This segment is not so much a lecture as a meta lecture or a, if you wish, a teaser. When one speaks about auction, the theory is so rich that it's very easy to speak about one-sided single good auctions and spend a lot of time on it. But one shouldn't forget that there are other types of auctions that are very interesting. And I just want to alert you here to examples of such auctions. So let's start with the multiple units version of auctions. So imagine that I have not one widget to sell but 10 widgets. So maybe I have 10 cameras for sale and I put them on. And imagine, for example, a seal bid auction. What would the bidding rules look here? You no longer specify only a price, you need to specify something else, for example, a price and a quantity. And so if you specify price and quantity, you might say I want three cameras and I'll pay you $100 for the three. And maybe you'd be happy to get also four cameras for the right price. For example, if I gave you willing to pay $100 for three cameras, perhaps you'd be happy to get four cameras for $110. Certainly for $100 you'd be happy to get four and three cameras, or is it the case? And so one needs to specify these more complicated rules here. In general, the most general way to specify things in a seal bid auction would be to specify what's called a schedule, a somewhat non-standard use of the term, but it's a whole curve that says for each price, how much am I willing to, how many units am I willing to buy for that price? And presumably, and this is not universal, but you might have a monotonicity rule. So that is the higher the price, the smaller the number of units I might want to buy. One can definitely find counter examples to this, but that would be a natural restriction in certain cases. Think now about an open outcry auction. So for example, think about the Japanese auction. And let's recall what it is, the Japanese auction, there is an ascending price called out by the auctioneer and people in the single unit good, in the single unit case, people start by all being in, so who will buy this camera for $0? Everybody's in. The price rises, one by one, the bidder is set down irrevocably and when there's only one person left standing, they get the camera for that price. What happens when I have now 10 cameras for sale? What would be the natural generalization of the Japanese auction? Well, here's a one natural generalization and it is specifying a quantity. So I have 10 cameras for sale and I ask, how many do you want at $0?  If it ever goes down to zero, then you're totally out of the game. And when does the auction stop?  So when the total demand by the agent at a given point is less than 10, we'll stop the auction. That raises an interesting question. What happens if in a discrete case when suddenly we went from a demand of 11 to a demand of 8, we stop the auction, what do we do with the two extra cameras? Well, there's an interesting topic for discussion that illustrates some of the subtleties that come up when we go from a single unit to multiple units. A different class of auctions is special because they're motivated by specific applications and ad auctions or advertising auctions are a good example. So we all have the experience of going to an online site, most famously Google, and doing, for example, a search, getting our search results, hopefully accurately reflecting the relevance of the sites to us. But then on the right, we have these sponsored links. And these are links that advertisers paid to give to us and they typically tied to the search term that we have. So for example, if we type in Paris hotels, then there are advertisers out there who paid for the right to display us their advertising when we type in those terms. And so how does one run an auction for such search terms knowing that they are run millions of times throughout the day all over the world? And they are run in different places multiple times. And so one needs to design an auction that is appropriate for this particular application. So sponsored search auctions or ad auctions are a very rich topic of research in both academia and industry. Finally, let me speak briefly about combinatorial auctions. Those are very interesting both because they are commercially very relevant and because they give rise to a host of very interesting intellectual issues. So what's an example of a combinatorial auction? Imagine an auction for shoes. You could have an auction for left shoes and you could have an auction for right shoes. But that'd be very weird because you really don't have a use for only a left shoe or only a right shoe. And so you really would want to be able to bid for both of those together. This is frivolous but now think about popular consumer auction sites such as eBay. Maybe you want to put together a home theater system. So you need a television, you need a DVD player and maybe you need some other items that go along to make a home entertainment system.  First off, you have many different televisions to bid on. Many different televisions are very same model and even more so of different models. You only need one television. So here you have these two televisions that you're bidding on but you don't want to get both of them. So how can you bid on and make sure you get the television you prefer but only one? At the same time, perhaps you have here a DVD player that you're bidding on. But you really wouldn't want to get the DVD player alone. A DVD without a television to display it on would be completely valueless to you. And yet these are different options. And so what you have here are examples of what are called complementarity and substitutability. The two TVs you may be looking at are substitutes. It doesn't mean that one is completely equivalent to the other but it means that the value of both of them together to you is much less than the sum of their individual value to you. In the extreme case, it's simply the value of the maximum among the two. On the other hand, the connection between the DVD player and the television is that of complementarity. The value of the DVD player plus the television is much more than the value of the DVD player alone plus the value of the TV alone because the value of the DVD alone is zero to you. And so those two goods are complementary. There are many examples of combinatorial auctions and most famous example are the spectrum auction. You can think of a spectrum auction. These are rights for you to use the electromagnetic spectrum. And you can think of those as a matrix where the rows are the regions in question. For example, geographical regions in the US. And the columns are certain bandwidths in megahertz, for say, that are being auctions. You could have a handful, say five different bands that are auctions and many hundreds of regions. And so each cell in this matrix is a good, but you have very strong complementarity among the rows and very strong substitutability among the columns. The columns are really interchangeable. All you need is one bandwidth and you don't care which it is. On the other hand, if you're a cellular operator, for example, getting a whole contiguous segment is something you essentially need in order to run a service. You don't want to get just a little pocket here and a little pocket there. And so you really need, say, all the metropolitan areas or all of the Midwest in order to justify rolling out a service there. That's an example of where a combinatorial auction would be very natural. Interestingly, it hasn't really run this way, although there are discussions about running it quite as a combinatorial auctions. Combinatorial auctions give rise to many issues. How to design the auction, to ensure the right economic properties. There are interesting computational issues. So finding the winner in a simple English auction or in a sealed bid auction is trivial. There's a maximum among a set of numbers. Computing the outcome of a combinatorial auction is distinctly more complicated. Both computing who the winner is and in particular computing the payments can be very complicated. In the technical language, the so-called winner determination problem is what's called NP-complete, which means that it's agreed upon difficult problem to compute. And there's been a literature about how to do it in a practical manner nonetheless. So these are examples of non-simple auctions, multiple unit, sponsored search auctions, combinatorial auctions. And all of this by way of wetting your appetite to learn more about these kinds of auctions.