# The Rational Fare Evader

Fare evading, i.e. not paying for the use of public transport, is a fairly significant issue for public transport providers. The Victorian Transport Authority has been tracking the estimated number of commuters who regularly fare evade and as you can see there is a sizeable portion who do this on a regular basis, ranging from roughly 5% to 20% of Victorian commuters (fig 1). There are potentially many causes for fare evading: I had the misfortune of forgetting to touch on to a crowded tram the other day which resulted in a discussion about these causes. This eventually led to the quip “if you get caught – just fare evade until you break even!”

So I thought, would that actually make sense?

Being the economist I was, I had to find out to make sure that we were making the rational choice in paying for our tickets instead of simply free riding the whole time. The main problem is, short of spending a lot of time collecting data there’d be no way to know for sure how likely one is to run into a Public Transport Authorised (PTA) officer, so we’ll have to make some simplifying assumptions.

Focusing on trams in particular, let’s say we have a full time student who travels 5 days a week, twice a day to and from university. This person also travels on weekends (which is half price compared to weekdays, so one ‘day’ of travel each weekend), and pays the full fare. In total, this person pays 2 trips x 6 days a week x 4 weeks a month = 48 ‘payments’ each month.

At \$3.50 for a full fee zone 1 trip this comes out to be \$168 per month, which is pretty comparable to what you might pay for petrol if you drove instead. This is in comparison to the \$207 fine for travelling without a valid ticket. From this it’s immediately obvious that rationally it only makes sense to evade if you encountered PTA officers less frequently than once a month, and from my observations this is likely not to be the case.

However in the event that you are a chronic evader, what would your chances of getting away with it be? If we assume the chances of meeting a PTA officer is equal on each trip and independent, we can divide 1 by 48 to get 2.1%, which is the probability of getting a fine for each individual trip. This turns into a binomial distribution with 48 trials (Fig 2), and from that we find that the probability of not being caught at all after a month is approximately 36.5%, only slightly better than 1 in 3. Applying our 36.5% probability gives us a nice 0.4% chance of not receiving any fines at all the entire year – not such great odds.

This got me thinking however, perhaps there is a way to reduce the total expected payout by trying a mixed strategy? What if you touched on every 2nd time, to both reduce the number of payments and reduce the chance of having to pay a fine?

In this case we reduce the number of ‘unpaid’ trips to 24 times a week, and paid trips also to 24. By halving  the No. of unpaid trips the average number of fines is halved, which means per month the cost is 84 (from the 24 paid trips) + 207*.5 = \$187, so still not as good as not fare evading. In fact we know the mean of this binomial distribution to be the probability (2%) x the number of times, and thus this increases linearly as we increase the number of unpaid trips.

Turns out we can’t find a better expected payoff, as it is directly proportional to the number of times you choose to evade, i.e. because expected number of times increases by 2% on average each time, the more times you evade the higher your expected cost becomes (Fig 3).

Thus rationally it makes no sense to be fare evading. Consider the unlucky time you were caught without a valid ticket a sunk cost.

Author Disclaimer: Of course, the results of this analysis is based purely on personal conjecture and is no ways mean to support further analysis of what it would take to make fare evasion ‘rational’. Purely for the sake of discussion, if we could improve the probability model of receiving fines then we could obtain a better answer.