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CS6998-3: Solutions to Problem Set # 2
S´ebastien Lahaie
Problem 1 (10 points)
(a) 5 points The proof is virtually identical to the proof that the VCG mechanism is truthful. Let v−i be some arbitrary reports by agents N − i, let vi be agent i’s true valuation, and let ˜vi be some other valuation. Let R be an efficient allocation with respect to (vi, v−i), and let R′^ be an efficient allocation with respect to (˜vi, v−i). If agent i reports vi its utility is ∑
j∈N
vj (Rj ) − hi(v−i), (1)
whereas if it reports ˜vi its utility is (^) ∑
j∈N
vj (R j′ ) − hi(v−i). (2)
Subtracting (2) from (1) we get (^) ∑
j∈N
vj (Rj ) −
j∈N
vj (R′ j ).
This is non-negative because R is efficient with respect to the profile (vi, v−i). Thus reporting vi maximizes i’s utility, since ˜vi was arbitrary.
(b) 5 points By definition, a Groves mechanism is individually rational if the utility to each agent i from truthfully reporting its value is non-negative. Let R be the efficient allocation selected if i reports truthfully. The utility to agent i is (^) ∑
j∈N
vj (Rj ) − hi(v−i),
so we must have hi(v−i) ≤
j∈N
vj (Rj ). (3)
This holds for any possible valuation of agent i, in particular the valuation where vi(S) = 0 for all S ⊆ M. In this case we can assume that Ri = ∅, and thus R is an efficient allocation among agents N − i. Condition (3) in this special case is
hi(v−i) ≤ max R′∈Γ
j∈N −i
vj (R′ j ). (4)
The Groves mechanism that maximizes the term hi(v−i) is the one that maximizes agent i’s payment. In view of (4), the VCG mechanism maximizes the payment because it achieves the upper bound.
Problem 2 (10 points)
(a) 7 points We first show that if R is an efficient allocation, then vi(Ri) = max j∈N vj (Rj ). (5)
Assume this does not hold, so that for some i ∈ N , there is a k 6 = i such that vk(Ri) is the maximum value for Ri over all agents. Note that this value must be positive. As vk(Ri) > 0, we have Ri ⊇ Sk. However, Ri ∩ Rk = ∅ by the feasibility of R. Thus Rk 6 ⊇ Sk and vk(Rk) = 0. Suppose that instead of giving Ri to i and Rk to k, we give Ri to k and ∅ to i. Then this changes the total value by vk(Ri) − vi(Ri) > 0. This is a contradiction because R is efficient. Hence vi(Ri) − p(Ri) = 0 for each i ∈ N , and vi(S) − p(S) ≤ 0 by the definition of p; the bundle Ri maximizes i’s utility, for all i ∈ N. It remains for us to show that R maximizes revenue at prices p. Let R′^ be a revenue-maximizing allocation such that the number of agents that receive ∅ is maxi- mized. Note that if we permute the bundles in R′, the revenue remains unchanged, because prices are anonymous. For each R′ i, let σ(i) ∈ N be an agent such that p(R′ i) = vσ(i)(R′ i). We claim that we must have σ(i) 6 = σ(j) when R i′ 6 = ∅ and R′ j 6 = ∅. Assume for the sake of contradiction that p(R′ i) + p(R′ j ) = vk(R′ i) + vk(R′ j ). Since R′ i ∩ R′ j = ∅ and vk is single-minded, the value of one of those bundles to agent k must be 0, say vk(R′ j ) = 0. Thus, vk(R′ i) + vk(R′ j ) ≤ vk(R i′ ∪ R′ j ) + vk(∅) ≤ p(R′ i ∪ R′ j ) + p(∅). We see that if we replace R′ i with R′ i ∪ R′ j , and R′ j with ∅, we get an allocation R′′^ with weakly greater revenue than R′. But since the latter is revenue-maximizing, so is R′′. This is a contradiction, because R′′^ contains one more ∅ than R. Thus we can permute the bundles in R′^ such that p(R′ i) = vi(R′ i) for R i′ 6 = ∅. For R i′ = ∅, we have p(∅) = vj (∅) for all j ∈ N. After the permutation, the revenue from R′^ is ∑
i∈N
p(R′ i) =
i∈N
vi(R′ i)
i∈N
vi(Ri)
i∈N
p(Ri)
where the second step follows because R is efficient, and the third from (5). As R′^ is revenue- maximizing, so is R, and this completes the proof. (b) 3 points Let Γ(S) be the set of all feasible allocations such that Ri = S for some i ∈ N. We have a variable xi(S) for each i ∈ N and S ⊆ M to denote whether i obtains bundle S. We have a variable z(R) for each feasible allocation R to denote whether R is selected. max x≥ 0 ,z≥ 0
i∈N
S⊆M
vi(S)xi(S)
subject to
i∈N
xi(S) =
R∈Γ(S)
z(R) (S ⊆ M )
∑
S⊆M
xi(S) = 1 (i ∈ N )
∑
R∈Γ
z(R) = 1
(d) 2 points
It is straightforward to check that ¯p satisfy the inequalities of part (b), so they are competitive equi- librium prices. We prove that they are minimal by induction. Let p be first-order CE prices. We have pn ≥ 0 by definition, and note that ¯pn = 0. Thus pn ≥ p¯n, establishing the base case. Assume pi ≥ p¯i where i ≤ n. For i = 2,... , n, we have vii − pi ≥ vii− 1 − pi− 1 which implies
pi− 1 ≥ vii− 1 − vii + pi ≥ vii− 1 − vii + ¯pi = ai(bi− 1 − bi) +
j>i
aj (bj− 1 − bj )
j>i− 1
aj (bj− 1 − bj )
= p¯i− 1.
The second inequality follows from the induction hypothesis, and the remaining from the definition of vii− 1 and ¯pi. This completes the proof.
(e) 1 point Fix agent i. With all agents present, the efficient allocation gives item 1 to agent 1, item 2 to agent 2, etc. by part (a). The total value to all the agents except i under this allocation is ∑
j 6 =i
vjj =
j 6 =i
aj bj. (8)
If agent i is removed, the efficient allocation gives item j to agent j for j < i, and item j − 1 to agent j for j > i (item n remains unallocated). This follows from the same reasoning as in part (a). The total value to all the agents except j in this case is ∑
j<i
vjj +
j>i
vjj− 1 =
j<i
aj bj +
j>i
aj bj− 1. (9)
By definition the VCG payment of agent i is (9) minus (8):
qˆi =
j<i
aj bj +
j>i
aj bj− 1 −
j 6 =i
aj bj
j>i
aj (bj− 1 − bj ).
Comparing with part (d), we find that ˆqi = ¯pi. That is, the VCG payment of agent i is the price of the item agent i receives at the lowest possible linear CE prices.
