Saturday, July 07, 2012

Heroes and Jerks

I was reading an article on the New York Times website this morning entitled Lifeguard Says He Chose Saving Man Over Saving Job. If you can't be bothered reading it. in short, a particularly heroic young man was employed as a lifeguard for a summer. He was at work, watching over his charges and he noticed that there was someone in trouble outside the area that he was hired to supervise. His employer's policy stated that, in the event that an emergency happens outside the supervised area, a lifeguard must wait for a supervisor to arrive to watch over that area before going to provide assistance. Uncowed by such risk-averse policy, our hero sprang into action and saved the day. His reward for his exertions: being fired.

I read the article and, like most people, I imagine, came to the conclusion that our risk averse world had gone crazy. Here's a guy doing the right thing and being punished for it. Outrageous! Then, I imagine, somewhat unlike most people, decided to prove, using statistics that this policy was ridiculous and is, as we all though, an example of our risk averse world going awry. When I ran the numbers though, things got a little more complicated.

There are three main variables at play here*:
  1. The length of time it takes a supervisor to arrive,
  2. How often emergencies arise within the supervised area, and
  3. What the tolerance of society is for emergencies in the supervised area to not be responded to in a timely manner.

Now, I don't actually know what reasonable figures are for any of these variables, so I made a table to analyse them.

  How Often an Emergency Occurs in the Supervised Area (1 in x times)

  Time for supervisor to arrive (minutes)
2 3 4 5 6 7 8 9 10
How often an emergency happens in the supervised area (every x minutes) 15 7.50 5.00 3.75 3.00 2.50 2.14 1.88 1.67 1.50
30 15.00 10.00 7.50 6.00 5.00 4.29 3.75 3.33 3.00
45 22.50 15.00 11.25 9.00 7.50 6.43 5.63 5.00 4.50
60 30.00 20.00 15.00 12.00 10.00 8.57 7.50 6.67 6.00
75 37.50 25.00 18.75 15.00 12.50 10.71 9.38 8.33 7.50
90 45.00 30.00 22.50 18.00 15.00 12.86 11.25 10.00 9.00
105 52.50 35.00 26.25 21.00 17.50 15.00 13.13 11.67 10.50
120 60.00 40.00 30.00 24.00 20.00 17.14 15.00 13.33 12.00
135 67.50 45.00 33.75 27.00 22.50 19.29 16.88 15.00 13.50
150 75.00 50.00 37.50 30.00 25.00 21.43 18.75 16.67 15.00
165 82.50 55.00 41.25 33.00 27.50 23.57 20.63 18.33 16.50
180 90.00 60.00 45.00 36.00 30.00 25.71 22.50 20.00 18.00

This table shows that, if it takes 5 minutes for a supervisor to arrive and an emergency happens in the supervised area every 90 minutes then, if a lifeguard takes off without waiting for their supervisor, they should expect an emergency to happen without someone to deal with it in the supervised area once every 18 times.

They say in the article that it takes "a few minutes" for a supervisor to show up. I don't know how many a few is. I thought 2-10 was a reasonable estimation of that (keeping in mind that the person being interviewed has a personal interest in making the period of time look short). I have no idea how often emergencies occur in supervised areas but they say it is about the size of two football pitches. Assuming that emergencies in the unsupervised area occur at the time when emergencies are most likely to occur in the supervised area, I thought a range of once every 15-180 minutes seemed reasonable.

Now we need to start looking at the risk tolerance of the society for having emergencies in the supervised area not responded to in a timely manner. Let's try taking a cutoff point of 1 in 20.

How Often an Emergency Occurs in the Supervised Area (1 in x times)
  Time for supervisor to arrive (minutes)
2 3 4 5 6 7 8 9 10
How often an emergency happens in the supervised area (every x minutes) 15 7.50 5.00 3.75 3.00 2.50 2.14 1.88 1.67 1.50
30 15.00 10.00 7.50 6.00 5.00 4.29 3.75 3.33 3.00
45 22.50 15.00 11.25 9.00 7.50 6.43 5.63 5.00 4.50
60 30.00 20.00 15.00 12.00 10.00 8.57 7.50 6.67 6.00
75 37.50 25.00 18.75 15.00 12.50 10.71 9.38 8.33 7.50
90 45.00 30.00 22.50 18.00 15.00 12.86 11.25 10.00 9.00
105 52.50 35.00 26.25 21.00 17.50 15.00 13.13 11.67 10.50
120 60.00 40.00 30.00 24.00 20.00 17.14 15.00 13.33 12.00
135 67.50 45.00 33.75 27.00 22.50 19.29 16.88 15.00 13.50
150 75.00 50.00 37.50 30.00 25.00 21.43 18.75 16.67 15.00
165 82.50 55.00 41.25 33.00 27.50 23.57 20.63 18.33 16.50
180 90.00 60.00 45.00 36.00 30.00 25.71 22.50 20.00 18.00
 
If we are in the red section, the policy was good. It means lifeguards should stay at their post until supervisors arrive. If we are in the green section, then the lifeguards are safe to run off as and when required. We can see here that as the time for the supervisor to arrive and as the frequency of the emergencies increases, it looks for the lifeguard and better for the policy.

Let's think about it for a second though. What does a cutoff point of 20 mean?

It means we're happy if an emergency goes unresponded to in the supervised area while a lifeguard responds to an emergency in the unsupervised area 1 in 20 times. What does that mean though? Well, at the beach, it could mean that someone drowns. Let's say that, while the lifeguard in the article ran off to save the person who was in trouble outside the supervised area, a young child named Sally who is swimming in the supervised area gets into trouble and drowns. Sally, who was doing the right thing, drowns because the lifeguard ran off to help some jerk who got in trouble in an area that he was told was unsafe. If we're happy with a cutoff point of 1 in 20, that means we're happy with Sally drowning while the lifeguard saves some irresponsible jerk 1 in 20 times.

How many emergencies happen near, but not in the supervised area over a summer? 20? 20 seems like a fairly conservative number*. If it's 20, then that little stretch of beach should expect to see one Sally drown every 2 years and two Sallys to drown in separate incidents in one summer every 6.5 years. It's important to note that these are Sallys that would not drown if lifeguards obeyed the policy.

If I were a policy maker, I reckon I could probably get away with one Sally drowning in this fashion every few years, but if we were to lose two Sallys in a single year because of boneheads getting in trouble and taking lifeguards away from their responsibilities of watching over people in the supervised area, we'd be in serious trouble. I reckon I could live with two Sallys dying in one summer every hundred years. For that to be the case, we need to set our cutoff rate to 44.18. This results in a table that looks like this:

How Often an Emergency Occurs in the Supervised Area (1 in x times)
  Time for supervisor to arrive (minutes)
2 3 4 5 6 7 8 9 10
How often an emergency happens in the supervised area (every x minutes) 15 7.50 5.00 3.75 3.00 2.50 2.14 1.88 1.67 1.50
30 15.00 10.00 7.50 6.00 5.00 4.29 3.75 3.33 3.00
45 22.50 15.00 11.25 9.00 7.50 6.43 5.63 5.00 4.50
60 30.00 20.00 15.00 12.00 10.00 8.57 7.50 6.67 6.00
75 37.50 25.00 18.75 15.00 12.50 10.71 9.38 8.33 7.50
90 45.00 30.00 22.50 18.00 15.00 12.86 11.25 10.00 9.00
105 52.50 35.00 26.25 21.00 17.50 15.00 13.13 11.67 10.50
120 60.00 40.00 30.00 24.00 20.00 17.14 15.00 13.33 12.00
135 67.50 45.00 33.75 27.00 22.50 19.29 16.88 15.00 13.50
150 75.00 50.00 37.50 30.00 25.00 21.43 18.75 16.67 15.00
165 82.50 55.00 41.25 33.00 27.50 23.57 20.63 18.33 16.50
180 90.00 60.00 45.00 36.00 30.00 25.71 22.50 20.00 18.00


Again, like I said, I don't know what the actual response time for supervisors is, but I'm pretty sure it's more than 4 minutes. Let's say, giving the lifeguard the benefit of the doubt that it's 4 minutes. Then the rate at which emergencies happen can't be any less than one every three hours, otherwise we're going to lose Sallys at a rate that is unacceptable. Again, I don't know how often emergencies happen on this particular stretch of beach, but it's not obvious to me that it's greater than one every 3 hours.

So, basically, this analysis shows that it's not obvious that the lifeguard was doing the right thing in running off to save some moron who should have known better than to swim outside the supervised area. The lifeguard was lucky that no one got into trouble while he was saving Mr. Moron's life, but luck isn't a basis upon which to make policy.

As much as I like to stick it to the man (especially the bureaucratic, needlessly risk-averse man), unless some better data comes to light, it looks like the policy is sound and the lifeguard should indeed have been sanctioned for breaking it.

* We could go crazy modelling the numbers of people going to the beach, the time they spend, the rate at which they get into trouble, the chances of clustering of emergencies, etc. etc. but for simplicity we'll exclude these. 
** For brevity, I didn't go into what happens if you have more or less incidents just outside the border of the supervised area each summer. As you would expect, as the number of incidents goes up, so should our cutoff. As it goes down, so should our cutoff. As an illustration, if you have 10 incidents, then your cutoff becomes 21.75, if you have 30 incidents, then your cutoff becomes 66.6.

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