08.0-pp-39-54 Lecture 4 Algorithmic Mechanism Design
Exercises
Exercise 4.1 Consider an arbitrary single-parameter environment, with feasible set X. Prove that the welfare-maximizing allocation rule
x(b) = argmax(x1,…,xn)∈X
n∑
i=1
bixi (4.2)
is monotone in the sense of Definition 3.6.
[Assume that ties are broken in a deterministic and consistent way,
such as lexicographically.]
Exercise 4.2 Continuing the previous exercise, restrict now to fea-sible sets X that contain only 0-1 vectors—that is, each bidder either wins or loses. We can identify each feasible outcome with a “feasible set” of bidders (the winners). Assume that for every bidder i, there is an outcome in which i does not win. Myerson’s payment formula (3.5) dictates that a winning bidder pays her “critical bid”—the infimum
of the bids at which she would continue to win.
Prove that, when S∗ is the set of winning bidders under the allocation rule (4.2) and i ∈ S∗, i’s critical bid equals the difference between (1) the maximum social welfare of a feasible set that excludes i; and (2) the social welfare ∑ j∈S∗\{i} vj of the bidders other than in the chosen outcome S∗.
[In this sense, each winning bidder pays her “externality”—the welfare
loss she imposes on others.]
Exercise 4.3 Continuing the previous exercise, consider a 0-1 single-parameter environment. Suppose you are given a subroutine that,given bids b, computes the outcome of the welfare-maximizing allocation rule (4.2).
(a) Explain how to implement a welfare-maximizing DSIC mechanism by invoking this subroutine n + 1 times, where n is the number of participants.
(b) Conclude that mechanisms that are ideal in the sense of Theorem 2.4 exist for precisely the families of single-parameter environments in which the welfare-maximization problem (given bas input, compute (4.2)) can be solved in polynomial time.
Exercise 4.4 Prove that the greedy algorithm in the proof of Theorem 4.2 always computes an optimal fractional knapsack solution.
Exercise 4.5 Prove that the three-step greedy knapsack auction allocation rule in Section 4.2.2 is monotone. Does it remain monotone with the two optimizations discussed in the footnotes?
Exercise 4.6 Consider a variant of a knapsack auction in which we have two knapsacks, with known capacities W1 and W2. Feasible sets of this single-parameter environment now correspond to subsets S of bidders that can be partitioned into sets S1 and S2 satisfying∑ i∈Sj wi ≤ Wj for j = 1, 2.
Consider the allocation rule that first uses the single-knapsack greedy allocation rule of Section 4.2.2 to pack the first knapsack, and then uses it again on the remaining bidders to pack the second knapsack. Does this algorithm define a monotone allocation rule?
Give either a proof of this fact or an explicit counterexample.
Exercise 4.7 (H) The revelation principle (Theorem 4.3) states that(direct-revelation) DSIC mechanisms can simulate all other mechanisms with dominant-strategy equilibria. Critique the revelation principle from a practical perspective. Name a specific situation where you might prefer a non-direct-revelation mechanism with a dominant- strategy equilibrium to the corresponding DSIC mechanism, and ex-plain your reasoning.
