able to use 78.6%. So his

revised target should be 78.6% of the Master’s

300 rounded up, ie 236 Daleks. So he needs a

Dalek rate of 8.6 over the rest of his career if he

is to triumph. A challenging target (more than

one Dalek per Earth year), but with eight regenerations in hand he will have the

opportunity to take some risks.

By now many of you will have worked out

that those two innovative statisticians were

Frank Duckworth and Tony Lewis, and that

their idea has far more down-to-earth applications

than the projection of time lords’ pension

benefits.

07_10_26-27.pdf