Overbooking

Why do airlines and hotels deliberately sell more seats and rooms than they have?

An airline sells a hundred and five tickets for a hundred seats, knowing perfectly well it cannot seat everyone who bought one. Stated that way it sounds like a swindle, or at best a gamble taken with the passenger's evening. But it is neither reckless nor secret, and the airline could stop tomorrow at the cost of flying emptier aeroplanes. So what does it know that makes selling more than it has the sober choice rather than the greedy one?

Start with the asset. A seat on a departed flight is the most perishable good there is: at the moment the door closes, an unsold seat is worth exactly nothing, forever, and every cost of carrying it has already been paid. So an empty seat is close to a pure loss of its fare.

Now the fact that creates the problem. Some passengers who hold a ticket do not turn up — a missed connection, a changed plan, a flexible fare that made not turning up cheap. If the airline sells exactly a hundred seats, it departs with fewer than a hundred passengers essentially every time. The empty seats are not a possibility to hedge against; they are a near-certainty to plan for.

So far this only argues for selling some extra. The interesting question is why it can be done without constant chaos, and that turns on the shape of the uncertainty rather than its size. Ask what governs whether any particular passenger shows up. Mostly private, unrelated things — a meeting overran, a child was ill. Their failures to appear are, to a good approximation, independent of one another.

Independence is the whole trick, because it makes the total far more predictable than any individual. Suppose the historical no-show rate on this route is 8 per cent and the airline sells 105 tickets. The expected number of no-shows is 8.4 — but the useful number is the spread, which for independent events grows only with the square root of the count: about 2.8 here. Work through it and no-shows land between 3 and 14 on roughly 97 per cent of departures. The airline is not betting on whether you show up, which it cannot predict at all. It is betting on a total whose plausible range is about eleven seats wide out of a hundred and five. On that assumption it is oversold on around 7 per cent of departures, and oversold by three seats or more on under 1 per cent.

Now the decision becomes an ordinary comparison of two costs the airline already knows. Selling one more ticket earns a fare if that seat would otherwise have flown empty, and costs a compensation payment if it turns out to be the seat too many: keep selling while the chance of an empty seat times the fare exceeds the chance of a denied boarding times what one costs. If a fare is worth 200 and getting someone off the flight costs about 800 all in, keep selling while the risk of bumping stays under about one in four. Change either number and the optimal overbooking level moves immediately.

One more piece completes it, and it is the part passengers experience. When a flight is oversold the airline does not pick a victim; it runs an auction, raising the voucher offer until enough people volunteer to travel later. That mechanism was proposed by the economist Julian Simon in 1968 and written into US practice a decade later, and it works because passengers differ enormously in what an evening is worth to them. Involuntary denied boarding — removal with no volunteer found — runs on the order of a few per hundred thousand passengers in US airline reporting, with volunteers outnumbering involuntary bumps by roughly ten to one.

The refinement that matters most is the one the analogy points at: independence is an assumption, and it is exactly the assumption that fails on the bad days. A snowstorm, a delayed inbound aircraft feeding a hundred connections, a conference block, a tour group booked as one party — each makes no-shows correlate. When they correlate, the narrow, comfortable distribution widens, and the airline is oversold by more than volunteers can absorb. Which is why denied boardings cluster on disrupted days rather than scattering evenly.

Two smaller corrections. This is not the airline gambling with someone else's evening for free — the passenger who gives up the seat is paid an amount they agreed to, which is what separates an auction from a lottery; the genuine grievance is with the involuntary case, where the price was set by regulation rather than by the person bearing the cost. And hotels sit in the same logic with different costs: a room perishes nightly too, but the remedy is to "walk" the guest to a comparable property, which is expensive and does not strand anyone, so hotels overbook more freely relative to capacity.

An honest limit on the numbers above: the 8 per cent is illustrative, not universal. Real forecasts are per-flight, built from the fare mix, the day of week, the season, and the connection pattern — and no-show rates have fallen substantially as non-refundable fares spread, which removes the slack that made overbooking so valuable in the first place.

• How a fixed statutory compensation scheme, rather than an auction price, changes what an airline chooses to sell.
• Why falling no-show rates, as fares became non-refundable, have made overbooking less useful than it once was.
• The related problem of deciding how many cheap seats to sell before holding the rest back for late, expensive bookings.

Every term the collection defines is gathered in the glossary.