Doing Lines

You’re at the grocery store, and you’ve just finished filling your cart with goodies. As you approach the check-out area, you see there’s a line at every counter. Which line do you join?

You might first decide to join the shortest line. If other customers tend to follow the same policy, the equilibrium outcome is equalization of customers per line. No line should differ from another by more than one customer. And note that this outcome will occur even if some customers choose randomly, so long as the number of customers who join the shortest line is sufficiently large. If random customers make some lines longer than others, the non-random customers will avoid the long lines and fill up the shorter lines until the lines are equalized.

But perhaps customers are more sophisticated. They might look into other customers’ carts to see how many items they’re buying, and then join the line with the smallest total number of items to be bought. The resulting equilibrium would involve equalization of items per line. And again, the predicted outcome occurs so long as a large enough number of customers pay attention to the number of items per line.

If customers are yet more sophisticated, they’ll realize that both number of customers and number of items matter. Every item requires one scan; every customer requires one payment phase. Given two lines with equal numbers of items, you’d prefer the line with a smaller number of customers (with more items each). Suppose that a payment phase takes m times as long as an item scan. Then in equilibrium, we should expect equalization across lines of the time function

T = i + mc

where i is the number of items and c the number of customers.

But there’s one more factor that matters: cashier speed. Let’s suppose customers know the speeds of the cashiers. Then if we let s stand for a given cashier’s speed, equilibrium implies equalization across lines of the time function

T = [i + mc]/s

This function assumes that, while cashiers may have different speeds, each cashier has the same ratio (m) of scanning speed to payment speed. Again, not all customers need to pay attention to this function for the equilibrium to occur, so long as enough customers do.

In all four cases, the interesting result is that you – the marginal customer – can probably do just fine without paying attention. Why? Because other customers’ behavior has already done the job of (nearly) equalizing the expected waiting time. If few or no other customers paid attention, then you could probably shorten your waiting time by looking around for the best line. But my experience is that, by the time I arrive at the front of a crowded grocery store, I might as well pick a line randomly.

Of course, most customers don’t typically know the relative speeds of cashiers, unless they happen to visit this grocery store a lot. I surmise that the number of such customers is not very large, and as a result, the actual equilibrium principle at work is probably closer to the second-to-last one (equalization of T = i + mc across lines), rather than the more complex one based on cashier speed. Still, as just another ignorant customer, your odds of shortening your waiting time are pretty low. Although some lines have shorter waiting time than others, you don’t know which ones. To improve your expected (ex ante) outcome, you’d have to know something the other customers didn’t – like the fact that cashier #3 has arthritis. And as soon as enough other customers knew it, too, the information would cease to be useful for shortening your waiting time.

An example. Suppose there are two cashiers, cashier #1 is 1.5 times faster than cashier #2, and all customers have the same number of items. Then to minimize the true waiting time, for every 10 customers, 6 should be in lane #1 and 4 in lane #2. But what if some customers don’t know about the fast cashier, so they choose the shorter line instead? These customers will keep choosing line #2 until the line lengths are equalized. However, customers “in the know” will then flock in greater numbers to line #1. If at least 6 out of 10 customers know about the fast cashier, we’ll end up with 6 customers in lane #1 and 4 in lane #2. And everyone’s waiting time will be about the same, whether they’re “in the know” or not.

But what if fewer than 6 in 10 customers know about the faster cashier? All those who do know will go to lane #1. The ignorant customers will then fill out the lines to the equalization point (5 and 5). As a result, any new customer who knows about the faster cashier can get a shorter waiting time by quickly sizing up the situation and choosing lane #1. Information only makes a difference if the number of people who have it is small.

If all of the above made sense to you, you’re well on your way to understanding the efficient capital markets hypothesis (read this for more).