Summary: CS7910 Homework 7
1. Does the revenue equivalence theorem hold even when bidders are risk-averse? Explain.
2. Is the Vickrey Clarke Groves mechanism vulnerable to collusion? Explain.
3. A combinatorial auction for edges in a graph. Suppose we are selling edges in an undirected graph
(perhaps representing network capacity). Each bidder i is interested in obtaining a set of edges that
constitute a path from some source node si to some target node ti. Any path will give the bidder the
same value vi. Receiving more than one path from si to ti is worthless (i.e. it will still give the bidder a
total value of only vi). Thus, a bid takes the form (si; ti; vi).
For example, consider the following graph.
Suppose we receive the following bids:
Bidder 1: (A;E; 4)
Bidder 2: (C; F; 2)
Bidder 3: (B;E; 1)
Then, the optimal allocation is to give edges AB and BE to bidder 1, and edges CD and DF to bidder 2, for a
total value of 4 + 2 = 6. Note that it is impossible to accept bidder 3's bid in addition: there are only two paths
from B to E, namely the one consisting of edge BE, and the one consisting of edges BC, CD, and DE. Since
edges BE and CD have been allocated already, neither of these paths can be allocated to bidder 3.
a. Compute the VCG payments for bidders 1 and 2.
b. Express each bidder's bid using the XOR language, i.e. as an XOR over
bundles of edges (with values for each bundle).