• Exercise adapted from Problem 4.3:

Consider a set M of distinct items. There are n bidders, and each bidder i has a publicly
known subset Ti ⊆ M of items that it wants, and a private valuation vi for getting them. If
bidder i is awarded a set Si of items at a total price of p, then her utility is vixi − p, where xi
is 1 if Si ⊇ Ti and 0 otherwise. Since each item can only be awarded to one bidder, a subset
W of bidders can all receive their desired subsets simultaneously if and only if Ti ∩ Tj = ∅
for each distinct i, j ∈ W .

(a) Is this a single-parameter environment? Explain fully.

(b) The allocation rule that maximizes social welfare is well known to be NP hard (as the
Knapsack auction was) and so we make a greedy allocation rule. Given a reported
truthful bid bi from each player i, here is a greedy allocation rule:

(i) Initialize the set of winners W = ∅, and the set of remaining items X = M.
(ii) Sort and re-index the bidders so that b1 ≥ b2 ≥ · · · ≥ bn.
(iii) For i = 1, 2, 3, . . . , n :
If Ti ⊆ X, then:
– Delete Ti from X.
– Add i to W .
(iv) Return W (and give the bidders in W their desired items).

Is this allocation rule monotone bidder smaller leads to a smaller cost? If so, find a
DSIC auction based on this allocation rule. Otherwise, provide an explicit counterex-
ample.

(c) Does the greedy allocation rule maximize social welfare? Prove the claim or construct
an explicit counterexample.