• 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
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• 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.
