CS6998-3: Problem Set # 2
Due in class, on Monday October 27
Problem 1
AGroves mechanism is a sealed-bid auction where each agent ireports a valuation ˜vi, the mechanism selects
an efficient allocation Rwith respect to ˜v, and charges each agent ia payment of
qi=hi(˜v−i)−X
j∈N−i
˜vj(Rj),
where hiis a function of the other agents’ reports. In the following, assume that agent valuations are drawn
from the domain of general valuations.
(a) Show that in a Groves mechanism, an agent maximizes its utility by reporting its true valuation.
(b) Show that among all individually-rational Groves mechanisms, the VCG mechanism maximizes the
payment of each agent.
Problem 2
Consider a set of nagents whose valuations v1,v2, . . . , vnare single-minded.
(a) Show that the following nonlinear, anonymous (i.e., order 2) prices are competitive equilibrium prices.
p(S) = max
i∈Nvi(S).
That is, given any efficient allocation R, the pair hR, piis a competitive equilibrium.
(b) Exhibit a linear program for the efficient allocation problem such that the dual variables corresponding
to the “supply equals demand” constraints are nonlinear, anonymous prices. (You do not need to
exhibit the dual, but might need to derive it for yourself to ensure your primal is correct.)
Problem 3
Consider a model with a set of agents Nand a set of items Mwhere the number of items equals the number
of agents, say n. For each agent i∈Nlet vij be the agent’s value for item j∈M. Each agent has a
unit-demand valuation, meaning that the value of bundle Sto a typical agent iis
vi(S) = max
j∈Svij.
In words, given a bundle S, the agent keeps the item it values the most in Sand discards the rest (hence
“unit-demand”). Assume further that vij is of the form vij =aibj, where a1> a2> . . . > an>0 and
b1> b2> . . . > bn>0. This means that each agent strictly prefers item 1 to item 2, item 2 to item 3, and
so on.
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