is no fun; bidders care about which other bidders win rackets as well. In such a setting, where bidders care about what other bidders win, we say that there areexternalities. Let us assume that each agent is awarded at most one racket, and that shuttlecocks and nets are freely available. In the most general bidding language for this setting, each bidder would specify, for every subset of the agents, what her value would be if exactly the agents in that subset won rackets. This is impractical because there are exponentially many subsets. Instead, we will consider more restricted bidding languages. Let us suppose that it is commonly known which agents live close enough to each other that they could play badminton together. This can be represented as a graph, which has an edge between two agents if and only if they live close enough to each other to play together. In the first bidding language, every agentisubmits a single valuevi. The semantics of this are as follows. If the agent does not win a racket, her utility is 0 regardless of who else wins a racket. If she does win a racket, her value is vtimes the number of her neighbors that also win a racket.

``````Alice
``````
``````Bob
``````
``````Carol Daniel
``````
``````Eva
``````
``````Frank
``````
``````Figure 1: Externas proximity graph.
``````
``````Suppose we receive the following bids:
``````
``````Alice
``````
``````Bob
``````
``````Carol Daniel
``````
``````Eva
``````
``````Frank
``````
``````4
``````
``````4
``````

``````Alice
``````
``````Bob
``````
``````Carol Daniel
``````
``````Frank
``````

(^12) 4 3 2 1 4 Eva 5 5 6 1 5 Figure 3: Graph with bids. Each number is the value that the closer agent on the edge has for playing with the further agent on the edge. Suppose we have four rackets for sale. One valid (but not optimal) outcome would be to pair Alice and Bob, and Daniel and Eva (and give them all rackets), for a total utility of 4 + 1 + 1 + 6 = 12. 2c. (12 points)Give the optimal outcome (pairing and allocation), as well as the VCG (Clarke) payment for each agent. 2d. (13 points)In general (general graphs, bids, numbers of rackets), is the problem of finding the optimal outcome solvable in polynomial time, or NP- hard? (Hint: think about theMaximum-Weighted-Matchingproblem. Keep in mind that the number of rackets is limited, though.)