*Note to non-Londoners: TfL is the umbrella organisation which runs London subways and busses, but the below applies to any system where fines are used to deter people from doing things which have the potential to save or make them money (fare-dodging, theft, tax evading etc)*

1) Some people don’t buy tickets for their travel, effectively attempting to steal a train journey, hoping they don’t get caught and fined. This is called fare-dodging and it costs TfL money

2) Fare dodging isn’t worth it. It costs more than it saves you once you’ve done it enough to get hit by a few of those £60 fines.

So my instinct is that these two statements accurately describes the world. In fact, if statement 1) is true statement 2) must be false, and vice versa.

So let’s take statement 2) and the fare-dodger’s case first: if buying a ticket (T) costs less than the probability of you getting caught (C(p)) times the cost of the fine(C(c)), then fare-dodging is more expensive than being an honest, upstanding citizen and statement 2) is true. Or:

i) if, and only if, T < C(p)*C(c) is true then statement 2) is true

For example, in Jonland, a train ticket costs £5

T = 5

your ticket gets checked, on average, once every ten journeys:

C(p) = 0.1 *

and if you’re checked and haven’t got a valid ticket, the Jonland ticket checker fines you £20

C(c) = 20

If we plug those number in to I) we get

5 < 0.1*20 which is

5 < 2

Which, as you may have noticed, isn’t true. So in Jonland, i) tells us, statement 2) is false and fare-dodging *is* ultimately worth it and saves you money.

Right, back to earth and time for a couple more rules. The total amount of money TfL *loses* due to fare-dodging every day is the number of people travelling that day (X) times the likelyhood they’re travelling without a ticket (Y)* times what a bought ticket would have cost (T). Or:

ii) X*Y*T = total lost in avoided ticket sales on a given day

The total amount of money TfL *makes* in a given day from people not buying tickets is the total number of people travelling that day (X) times the likelyhood they’re travelling without a ticket (Y)* times the likelyhood they get caught (C(p)) times the cost of the fine (C(c)). Or:

iii) X*Y*C(p)*C(c) = total made from fined fare-dodgers on a given day

If the money they lose from fare dodgers (ii) is more than the money they make in fines (iii) then they wind up out of pocket. Or:

iv) if, and only if, X*Y*T > X*Y*C(p)*C(c) is true then statement 1) is true

Let’s assume that at least one person travels per day, that X>=1, and that people aren’t perfectly honest, that there’s *some* chance of fare dodging, meaning Y>0. That means X and Y are both positive, and as two positive number multiplied together will always give another positive number, X*Y must be positive too, regardless of how huge or tiny X and Y are. This means we know:

X*Y*5 > X*Y*3 is true because 5>3 is true

X*Y*2 > X*Y*6 is false because 2>6 is false

X*Y*9 > X*Y*9 is false because 9>9 is false

The whole ‘X*Y’ bit of the equation is starting to look a bit obsolete. And it is – we can drop it from our equation**. Thus iv) can be rewritten as:

v) if, and only if, T > C(p)*C(c) is true then statement 1) is true

Remembering that

i) if, and only if, T < C(p)*C(c) is true then statement 2) is true

We see that they statements 1) and 2) cannot be true at the same time – T would have to be both greater and less than C(p)*C(c)

(mic drop)

*we’re scaling this from 0 to one, so 0.1=10% chance, 1=100% chance etc

**this is significant beyond just tidying up the math. It means that whether TfL falls into statement 1) or statement 2) depends on the cost of a ticket, the cost of a fine and how likely you are to have your ticket checked (T, C(c) and C(p)). It *does not* depend on the number of people using the service or the percentage of them who are actually attempting to fare dodge. In other words, if statement 1) is true for a given system, you can quadruple the number of travellers and take the percentage of fare dodgers from 99.9% to 0.1% and statement 1) will still be true.

NB – stealing is shitty behaviour and erodes trust in societies. Don’t do it.

And if you *are* going to do it, steal from something besides a public service.

like maybe tesco