Editor's puzzle

Abstract

As usual, you are allowed to use computers to help you solve this puzzle, but the final argument should be human-readable and as strong as possible. Solutions should be sent to the editor at conitzer@cs.duke.edu with subject header SIGecom Exchanges Puzzle. The author(s) of the most elegant solution (as judged by the editor) will be allowed to publish his or her or their proof in the next issue of the Exchanges (ties broken towards earlier submissions). To make the solution accessible to a wide audience, try to minimize technical jargon in the proof. The editor will not give any feedback on submitted solutions and ignore any requests for hints, etc. Also, the previous issue’s puzzle on combinatorial auction winner determination was solved by each of Daniel Lehmann, Peter Stone, and Mukund Sundararajan, who submitted very similar solutions, but they did not consider their solutions elegant enough to publish. If someone submits a more elegant solution to that puzzle, I will be happy to publish it in the next issue. Hint (for that puzzle): think about prices and duality, though integrality plays a role... Now, let us get to this issue’s puzzle: