The First One Now Will Later Be Last

How the dismal science applies to your life.
Feb. 8 2001 11:30 PM

The First One Now Will Later Be Last

A foolproof method to shorten queues.

Illustration by Robert Neubecker

You spend too much time waiting in lines. "Too much" isn't some vague value judgment—it's a precise economic calculation. A good place in line is a valuable commodity, but it's not ordinarily traded in the marketplace. And this "missing market" inevitably produces inefficient outcomes.

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If a line-placement market existed, you could pay the guy in front of you to leave, or take up a collection among the people behind you and then pay the guy in front to leave. But you don't because 1) social conventions make paying for a place in line awkward; 2) negotiating a price is a hassle; and 3) you're worried about the "free riders" behind you mooching off your investment. Because there is no market, you and the guy up front miss out on a mutually beneficial exchange, which is the precise economic definition of inefficiency.

Under the current rules, line formation suffers from economic inefficiencies because we enter lines without regard to the interests of later arrivals who queue behind us. How to make line formation more efficient? Change the rules so that new arrivals go to the front of the line instead of the back. Then the addition of a new person in line would impose no costs at all on those who come later. With that simple reform, lines would be a lot shorter. People who got pushed back beyond a certain point would give up and go home. (Well, actually they'd leave the line and try to re-enter as newcomers, but let's suppose for the moment that we can effectively prohibit that behavior.) On average, we'd spend less time waiting, and we'd be happier.

Before you start objecting, let me work through a stylized example. Imagine a water fountain in a city park where a steady gaggle of equally thirsty joggers run by. Under the current go-to-the-back-of-the-line system, each new jogger observes the line length and decides whether to join the line or run on. Because they're all equally thirsty, there's some maximum line length they're all willing to tolerate; say they're willing to wait in lines of up to 12 people. Whenever the line length falls below 12, some newcomer instantly joins and restores the length to 12.

That's disastrous. It means the line is always at the maximum length anyone's willing to tolerate. As a result, those who join the line can be no happier than those who jog on past—if they were happier, the line would grow even longer. Since the water fountain makes no one any happier, it might as well not be there in the first place.

But what if we sent newcomers to the front of the line? Because we've assumed a steady stream of arrivals, you'd never get in line while someone's still drinking. If you did, someone else would be sure to get in front of you, and then someone else would get in front of him, and you'd never get your drink. But if you're lucky enough to arrive just as someone else is finishing, you immediately take his place.

This system has the advantage that nobody ever wastes time in line. You might think it has the offsetting disadvantage that a lot of people never get to drink. But that disadvantage is illusory. Under the current system there are also a lot of people who never get to drink—namely the ones who choose not to join the line because it's too long. Under either system the fountain is in constant use, so either system serves exactly the same number of drinkers.

OK, now let's tweak the example to make it a little more realistic: Suppose newcomers arrive not in a steady stream but sporadically and unpredictably. (Note to the terminally nerdy: To make this argument precise, assume that both the arrival times and the time it takes to drink are both Poisson distributed.) Then if the line is short enough, you'll enter it, at least provisionally. But if a flood of new arrivals pushes you far enough back, you'll give up and go home.

That's exactly the behavior a benevolent social planner would prescribe. If the line is short, there's a chance it will clear out altogether, in which case your willingness to wait is socially valuable—it's insurance against the fountain's sitting idle. That's when the planner wants you to stick around, and that's when you do stick around. If the line is long, the fountain's almost sure to be in constant use anyway, so your presence serves no social purpose. That's when the planner wants you to leave, and that's when you do leave. Your private incentives lead you to act as the planner would dictate, and therefore render the planner unnecessary. That's a recipe not just for a better outcome, but for the best possible outcome. (For this wonderfully clever argument, I am indebted to Professor Rafael Hassin of Tel Aviv University.)

There's a nice analogy here with the textbook argument for free markets. The reason markets (usually) work so well is that private incentives (e.g., the profit motive) lead individuals to behave in socially valuable ways (e.g., producing products that consumers want). Letting people go to the front of the water fountain line does not create a market, but it still creates a private incentive that's directly in line with the greater social good.