Figure 10.15 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 the person would have to walk from their apartment to the parking space.
Thus, for example, z is at a distance of 2 from space c, whereas y is at a distance of 5 from c and x is at a distance of 6 from c. (We ignore the fact that the block between Gloucester and Hereford is a bit shorter than the others; all blocks are treated as the same in counting distance.)
Suppose that a person has a valuation for a potential parking space equal to
(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 setup, 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 part (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..
[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 part (b) relate to the intuitive “attractiveness” these spaces would have for the people in apartments x, y, and z? Explain.
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