Now the first two auction types I talked about, English auctions and Dutch auctions, are what are called open auctions. Everyone just sits there and kind of submits their bid, and then kind of look around them. And maybe even make guesses about other people that might bid against them and what they might bid, right? That certainly happens in those kinds of auctions. There are other types of auctions that are called sealed-bid auctions. In those situations, bidders submit their bid independently and they're not privy to the other bid values. So you can imagine it, literally writing down how much you're bidding on something, sealing it in an envelope, turning it into the auctioneer, and then the auctioneer opens all the envelopes. Now in the real world, of course, you may not have to put it in an envelope, but that's kind of the way you want to think about it in your head. So you really don't know what's going on. You may not even know how many other bidders there are. So there's a couple of different types of sealed-bid auctions. A first-price sealed-bid auction, where the highest bidder wins, and a second-price sealed-bid auction, where the highest bidder also wins but doesn't pay the amount that they wrote down. They pay the second highest bidder's bid. Now recall from just a little while ago, that's what I put you through when you did the auction pricing for the $20 bills. I had you do a first-price sealed-bid auction for that $20 bill, and I had you do a second-price sealed-bid auction for that $20 bill. And I said, I generally make more money on the second-price sealed-bid auction. So let's bore down into that a little bit and find out why that's the case, and maybe when first-price may be better than second-price and vice versa. So let's consider a simple example. So we have four people, Bob, Raj, Sally, and Nora. And the blue bar on the graph in front of you shows the actual value they have for the item that they're biding on. Now, of course, they're not telling you what their actual value is. They may just know this in their head, right? Now suppose they’re putting in a bid for a first-price sealed-bid auction. They're not going to bid their full private valuation of the item. They're going to bid a little bit less than that, hoping they can get a deal on the item. So the orange bar in front of you is kind of what they might put in as a bid. So in this case, Nora bids the most, she has the highest valuation. But she doesn't bid all the way up to her $4 valuation, and so forth for each of these individuals. Now what happens in the outcome? Nora's going to win, but she's not going to pay $4. She's going to pay something like $3.20 because her bid is the highest, but there's a gap. She gets sort of 80% surplus, $4 minus 3.20, because she had an incentive to shave her bid down. Now if we had the same private valuations but asked them to go through a second-price seal-bid auction, everyone would bid their full private valuation. I mean, why not? You know you don't have to pay what you bid. So you might as well bid your full private valuation in that situation. Now, with these underlying preferences or underlying valuations, Nora would still win. She would bid $4, but she would pay what Sally bid, which is $3. Now, if you notice, what happened here is that the first-price sealed-bid auction actually yielded more money for the sellers than the second-price sealed-bid auction, for this particular circumstance. So, right, first-price beats second-price. Now let's look at one other situation, okay, that I think will illustrate the differences between the two auction types. We have Bob, Raj, Sally, and Nora, again. But this time, I've changed their private valuations. So they range kind of in the $2 range, but there's not much variance in their private valuations. Look, before there was quite a bit of variance. They range from $1 all the way up to $4. In this case, it's much less than that, right, just in that $2 range.. If we put them through a first-price sealed-bid auction, what will they do? Of course, they'll shave their bids down just a little bit. And Nora will win and she'll get the item for something like $2 and, let's say, 8 cents. However, if we put them through a second-price sealed-bid auction, what happens? The left hand side is scaled a little bit differently, so don't let that fool you. What'll happen is they'll bid their full private valuation. Nora will win and pay Sally's valuation or bid. In this case, Nora will win and pay $2.50. Notice that in this particular example, the second-price sealed-bid auction yielded more money for the sellers than the first-price sealed-bid auction. Why was that the case? Well, first-price works well when there's a lot of variance in the private valuations. And the reason that's the case is because if there's a lot of variance, the difference between the value of the person who values it the most and second most might be quite large. So if you do a second-price sealed-bid auction, that first person, the person that values it the most, is going to get it for the second highest bidder's valuation. That gap could be very large and you're giving away a lot of money. However, if there's less variance in the underlying valuation, that means Nora and Raj's valuations are very close to each other. So even if the highest bidder gets it at the second highest bidder's bid, the bids won't be very far apart, so you're giving up less money. At the same time, you're incentivizing each of the bidders not to shade their bid down, but to bid their full private valuation. So that second-price sealed-bid auction generally does well in those low variance situations. And one of the low variance situations was that $20 bill, right? In fact, the variance on the value of that is absolutely zero, and that's why you make more money on a second-price sealed bid auction when you're auctioning off that $20 bill.
