This segment is not so much a lecture, as a meta lecture or if you wish a teaser. When one speaks about auctions, 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 ten widgets. So, maybe I have ten cameras for sale and I put them on. And imagine, for example, a sealed bid auction. What would the bidding rule look like here? You would no longer specify only a price. You would need to specify something else, for example, the price of a quantity. And so if you specify price in a quantity, you might say, I want three cameras and I'll pay you a hundred dollars for the three, but maybe you'd be happy to get also four cameras for the right price. For example, if I gave you willing to pay a $100 for three cameras, perhaps you'd be happy to get four cameras for $110. Certainly for a hundred dollars, you'd be happy to get four out of three camera or is that the case. So, one needs to specify this more complicated rules here. In general, the most general way to specify things in a sealed bid auction would be to specify what's called a schedule. A some what not 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. I can definitely find counterexamples 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 lets recall what it is. The Japanese auction, there is an ascending price called out by the auctioneer. And people in a single unit case, people start by all being in. So who will buy this camera for $0 dollars, everybody's in? And as the price rises, one by one, the bidders sit down irrevocably. And when there's only one person left standing, they get the camera for that price. What happens when I have now ten cameras for sale? What would be the natural generalization the Japanese auction? Well, here's a one natural generalization and it is specifying a quantity. So I have ten cameras for sale and I ask, how many do you want at $0? Well, presumably you want all ten. And as the price rises, you reduce your demand. And whenever you reduces, you can never bring it back up again. If it goes down to zero, then you're totally out of the game and when does the auction stop? Well, as soon as the demand is less than supply it's stop. So when the total demand by the agent at a given point is less than ten, we'll stop the auction that raises an interesting question. What happens if in a discreet case when suddenly we went from a demand of 11 to a demand of 8? To stop the auction, why do we do the two extra cameras? Well, there's a interesting topic for discussion. Let's illustrate some of the subtleties that come up when we go from a single unit to multiple units. A different class of auction is special, because they're motivated by specific applications and at auctions or advertising auctions are a good example. So we will 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 get 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 won an auction for such search terms knowing that they are run millions of times throughout the day all over the world and they're on in different places, multiple times. And so, I need to design an auction that is appropriate for this particular application. So, sponsored search options or ad options are very rich topic of research in both academia and industry. Finally, let me speak briefly about combinatorial options. Those are very interesting boats, because they are commercially very irrelevant 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 any use for only a left shoe or only a right shoe. And so, what you really want to be able to bid for both of those together. This seems frivolous, but now think about a popular consumer auction site 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 to go along to make a home entertainment system. Well, you go out there and you bid for televisions. First off, you have many televisions to bid on. Many different televisions of the same model and even more so of different models. You only need one television. So here, you are. 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 the television to display it on would be completely valueless to you and at different auctions. And so what you have here are examples of what are called complementarity and substitutability. The two TVs that 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. 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, that's the way those two goods are complementary. There are many examples of combinatorial auctions and a most famous example are the spectrum auction. You can think of the spectrum auction, these are rights to use the electromagnetical 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, say that are being auctioned. So you could have a handful say, five different bands at our auctions and many hundreds of regions. And so each sell in this matrix is a good, but you have very strong complement parity among the rows and very strong subjectability among the columns. The columns are really interchangeable, only needs one bandwidth and you don't care which it is. On the other hand, if you're a setter or 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 packet here and a little packet 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 combinatorial auction will 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 your auction to ensure the right economic properties. There are interesting computational issues. So, finding the winner in a simple English auction or the sealed-bid auction is trivial, you look at the maximum among the sealed numbers. Computing the outcome of the combinatoria 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 is called NP-complete, which means that it's an agreed upon difficult problem to compute and there's been the literature about how to do it a practical manner nonetheless. So these are examples of non-simple auctions, multiple units, sponsor search auctions, combinatorial auctions and all this by the way of whetting your appetite to learn more about these kinds of auctions.