Stat 155 Fall 2022 Homework 4
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Homework 4
Stat 155 Fall 2022
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• 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 vi xi − 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:
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:
2022-10-21