The Vickrey Auction

You walk into the room and scan the seats, looking for the two girls that have caught your eye, but they’re nowhere to be seen. However, you spy two bags next to Jack that looks like theirs, and you grudgingly walk up to figure out what’s up.

“You looking for the girls?” Jack asks, staring at his phone as you approach. “Helen and Mal just stepped out a minute ago with the prof.”

“Oh cool, so they’re coming back,” you say, plopping down in a free chair next to theirs.

{if Jack < 15} “Yeah, but not for you,” he scoffs. “So stay away from Helen. She’s mine.”{else} “Friendly tip?” he says. “Stay away from Helen. She’s mine.”{endif}

Just as Jack finishes speaking, a door by the front of the lecture hall swings open, and Professor Ward steps up, laughing about something with Helen and Mal. The two girls make their way up the steps, greeting you and Jack with hugs, and then class begins.

“Let’s move on to some game theory now,” Professor Ward says. “We’ll need it to do a proper study of firm dynamics, but the solution methods and techniques are broadly applicable to virtually any problem you can think of…”

The lecture quickly progresses to a discussion of dominant strategies and Nash equilibria, which sounds intimidating when the professor first brings it up, but as she explains the concepts, you realise it’s really not that hard at all.

“Now comes the fun part of the class,” Professor Ward says, a smile crossing her face. It’s certainly not her regular expression, but you realise she looks pretty darn cute when it is there. That, and the button-down she’s got on does nothing to hide the shape of her bust. “I do this every year, but it’s usually something a little tamer… I figured we would do something more exciting. We’re going to auction off two dates.”

The class freezes for a second, trying to understand what was just said. You look over at Jack, who has a more-than-excited look on his face on, and then at Mal, who winks at you, and then at Helen, who with her coy smile, refuses to even meet your eye.

“We’ll be illustrating several concepts in class at once. The first, relative price—now, I can’t make you students actually pay me for stuff; some stupid rule from administration there. But we’re economists here, so we know that people value things other than money… and so instead of auctioning off dates for money, it’s going to be dates for detention time.”

There’s a little more stirring in the classroom at that, people muttering to each other about what that could actually mean.

“When you bid for the date, you will write down a number. This number will be your price, but it will be in minutes of private detention that you have to spend with me, not dollars. The dates will last one hour. How much more time are you willing to give up for it?

“The second concept that we’ll use here is auction design. Most of you are aware of the fact that the standard English auction—people bidding up one another until we get to a final price—is efficient in that the bidder with the highest valuation of the object will get it. What’s less obvious, but really actually pretty obvious when you think about it, is that the winner pays the valuation of the second highest-bidder.

“Today, we’re going to be using a different mechanism—the second-price sealed bid auction, otherwise known as the Vickrey auction—which yields identical results. What that means is that you will each write in your valuation of a date—and you can only pick one person to go out with—and the winner will pay the ‘price’ of the second highest bidder.

“Now, for the moment you’ve all been waiting for. Who are our fine girls for the dates? Come up please!”

With that, Helen and Mallory make their way to the front of the classroom, beaming and curtsying. The guys in the room look impressed. The girls, a little less so, but you see a couple of them eyeing the two up. A chance for a threesome? The idea flashes through your head, but you’ll worry about that if you can actually get one of the girls, first.

As the two girls walk up and down the aisles, distributing envelopes, you ponder long and hard on what your move should be. You can only pick one name. Who should you pick? And… you could put down any time you want, but some would obviously be too low (you’d sacrifice more than a minute of detention time to go out with one of them), but others would be too high (you wouldn’t do three days, surely, or even twelve hours… the opportunity cost would be too high; think of all the other girls you could meet in the meantime!)

Economic theory, according to Professor Ward, says that you should bid your true value—the amount of time spent in detention that would make you indifferent between going out on the date and serving detention, or getting neither. So what is that amount, and who will it be?

Helen +1
Mallory +1
Jack +1