Is this a single-parameter environment? Explain fully. Does the greedy allocation rule maximize social welfare? Prove the claim or construct an explicit counterexample.

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:

1. (i) Initialize the set of winners W = ∅, and the set of remaining items X = M.
2. (ii) Sort and re-index the bidders so that b1 ≥ b2 ≥ · · · ≥ bn.
3. (iii) For i = 1, 2, 3, . . . , n :

If Ti ⊆ X, then:
– Delete Ti from X.
– Add i to W .

       4. (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 counterexample.

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