Mea Culpa on the Sneeze Guy, and a Riddle!

This is rare: I think my earlier post about the sneeze guy was wrong, or at least didn't contain the full picture. What's even rarer, is that it took an email from Tyler Cowen to prod me to reverse myself.

Although Mike in the comments of my last post had talked about the sneeze working its effects backwards through the line--much like a traffic accident can cause a delay to ripple backwards through the highway--I don't think that's the way to analyze this situation. Or at least, that's not what the sneeze guy had in mind.

Suppose the security line originally has 10 people in it. After 6 seconds one person walks through the metal detector, and a new person arrives at the end of the line. So the line is a constant 10 people long, and each person has to wait a total of one minute to get through security.

Now some guy at the front has a sneezing fit. He ends up taking 12 total seconds to go through; i.e. he "wastes" 6 seconds. This means that the line now grows to 11 people in length, and it stays like that.

So because of that guy's 6-second delay, every single person who ever joins this line has to wait for an extra person to go through, i.e. is delayed an additional 6 seconds. If a million people go through in the next year, that's 6 million people-seconds wasted.

Now relax the assumption that the line is fixed in length, but still insist that it never hits zero. Even so, the initial 6-second delay causes the line to always have one more person in it, than it otherwise would have--so long as we assume that people entering the line don't take its length into account.

I think the above analysis is correct, insofar as it goes. Since that's all the sneeze guy was claiming, I think we have to admit he was right.

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RIDDLE

There are still weird things with this. I think we've established above that if one million people go through that line, they each lose 6 seconds. But now suppose that it's the same 10 people who keep going around and around the security gate. I.e. once you go through the metal detector, you go around to the back of the line.

To get rid of the infinities, suppose for some reason that 10 people need to go through the metal detector a total of 1,000 times each.

Now if we had just said there were an initial pool of 10,000 people who had to go through the metal detector once, the analysis above shows the sneeze costs a total of 60,000 person-seconds in lost time.

But what in our new scenario? Does it cost the same amount?

That seems weird. Just imagine you're watching the group of ten people from the ceiling. They form a ring that is rotating, where every 6 seconds the people all walk a little bit around the ring, with one of them going through the gate. Do you mean to tell me that if this rotation pauses for six additional seconds, that each person loses 6,000 seconds of time? No way.

And yet, this seems to be equivalent to the original scenario, with 10,000 strangers forming the line. Surely the amount of time wasted shouldn't depend on the identity of the person in line. If the guy sneezing at the front causes the 8,000th person going through to lose 6 seconds, why does it matter who the 8,000th person is?

I'm pretty sure I know the solution to this riddle, but it's just beyond the reach of perfect obviousness in my mind. So I'm hoping someone can spell it out clearly and I can say, "Yep, you got it!"

1. Start by eliminating the queue and the other 9 people. Now we can see the essentials: I sneeze coming out of the womb and am born 6 seconds late.

What is the cost? Is it a single lost 6 seconds, or is it the summation of the 6 seconds behind I run in everything for the rest of my life?

Returning to the queue: Why fizate on the metal detector? Why not sum the 6 seconds late every step forward in the line is for every person?

Because real cost, and the real measure of "seconds wasted," is opportunity cost. Did someone miss out on something of value due to the sneeze? Then that is a cost. In a world where everything appears in continuous magnitudes, there are eternal effects from every micro-second of opportunity missed. But the real world isn't like that -- there isn't an infinitesmal plane perpetually leaving from every gate in the world -- there is a discrete plane I hope to catch, and if I catch it, the sneeze "wasted" no seconds, and if I don't, it may have wasted hours.

2. Eh, I don't think this is really it, Gene. Even if you make your flight, it still s*cks to wait in the security line. If you want, suppose there are hot coals that people stand on.

So if there are 1,000 people in line, and they all have to stand on the coals 6 seconds longer because the guy in front sneezes, I think there is a very real sense in which his sneeze has imposed more harm (we can drop the economics term "costs" if you want) than if there had only been 3 people in line.

Are you with me so far? Then how does it change things if the 1,000 people aren't standing there the whole time, but instead trickle in at a rate independent of line length? I say it doesn't alter things, and that the delay causes every person who joins the line to stand 6 seconds longer on the coals than he otherwise would have.

And now, if we switch it to the same 10 people cycling through 100 times each, I don't see why the logic should change.

Yes, if we reduce it to one person, the anomaly disappears, but that's my point. It doesn't seem true for one person, and yet it seems true for multiple people. So what's the difference?

E.g. if I say, "Joe punched ten people in the face," that is bad. If instead I say, "Joe punched one person in the face ten times," then we also know that is bad, and that the 10 punches are worse than just one punch for that person.

Yet it seems we are saying that standing longer on the hot coals for the 50th person in line is worse, if you haven't gone through yet.

3. This is a good point. Looking at it from a mole balance perspective (I'm not sure what you economists call this...unit balance?), In - Out = Accumulation, and if the rate of people leaving the line due to the sneeze is reduced, if only for a few seconds, then there will be some accumulation, resulting in a longer wait.

I still maintain that if someone far enough back from the head of the line were to sneeze to be able to walk up to the next person before that person left the line, there would be no net change in wait time, because neither the "In" or "Out" term would change.

This is, I think, the answer to your riddle. In the scenario where there are a thousand people trickling into the line at a constant rate, the "In" term is independent of the "Out" term. In the 10 people continuously cycling scenario, "In" necessarily equals "Out," so it would be impossible to accumulate any extra time. The only time lost would be the 6 seconds the line stood still while the guy was sneezing.

Also, my apologies for posting this comment so late. It was a busy weekend.

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