Networks: Spring 2010 Homework 3 David Easley and Jon Kleinberg Due February 26, 2010 As noted on the course home page, homework solutions must be submitted by upload to the CMS site, at https://cms.csuglab.cornell.edu/. This means that you should write these up in an electronic format (Word files, PDF files, and most other formats can be uploaded to CMS). Homework will be due at the start of class on the due date, and the CMS site will stop accepting homework uploads after this point. We cannot accept late homework except for University-approved excuses (which include illness, a family emergency, or travel as part of a University sports team or other University activity). Reading: The questions below are primarily based on the material in Chapters 9, 10 and 11 of the book. (1) In this problem we will examine a second-price, sealed-bid auction for a single item. Assume that there are three bidders who have independent, private values v i ; each is a random number independently and uniformly distributed on the interval [0, 1], and each bidder knows his or her own value, but not the other values. You are bidder 1. You know that bidders 2 and 3 bid according to bidding rules in which they bid below their true value by certain factors. Specifically, when bidder 2 has a true value of v 2, she bids c 2 v 2, for a constant number c 2 that is greater than 0 and less than 1. Similarly, when bidder 3 has a true value of v 3, he bids c 3 v 3, for a constant number c 3 that is greater than 0 and less than 1. You know your own value v 1, and you know these multipliers c 2 and c 3, but you do not know the true values v 2 and v 3 of the other bidders. How much should you bid? Provide an explanation for your answer; a formal proof is not necessary. (2) In this problem we will examine a second-price, sealed-bid auction for a single item. We ll consider a case in which true values for the item may differ across bidders, and it requires extensive research by a bidder to determine her own true value for an item maybe this is because the bidder needs to determine her ability to extract value from the item after purchasing it (and this ability may differ from bidder to bidder). There are three bidders. Bidders 1 and 2 have values v 1 and v 2, each of which is a random number independently and uniformly distributed on the interval [0, 1]. Through having performed the requisite level of research, bidders 1 and 2 know their own values for the item, v 1 and v 2, respectively, but they do not know each other s value for item. Bidder 3 has not performed enough research to know his own true value for the item. He does know that he and bidder 2 are extremely similar, and therefore that his true value v 3 is
exactly equal to the true value v 2 of bidder 2. The problem is that bidder 3 does not know this value v 2 (nor does he know v 1 ). (a) How should bidder 1 bid in this auction? How should bidder 2 bid? (b) How should bidder 3 behave in this auction? Provide an explanation for your answer; a formal proof is not necessary. x b z c y a Figure 1: The map for a parking-space market. (Image from Google Maps, http://maps.google.com/) (3) Figure 1 shows a map of part of the Back Bay section of Boston. Suppose that the dark circles labeled x, y, and z represent people living in apartments in Back Bay who want to rent parking spaces by the month for parking their cars. (Due to the density of buildings, these parking spaces may be a short walk from where they live, rather than right at their apartment.) The dark circles labeled a, b, and c represent parking spaces available for rent. Let s define the distance between a person and a parking space to be the number of blocks
they d have to walk from their apartment to the parking space. Thus, for example, z is at a distance of 2 from space c, while y is at a distance of 1 from c and x is at a distance of 4 from c. (We ll ignore the fact that the block between Gloucester and Hereford is a bit shorter than the others; all blocks will be treated as the same in counting distance.) Suppose that a person has a valuation for a potential parking space equal to 8 (their distance to the parking space). (Notice that this formula gives higher valuations to closer parking spaces.) In terms of these valuations, we d like to think about prices that could be charged for the parking spaces. (a) Describe how you would set up this question as a matching market in the style of Chapter 10. Say who the sellers and buyers would be in your set-up, as well as the valuation each buyer has for the item offered by each seller. (b) Describe what happens if we run the bipartite graph auction procedure from Chapter 10 on the matching market you set up in (a), by saying what the prices are at the end of each round of the auction, including what the final market-clearing prices are when the auction comes to an end. You can either draw the preferred-seller graph from each round, or it is also fine to simply list the prices for each seller at the end of each round of the auction. (Note: In some rounds, you may notice that there are multiple choices for the constricted set of buyers. Under the rules of the auction, you can choose any such constricted set. It s interesting to consider though not necessary for this question how the eventual set of market-clearing prices depends on how one chooses among the possible constricted sets.) (c) At a more informal level, how do the prices you determined for the parking spaces in (b) relate to these spaces intuitive attractiveness to the people in apartments x, y, and z? Explain. (4) Please do Exercise 6 from Chapter 11 of the textbook (in Section 11.8). You do not need to hand in an answer to part (a), just (b) and (c); it is enough to draw the network for yourself without handing it in, provided you are able to provide a description and explanation for (b) and (c). (5) Let s continue thinking about the model of matching markets from Chapter 10. Since we ve seen that a single market can have several different sets of market-clearing prices, it s natural to ask how we should think about the set of all possible market-clearing prices for a given market. It turns out that there is a natural geometric view of the market-clearing prices, which can be useful in developing a more global picture of them. In this question, we give an illustration of how the geometric view works in a very simple setting. It is simple because we ll assume that there are only two sellers and two buyers, and
6 5 4 Point (1,2): seller a charges 1 and seller b charges 2. 3 2 1 0 0 1 2 3 4 5 6 Figure 2: A grid of possible prices for two items. that the two buyers agree on which item they value more; also, we ll only consider prices that are lower than the valuation that any buyer has for any item (so any buyer would receive a positive payoff from buying anything). Let s begin with the following example. Suppose we have sellers named a and b, and buyers named v and w. Each seller is offering a distinct item for sale, and the valuations of the buyers for the item are as follows. Buyer Value for Value a s item b s item v 7 10 w 8 9 for Now, let s say that a pricing of the two items is simply a choice of a price by each seller: a price p chosen by seller a, and a price q chosen by seller b. Consider the set of all possible pricings, where the price of each item is between 0 and 6 (and hence cheaper than any valuation). These pricings can be drawn as a grid of points, as shown in Figure 2, where the point (p, q) at x-coordinate p and y-coordinate q corresponds to the pricing in which seller a charges p and seller b charges q. So for instance, the point (1, 2) indicated as an example
6 5 4 3 2 1 0 0 1 2 3 4 5 6 Figure 3: Shading in the market-clearing prices in the grid. in the figure corresponds to the pricing in which seller a charges 1 for his item and seller b charges 2 for her item. Given this grid of points, we can shade in all the points that correspond to pricings that are market-clearing. If we do this for our current example, we get the shading of points shown in Figure 3. This corresponds to the set of 15 market-clearing prices (0, 1), (0, 2), (0, 3), (1, 2), (1, 3), (1, 4), (2, 3), (2, 4), (2, 5), (3, 4), (3, 5), (3, 6), (4, 5), (4, 6), (5, 6). (a) Try doing the same process for the following different set of valuations. Again, suppose we have sellers named a and b, and buyers named v and w. Each seller is offering a distinct item for sale, and the valuations of the buyers for the item are as follows. Buyer Value for Value a s item b s item v 8 11 w 7 9 for
6 5 4 3 2 1 0 0 1 2 3 4 5 6 Figure 4: The market-clearing prices in the grid lie between two parallel diagonal lines. Again, let s only consider prices between 0 and 6 (again, chosen so that the prices lie below all the valuations). Which grid points in this new example correspond to market-clearing prices? (In your answer, you can either draw a shaded grid as in Figure 3, or if it s easier, you can write out the list of grid points as we did above for our original example.) (b) If you did part (a) correctly, you should find that the set of shaded grid points corresponding to market-clearing prices consists of all the points that lie between two parallel diagonal lines. Our original example had this property too: the points corresponding to market-clearing prices lay neatly between two parallel diagonal lines, as shown in Figure 4. In fact, this is a general phenomenon. Give an informal explanation for why the prices in these two examples lie between two diagonal lines. In particular, interpret the meaning of the upper diagonal line by saying why no pair of prices above this line can be market-clearing. Then, interpret the meaning of the lower diagonal line by saying why no pair of prices below this line can be market-clearing.