Networks: Fall 2010 Homework 3 David Easley and Jon Kleinberg Due in Class September 29, 2010

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1 Networks: Fall 00 Homework David Easley and Jon Kleinberg Due in Class September 9, 00 As noted on the course home page, homework solutions must be submitted by upload to the CMS site, at 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 Universityapproved 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, 0 and of the book. () 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, ], and each bidder knows his or her own value, but not the other values. You are bidder and your value for the item is v = /. You know that bidder bids optimally given her value for the item. You know that bidder seems to like to win the auction even if winning requires him to pay more than his value for the item. In particular, bidder always bids / more than his value for the item. How much should you bid? Provide an explanation for your answer; a formal proof is not necessary. () (This is Exercise 0 from Chapter 9.) 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 and have values v and v, each of which is a random number independently and uniformly distributed on the interval [0, ]. Through having performed the requisite level of research, bidders and know their own values for the item, v and v, respectively, but they do not know each other s value for the item. Bidder has not performed enough research to know his own true value for the item. He does know that he and bidder are extremely similar, and therefore that his true value v is exactly equal to the true value v of bidder. The problem is that bidder does not know this value v (nor does he know v ).

2 (a) How should bidder bid in this auction? How should bidder bid? (b) How should bidder behave in this auction? Provide an explanation for your answer; a formal proof is not necessary. () Consider a trading network with intermediaries in which there are two sellers S, S, two buyers B, B and two traders (intermediaries) T, T. The sellers can each trade with only one of the traders: seller S can only trade with trader T ; and seller S can only trade with trader T. The buyers can each trade with only one of the traders: buyer B can only trade with trader T ; and buyer B can only trade with trader T. The sellers each have one unit of the object and value it at 0; the buyers are not endowed with the object. Buyer B values a unit at and buyer B values a unit at. To answer the questions below it will help if you draw the trading network, with the traders T and T, the buyers B and B, and the sellers S and S as nodes, and with edges connecting nodes who are able to trade with each other. (You do not need to hand in the pictures of the network; it is enough to draw the network for yourself without handing it in, provided you are able to provide a full description and explanation for (a) and (b).) (a) Find Nash equilibrium bid and ask prices for this trading network. How much profit do the traders make? (b) Suppose now that we allow both sellers to trade with either trader and both buyers to trade with either trader. Is there a Nash equilibrium in this new trading network in which one or both of the traders makes a profit? Explain. () Consider a popular tourist location where residents often try to rent out their houses for months when they will be away. Thus, particular houses are empty for particular month-long periods, and potential renters express an interest in renting them. Each renter has a month when they would ideally like a house, but they are willing to consider renting for other months, with their interest falling off for months that are further from their ideal month. Suppose there are three people who have expressed an interest in renting. Person x would ideally like to rent a house for June; Person y would ideally like to rent a house for July; and Person z would ideally like to rent a house for August. There are three houses currently available: the first is empty in February, the second is empty in March, and the third is empty in September. We ll assume the three houses are essentially identical except for the different months in which they re available to rent. For each person, their valuation for a house that is available in month M is equal to 7 (the gap in time between the start of M and the start of their ideal month). This gives higher valuations to months that are closer to someone s ideal time; for example, person x s valuation for September would be (since there is a -month gap between June

3 and September, and 7 = ), while person y s valuation for September would be (since there is a -month gap between July and September, and 7 = ). (a) Describe how you would set up this question as a matching market in the style of Chapter 0. 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 0 on the matching market you set up in (a), by saying which constricted set you find in each round of the auction, what the prices are at the end of each round, and what the final market-clearing prices are when the auction comes to an end. 6 Point (,): seller a charges and seller b charges Figure : A grid of possible prices for two items. () Let s continue thinking about the model of matching markets from Chapter 0. 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.

4 Figure : Shading in the market-clearing prices in the grid. 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 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 0 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, where the point (p, q)

5 Figure : The market-clearing prices in the grid lie between two parallel diagonal lines. 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 (, ) indicated as an example in the figure corresponds to the pricing in which seller a charges for his item and seller b charges 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. This corresponds to the set of market-clearing prices (0, ), (0, ), (0, ), (, ), (, ), (, ), (, ), (, ), (, ), (, ), (, ), (, 6), (, ), (, 6), (, 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.

6 Buyer Value for Value a s item b s item v 0 8 w for 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, 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 marketclearing prices lay neatly between two parallel diagonal lines, as shown in Figure. 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. (6) (This is Exercise from Chapter 0.) In Chapter 0, we discussed the notion of social-welfare maximization for matching markets: finding a matching M that maximizes the sum of buyers valuations for what they get, over all possible perfect matchings. We can call such a matching social-welfare-maximizing. However, the sum of buyers valuations is not the only quantity one might want to maximize; another natural goal might be to make sure that no individual buyer gets a valuation that is too small. With this in mind, let s define the baseline of a perfect matching M to be the minimum valuation that any buyer has for the item they get in M. We could then seek a perfect matching M whose baseline is as large as possible, over all possible perfect matchings. We will call such a matching baseline-maximizing. For example, in the following set of valuations, Buyer Value for Value for Value for a s house b s house c s house x 9 7 y 9 7 z 0 8 the matching M consisting of the pairs a-x, b-y, and c-z has a baseline of 8 (this is the valuation of z for what she gets, which is lower than the valuations of x and y for what they get), while the matching M consisting of the pairs b-x, c-y, and a-z has a baseline of 7. In fact the first of these example matchings, M, is baseline-maximizing for this sample set of valuations. 6

7 Now, finding a perfect matching that is baseline-maximizing is grounded in a kind of egalitarian motivation no one should be left too badly off. This may sometimes be at odds with the goal of social-welfare maximization. We now explore this tension further. (a) Give an example of equal-sized sets of sellers and buyers, with valuations on the buyers, so that there is no perfect matching that is both social-welfare-maximizing and baselinemaximizing. (In other words, in your example, social-welfare maximization and baseline maximization should only occur with different matchings.) (b) It is also natural to ask whether a baseline-maximizing matching can always be supported by market-clearing prices. Here is a precise way to ask the question. For any equal-sized sets of sellers and buyers, with valuations on the buyers, is there always a set of market-clearing prices so that the resulting preferred-seller graph contains a baseline-maximizing perfect matching M? Give a yes/no answer to this question, together with a justification of your answer. (If you answer yes, you should explain why there must always exist such a set of market-clearing prices; if you answer no, you should explain why there can be examples in which a baselinemaximizing matching cannot be found in the preferred-seller graph resulting from marketclearing prices.) 7