Last week United overbooked a flight and needed passengers in Chicago to give up a seat. After United was unable to find volunteers, airport security forcibly and violently removed a passenger.

William Spaniel, of GameTheory101, asked why United didn’t consider using game theory to solve the problem.

Thanks @united for giving me a lesson plan! pic.twitter.com/5hFi1OHFt5 — William Spaniel (@gametheory101) April 11, 2017

If you can’t read the tweet, here is the scheme. United could have asked each passenger to write down how much they valued being on the flight. The person with the lowest value is bumped and then paid the second-lowest value.

This is an example of a Vickrey auction (second-bid auction), and it can be proven to be efficient–the logic is explained below.

So why didn’t United use such a procedure, and why do we rarely see Vickrey auctions in practice? I’ll review the theoretical advantage of the procedure and then mention a few practical issues.

But not all is lost. Since 2011 Delta has employed a very similar auction that has led to fewer involuntarily bumped passengers. I explain this system and how it gets around the problems of the Vickrey auction below.

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"All will be well if you use your mind for your decisions, and mind only your decisions." Since 2007, I have devoted my life to sharing the joy of game theory and mathematics. MindYourDecisions now has over 1,000 free articles with no ads thanks to community support! Help out and get early access to posts with a pledge on Patreon. .

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Theoretical advantages of a Vickrey auction

A Vickrey auction involves everyone submitting a bid for an item, in secret, and then awarding the item to the highest bid. That person pays the price of the second highest bid.

When bumping passengers, the “bids” are the amounts each passenger is asking for compensation. From United’s perspective, this is a cost, so each bid is a negative number. The passenger asking for the least amount of money is the lowest negative number (so this is the largest bid from United’s perspective). So the Vickrey auction works the same way: the passenger asking for the least money gets awarded the amount of the passenger asking for the second least amount.

In theory, there is no incentive to for any passenger to lie. Each passenger should honestly write down how much they value being on the flight. Why is that?

Obviously you would not want to write too small of a number, or else you might be bumped and compensated less than you value the flight.

Similarly, you do not want to inflate your bid. If you write down an absurdly high value (say $100,000), when you actually valued it at $1,000, then there will be times you might have been bumped and paid more than $1,000.

The amount you bid exactly determines when you get bumped: if you bid your value, you will only get bumped if you can get paid at least your value.

Every passenger has a weakly dominant strategy to write down their honest value.

Furthermore, if all passengers do this, then the person with the lowest value is bumped. This is economically the most efficient outcome as the person most willing to de-board is identified and compensated.

So this idea should work. But why don’t we see it in practice more?

Practical issues

Two auction theory professors, Lawrence M. Ausubel and Paul Milgrom, write about The Lovely but Lonely Vickrey Auction:

“Despite the enthusiasm that the Vickrey mechanism and its extensions generate among economists, practical applications of Vickrey’s design are rare at best.”

So why don’t more people use a second-bid auction?

One problem is people may not trust the seller or auctioneer to be truthful. Imagine you wrote $1,000 for your value as the lowest, and United is supposed to pay you $10,000 which is the next lowest. But United is the only one that sees all the values, so it could make up that $1,050 was the next highest value and stiff you on compensation. In other auctions, a seller might organize shill bids to artificially raise the selling price. If people do not trust the auction, then there is no reason to bid honestly, which ruins some of the point of a Vickrey auction.

A second issue is bidders can collude. For example, if enough people on the flight wrote down absurdly large values like $1 million, then United might get stuck with paying a ridiculously large bill.

This second issue is a particularly worrisome possibility in practice. In theory, second price auctions should raise the same amount as first price auctions: in the first price auction bidders “shave” their bids to about what the next highest value would be. But in practice, sellers are scared of low bids in auctions, and Vickrey auctions may generate less money than first price auction where people overbid irrationally (in a first price auction, a seller benefits when a single person drastically overbids–in a second-price auction the overbid is negated if the next highest bid is reasonable).

Vickrey auctions are interesting theoretically, but they are not so easy to implement in practice. That’s why Delta implemented a slightly different auction.

A modified auction from Delta

Delta airlines does a kind of blind auction: when a flight is overbooked, it asks passengers during check-in how much they are willing to be compensated to be put on a later flight. The instructions indicate that Delta accepts lower bids first.

This is a first-price auction–people offer an amount and they actually get that amount. Passengers can offer a fair value and if Delta accepts the offer they get that amount–there is no worry Delta will manipulate the bids. This takes care of one concern of the Vickrey auction.

Second, passengers will find it harder to collude as each passenger bids secretly while checking in.

As an added benefit, the bidding process happens before people are on the flight, so there is an efficiency to bumping passengers earlier and allowing them time to re-arrange travel plans.

(Incidentally, what is your best strategy for this auction? Let’s say you would be willing to get bumped for $1,000 minimally. If you accept this offer, you don’t profit from winning, you just break even. If you ask for an absurdly high amount, on the other hand, you are unlikely to win at all. What you want to do is pad your offer to be slightly lower than the next lowest person’s offer. This way you get your offer, and you profit as much as you can.)

Has Delta’s auction method worked? Per PBS, in 2014 Delta bumped 3 passengers involuntarily per 100,000 versus American bumping 5 and United bumping 11. This translates into thousands of passengers fewer per year, and fewer chances for a controversy when bumping a passenger involuntarily.

United is likely going to try something to fix its reputation and avoid another incident. Perhaps United will also use the lessons of auction theory and strive for a more efficient outcome for everyone.