Going once, going twice

A rare trading card is up for sale. Four bidders want it: you, Robo, Robo 2, and Robo 3. Every bidder has a private value: the most they would pay for the card and still be glad they bought it. Your value is shown to you. The robots' values are secret. Each auction, every robot gets a fresh value somewhere between $10 and $100, every amount equally likely.

Winning is not the goal. Profit is. If you win at price p, your profit is your value minus p. If you do not win, your profit is 0. Paying $80 for a card you value at $60 is not a victory. It is a loss of $20.

The auction you probably picture is the English auction: the auctioneer calls a price, everyone willing to pay it stays in, and the price climbs by $5 until one bidder is left. That bidder pays the last price called. Play a few. Each time a price is called, stay in or drop out.

Every robot follows one rule: stay in while the price is below my value, drop out the moment it is not. It is the right rule for you too. Say your value is $60 and the price called is $45. Staying in costs nothing yet, and if everyone else drops out the card is yours for $45, a $15 profit. Dropping out throws that away. Once the price called reaches $60, staying in can only win you the card at $60 or more: profit zero, or a loss. So the plan is: stay until the price passes your value. Nothing the other bidders do changes it.

Look at what the winner pays. Not their own value. They pay the price at which the second-to-last bidder gave up, which is roughly the second-highest value in the room. Hold onto that.

Change the rules. Each bidder writes one number in an envelope. The envelopes are opened, the highest number wins, and the winner pays the number they wrote. This is a first-price sealed-bid auction.

The English auction had an easy rule. This one does not. Say your value is $60. If you write 60 and win, you pay $60 for a card worth $60 to you. Profit: zero, guaranteed. Bidding your full value can never earn you a cent. So you must bid less. But bidding less means you might lose to a robot that wrote $52 when you would gladly have paid $55. A lower bid gives you a bigger profit if you win and a smaller chance of winning. That tradeoff is the whole game.

The robots know this. Each one shades its bid: it writes down three quarters of its value. A robot that values the card at $80 writes $60. Your job: pick a bid, ten rounds in a row, and finish ahead.

Here is a small change with a huge effect. Envelopes again, and the highest number still wins. But the winner pays the second-highest number, not their own. This is a second-price auction, also called a Vickrey auction (say: VICK-ree) after the economist who worked out what it does.

In a second-price auction, writing down exactly your true value is a dominant strategy, Chapter 2's word: it is at least as good as every other bid no matter what the other bidders write. No shading. No guessing what Robo will do. No tradeoff. Just tell the truth.

That is a strong claim, so test it before you believe it. Set your value V. Set R, the highest number anyone else wrote (only the highest matters, because that is the price you would pay). Then slide your own bid B around and watch your profit.

Slide R anywhere. The yellow dot, your profit when you bid V, is never below any other point on the curve. Other bids sometimes tie it. None beat it. Here is the proof, in two cases.

Notice the trick. Your bid decides whether you win, but never what you pay. Since your bid cannot touch your price, there is nothing to gain by lying about it. Now play it for real. The robots bid their true values; they have read the proof.

One more. In a Dutch auction the price starts high and falls. A clock ticks down from $100, and the first bidder to shout "Mine" takes the card at whatever the clock says. Flower sellers in the Netherlands have sold their flowers this way for more than a century, which is where the name comes from.

It looks like a different game from sealed envelopes. It is not. Before the clock starts you have to decide the number you are waiting for: the price at which you will shout. Shout too early and you pay too much; wait too long and a robot beats you to it. That waiting number is your sealed bid. The first-price auction and the Dutch auction are the same game in different clothes: the highest number wins, and the winner pays their own number. The robots below wait for three quarters of their value, just as before. Try it by hand first, then type a waiting number and let the clock shout for you.