Prove that the greedy algorithm in the proof of Theorem 4.2 always computes an optimal fractional knapsack solution. 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?
Algorithmic Mechanism Design
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
[Assume that ties are broken in a deterministic and consistent way, such as lexicographically.]
Exercise 4.2
Continuing the previous exercise, restrict now to feasible 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 i 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 b as 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.
Problems
Problem 4.1
Consider a variant of a knapsack auction in which both the valuation vi and the size wi of each bidder i are private. A mechanism now receives both bids b and reported sizes a from the bidders. An allocation rule x(b, a) now specifies the amount of capacity allocated to each bidder, as a function of the bids and reported sizes. Feasibility dictates that ∑n i=1 xi(b, a) ≤ W for every b and a, where W is the total capacity of the shared resource. We define the utility of a bidder i as vi − pi(b, a) if she gets her required capacity (i.e., xi(b, a) ≥ wi) and as −pi(b, a) otherwise. This is not a single-parameter environment. Consider the following mechanism. Given bids b and reported sizes a, the mechanism runs the greedy knapsack auction of Sec- tion 4.2.2, taking the reported sizes a at face value, to obtain a subset of winning bidders and prices p. The mechanism concludes by awarding each winning bidder capacity equal to her reported size ai, at a price of pi; losing bidders receive and pay nothing. Is this mechanism DSIC? Prove it or give an explicit counterexample.
Problem 4.2
Section 4.2.2 gives an allocation rule for knapsack auctions that is monotone, guarantees at least 50% of the maximum social welfare, and runs in polynomial time. Can we do better? We first describe a classical fully polynomial-time approximation scheme (FPTAS) for the knapsack problem. The input to the problem is item values v1, . . . , vn, item sizes w1, . . . , wn, and a knapsack capacity W . For a user-supplied parameter ǫ > 0, we consider the following algorithm Aǫ; m is a parameter that will be chosen shortly.
• Round each vi up to the nearest multiple of m, call it v′ i.
• Divide the v′ i’s through by m to obtain integers ̃v1, . . . , ̃vn.
• For item values ̃v1, . . . , ̃vn, compute the optimal solution using a pseudopolynomial-time algorithm.
[You can assume that there exists such an algorithm with run- ning time polynomial in n and maxn
i=1 ̃vi.]
(a) Prove that if we run algorithm Aǫ with the parameter m set to ǫ(maxn i=1 vi)/n, then the running time of the algorithm is polynomial in n and 1 ǫ (independent of the vi’s).
(b) (H) Prove that if we run algorithm Aǫ with the parameter m set to ǫ(maxn i=1 vi)/n, then the algorithm outputs a solution with total value at least 1 − ǫ times the maximum possible.
(c) Prove that if we run algorithm Aǫ with the parameter m set to a fixed constant, independent of the vi’s, then the algorithm yields a monotone allocation rule.
(d) Prove that if we run algorithm Aǫ with the parameter m set as in (a) and (b), then the algorithm need not yield a monotone allocation rule.
(e) (H) Give a DSIC mechanism for knapsack auctions that, for a user-specified parameter ǫ and assuming truthful bids, outputs an outcome with social welfare at least 1−ǫ times the maximum possible, in time polynomial in n and 1
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. This is a single parameter environment. 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 if Ti ∩ Tj = ∅ for each distinct i, j ∈ W .
(a) (H) Prove that the problem of computing a welfare-maximizing feasible outcome, given the vi’s and Ti’s as input, is N P-hard.
(b) Here is a greedy algorithm for the social welfare maximization problem, given bids b from the bidders. initialize W = ∅ and X = M sort and re-index the bidders so that b1 ≥ b2 ≥ · · · ≥ bn for i = 1, 2, 3, . . . , n do if Ti ⊆ X then remove Ti from X and add i to W return winning bidders W
54 Algorithmic Mechanism Design Does this algorithm define a monotone allocation rule? Prove it or give an explicit counterexample.
(c) (H) Prove that if all bidders report truthfully and have sets Ti of cardinality at most d, then the outcome of the allocation rule in (b) has social welfare at least 1 d times that of the maximum possible
