Auctions

Activity (20 minutes)

Goal. Contrast a sealed second-price auction with the open £5 auction from the first class and set up the question of truthful bidding.

The deck. There is a write-on deck at /decks/auctions/main.pdf: a sheet for the sealed bids, the rules left blank, and the two cases of the weak dominance argument side by side.

Ask the class to repeat the £5 auction, but this time the highest bid wins and pays the amount of the second bid. Run it sealed: get everyone to write their bid on a piece of paper and reveal at the end.

Discussion (20 minutes)

Work through the Auctions chapter.

Discussion Point: After the definition of an auction, ask what the rules are for the sealed auction we just played.

Discussion Point: After two theorems ask why I chose the auction I did?

Answers for the deck

The rules. Everyone submits a sealed bid without seeing anyone else's; the highest bidder wins; the winner pays the second highest bid.

Should you bid your value? Write \(v\) for your value and \(b\) for the highest bid among everybody else. You win when your bid beats \(b\), and you then pay \(b\) whatever you bid, so your bid only changes whether you win, not what you pay.

Bidding above \(v\) only wins you the extra auctions where \(b\) lies between \(v\) and your bid. In those you pay \(b > v\) and make a loss.
Bidding below \(v\) only loses you auctions where \(b\) lies between your bid and \(v\), which you would have won at a profit. When you still win you pay the same \(b\) as before.

Neither can do better, so bidding your true value is weakly dominant.

Why that matters. You do not need to know anything about the other bidders to bid well, and the item reaches the person who values it most.

First price. No. If you pay your own bid then bidding your value gives you a payoff of zero whenever you win, so you should shade below it, and how far below depends on what you think the others will do.

From the activity to the exam answer

The activity above is written up as a marked exam question: Question 1 (the in-class activity) on the Auctions page, with a full worked solution. Closing the loop here is the step that helps students who find exams hard: work through that question together, or set it as the immediate follow-up, so they see the game they just played turned into a full-mark answer.

Communications

General email templates to send before and after this class. Fill in the bracketed placeholders before sending.

Before class

Hi all,

A reminder that our next Game Theory class covers Auctions.

All of the course materials, including the relevant chapter, are available
at https://vknight.org/gt/. It is worth skimming the chapter beforehand.

See you in class,
Vince

After class

Dear all,

Thanks for your work in today's class on Auctions.

A recording is available here [RECORDING LINK] and on Learning Central.

All class resources are available at https://vknight.org/gt/.

Thanks,
Vince