Significance magazine (link) has a fun article about parking in front of fire hydrants in New York City. It's great data journalism.

The researchers conducted a "field experiment" in which they observed "six routes through Manhattan streets" on July 11, 2022 from 1-5 pm, during which they observed 138 violations. They then merged the hand-collected data with NYC's open data recording all tickets, matching license-plate numbers to learn that only 7 of those drivers have been served tickets.

Based on that, they deduced that most drivers who park in front of hydrants get away with the violation. They computed a detection rate of 5% (= 7/138). During that period of time, the traffic cops issued 423 tickets, and therefore, they deduced that the total violations to be 423/5% = 8,460. This number includes both the 423 tickets and a further 8,037 instances for which no tickets were issued.

They now model the cat-and-mouse game between the rule-bending drivers and the cops as an "exponential game". A timer is started each time some driver parks at a hydrant. The game is to see whether the cop shows up to flag the violation first, or the driver returns to pick up the car first. The model assumes that the time until the next check is exponentially distributed. An exponential distribution is where most values are small, but a few values can be very large. Such a distribution is fully described by its average value, which is estimated to be 6 hours. On average, a cop checks a given hydrant once every 6 hours. (That's the average interval, but each interval has a different value.)

They apply the same exponential model structure for the rule-bending drivers. In this case, the time until the driver returns to a given hydrant - i.e. the duration of violation - is exponentially distributed, with a mean of 18 minutes (0.3 hours). Comparing the two exponential distributions, we can already see that drivers who leave their cars for a short period of time (say under 60 minutes) have a low chance of getting ticketed but if they park their cars illegally for hours at a time, their risk of detection is much higher. The authors did these calculations: the probability of detection is 16% for one hour, and increases to 99% for 24 hours.

Given the above model - and we assume also that the cops and the drivers act independently, the probability that the cop arrives before the driver returns is the key statistic we care about. That's the chance of the driver receiving a ticket, which as we know from above, is 5% based on the observed data. A consequence of the exponential game model is that this number is the average time until a cop arrives divided by the sum of the average times of a cop arriving and of the driver returning. Thus, 0.05 = 0.3/(0.3+6). [This verifies that the above calculation is correct.]

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This type of model has been deployed to analyze gambling. If one buys lottery tickets every day, what is the chance that one hits the jackpot before one goes broke?

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What's left unexplained is how they got the 6-hour estimate for the time until detection. This is a clever bit of work. Remember the researchers matched a subset of the observed violations to actual tickets issued. For these violations, the researchers determined the duration between observing the violation and issuing the ticket to be 6 hours on average.

But... that duration is counted from the moment the researchers found the illegally-parked car, not from the moment the driver parked illegally. What's unknown is the passage of time before the researchers documented the violation.

Conveniently, that unknown duration can be ignored. This convenience comes with using the exponential model. In the exponential world, the future does not depend on the past. Let's say the car has been illegally parked for 10 minutes already. How much time will elapse before a cop shows up? The answer is the same as before: it's exponentially distributed with average of 6 hours.

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Based on the math above, the researchers found that illegally parking makes sense. The penalty for getting caught is $115. Returning within 5 minutes limits the expected penalty to under $2, which is similar to the cost of feeding a parking meter.

P.S. [7/10/2023] Corrected "flout" in the title of the post. Thanks to the reader who alerted me to this.

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