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Friday, April 11, 2008

The Infinitely Bad Argument

OK this intriguing blog post (HT2MR) has taken up a good 15 minutes of my day. Summary:

You're in an airport, about to go through the security line. You sneeze, which delays you by two seconds. It doesn't just delay you by two seconds, though; it also delays everyone waiting in line behind you. And everyone who will show up while the people currently in line haven't gone through yet. In fact, if you assume that the queue is never empty...we're talking about arbitrarily large quantities of wasted time.

Some of the responses at MR pointed out holes in this logic. There are all sorts of ways that eventually people behind the sneezer end up at the same position, at the same time, that they would have in the absence of the sneeze. E.g. if the sneezer is behind someone else and the line stops moving for a few seconds, then obviously his sneeze does nothing. And even if his sneeze delayed himself and 30 people behind him, so long as there would have been a few seconds when the line weren't moving forward for the 31st person, then the sneeze did nothing.

I think this is probably "the" solution in this case to deal with the apparent paradox. However, I'm wondering if there are other, deeper flaws with the logic. For example, suppose that the line always moves, and is never empty. Then it seems that the sneeze is infinitely costly, and this just seems crazy. After all, it's not as if people start arriving 400 hours late to their destination, so something is screwy.

At first I was tempted to deal with this by saying, "Oh, it's because this 'infinite' cost is dispersed among so many different people. If you have an arbitrarily large number of people suffering a tiny cost, then you obviously get arbitrarily large aggregate costs. No paradox."

But that's not good enough. Suppose that the queue is never empty, but that it consists of the same 10,000 people who fly very often. Then you get these arbitrarily large delays spread among a finite group of people.

Indeed, if our sneezer himself will end up at the end of the queue before it empties out, his sneeze will delay HIM many times over.

So something is really screwy with this logic. And I don't think we should dispose of it with appeal to time preference. That's a definite complication and it mitigates whatever the "total cost" is of the sneeze, but again it seems to be missing what (must be) fundamentally wrong with the analysis.

One thing that occurs to me is the notion of "waste." After all, everything that happens in the queue slows people down. When they were deciding how much time to allow for the queue, they presumably took into account real world possibilities, such as people sneezing and elderly people walking slowly. So it seems to me that a guy sneezing on January 1 couldn't possibly mess up the plans of people traveling on March 15, if for no other reason than that these later travelers would take the sneeze's effects into account when driving to the airport etc.

7 comments:

  1. The main problem with this analysis is that it assumes the pause caused by the sneeze is simply additive. In reality, the pause will propagate through the line as a wave, ending its effect once it reaches the end of the line.

    As for an always-moving, never empty line, the pause would still reach the end of the line, unless the line was growing at a faster rate than the pause could propagate.

    It's the same deal as trying to get into a crowded bar. The bouncer will let one or two people in, then the line shifts forward, one or two people at a time, until the wave of movement reaches the end of the line.

    You raise an interesting point with the 10,000-people-continuously-flying scenario. In this case, there could definitely be some very large delays, because a wave traveling through a system with a feedback loop can create constructive and destructive interference. In such a scenario, given the right conditions, the security line could be hours long or non-existent. It would be possible that the sneezer could create interference in such a way so as to not wait in any line next time! Or! If a constant line speed is just an unstable solution to the line's dynamics equation, the sneeze could make the line speed (and therefore, wait time) completely unpredictable!

    I would love to see the TSA solve that problem. No 4oz bottles, and NO NON-LINEAR DYNAMICS!

    ReplyDelete
  2. Anonymous9:26 AM

    Nice hair-do, Mike!

    ReplyDelete
  3. Brian N.10:59 AM

    "The main problem with this analysis is that it assumes the pause caused by the sneeze is simply additive. In reality, the pause will propagate through the line as a wave, ending its effect once it reaches the end of the line."

    That was my objection; unless everyone sneezes, not all at once but under the same condition sneezer one did, then we have no reason to consider anything other than that single, solitary 2-second delay.

    ReplyDelete
  4. Mike,

    OK I love your point about the sneeze hitting the end of the line (unless the line is growing, which nobody asserted). I can't believe I didn't think of that--and I don't think anyone at MR did either. In any event, I'm sending this to Cowen.

    Brian N: I'm not yet convinced on your extension. If the line is always 50 people long, but it's always moving--so one person always shows up at the end of the line just as the first person goes through security--then the 2-second delay causes a total of 100-person-seconds of lost time, given the setup. Are you disputing that? I.e. are you saying it's just the first person who is delayed?

    ReplyDelete
  5. Brian N.8:05 AM

    No. I'm only saying that there's the one delay. It may, in effect, become a 100 person/second delay, considering everyone hindered, but it doesn't have the self-propagating (probably not the right word) effect. In order for that to occur, as I've said, everybody would have to, once they got to the front of the line, sneeze. But since TSA officers are known sadists and would obviously not be averse to throwing sneeze powder at their victims, it's a given.

    ReplyDelete
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