A couple of weeks ago, I promised a couple of new ideas for increasing the blog’s level of activity. Here is one of them: more half-baked posts! This post’s topic is one I’ve been wrestling with for a couple of days, and it’s turned out to be surprisingly difficult to wrap my head around. I’m not sure at all I’ve reached the right conclusion, but I’m posting it anyway to see if my readers (especially the mathematically and economically inclined) can confirm or correct my thinking.
As I stood in line to buy popcorn at the movie theater on Saturday, I noticed that the young couple in front of me had split between two lines: the girl stood in my line, the boy in the next line over. When it became clear that the boy’s line was moving faster, she joined him. This is a strategy that I often use myself when I have a companion. I have a vague impression, however, that there’s a mild social stigma against it. No one will actually call you out for it, but they might give you the stink-eye. When I do it, I generally try to be subtle – no words, just gestures.
But as I stood in line Saturday, I began to think the stigma is unjustified. The line-splitting strategy does not just help those who employ it; it also performs a useful social service. The line-splitters improve the efficiency of the system by allocating more customers to the faster checkers, thereby lowering the average waiting time. They also tend to equalize the waiting time across lines, much as they would if they knew in advance which checker was fastest and simply chose that line – yet the effect occurs without their actually possessing such knowledge.
Or at least, that was my logic while standing in line. I later formalized my logic like so. Suppose there is one fast lane and one slow lane. The lines are currently equal in length because customers don’t know which lane is faster. Just arriving at the lines is one couple (C), followed by two singles (S1 and S2). Assume the singles will not switch lines after joining, perhaps because new customers fill in behind them. Are S1 and S2 better off when splitting is allowed or banned? The key insight is that regardless of the policy, someone, either S1 or S2, will have to wait behind C to get served. If the policy allows splitting, C will always end up getting served in the fast lane, so either S1 or S2 (whoever joined the fast lane) will have to wait through one more fast transaction. If the policy bans splitting, then about half the time C will choose the fast lane by pure luck, and the result be just as if they had split between lines. But the rest of the time C will unluckily choose the slow lane, and that means either S1 or S2 (whoever joined the slow lane) will have to wait through one more slow transaction.
More simply: Someone has to wait for the couple to get served. But how much time it takes for them to get served depends on where they get served, and splitting assures that they get served in the fast lane. Therefore the expected waiting time for whoever’s behind them goes down.
Now, however, I think this logic is incomplete. Why? Because it doesn’t extend easily to the case in which lots more customers fill in the lines behind C, S1, and S2. If the new arrivals simply filled in the lines in equal numbers, then the analysis would clearly apply. In fact, it would be strengthened, because more people would share in the benefit of assuring the couple gets served in the faster lane. But if one lane is faster than the other, then it will get shorter more quickly, and thus more new arrivals will join that lane (without even knowing that it’s faster). That means a larger number of people will end up waiting behind the couple in the fast lane, while a smaller number will benefit from not having to wait behind the couple in the slow lane. It turns out that this effect cancels out the efficiency gain I showed above. For instance, if the fast line is twice as fast, it will serve twice as many customers. Moving a couple from the slow lane to the fast lane will save each slow-lane customer twice as much time as each fast-lane customer loses (this is why it appeared, in the two-singles case above, that switching allowed an overall gain in efficiency). But twice as many fast-lane customers suffer the loss, so the scale balances.
And now I may have an explanation for the taboo. Line-splitting, while having no efficiency effect, does have a distributional impact. The couple assures itself of getting served in the fast lane. As the single customers fill in the lines behind, they will be just slightly more likely to end up in the slow lane than they would have otherwise, because that’s necessary to balance the ratios. So the couple’s gain comes at the singles’ expense, although on average waiting time does not change.
Then again... Well, let me stop here. I think there’s still a flaw in my analysis, but I can’t quite put a finger on it, so I’ll leave that part to you. Bonus points if you see the analogy with insider trading.