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.
Exercise 6.4
Exercise 7.4
Exercise 9.5
Exercise 10.5
Exercise 10.6
Comment on Exercise 9.5 Here is a more clear description of the Random Serial Dictatorship algorithm:
(i) Each agent submits a ranked list of house preferences.
(ii) The agents are randomly ordered (independent of their ranked list of house preferences).
(iii) The agents are considered in order. When agent i is considered, she receives her top ranked option that is still available.
