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Limitations of VCG-Based Mechanisms

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Limitations of VCG-Based Mechanisms

Shahar Dobzinski

Joint work with Noam Nisan

- m items, n bidders, each bidder i has a valuation function vi:2M->R+.
Common assumptions:

- Normalization: vi()=0
- Monotonicity: ST vi(T) ≥ vi(S)

for every S,T M: v(S)+v(T) ≥ v(ST)

(but all of our results also hold for submodular valuations)

- A 2 approximation algorithm exists [Feige], and a matching lower bound is also known [Dobzinski-Nisan-Schapira].
- A deterministic O(m½)-truthful approximation algorithm exists [Dobzinski-Nisan-Schapira].
- Our Goal: lower bounds on the power of polynomial time truthful mechanisms

- A truthful mechanism for combinatorial auctions (VCG):
- Find the optimal allocation (O1,…,On). Assign the bidders items accordingly.
- Pay each bidder i: Sj≠ivj(Oj).

- Proof (of truthfulness):
- The utility of a bidder is the welfare of the allocation: e.g., Bidder 1’s utility is v1(O1)+Sj>1vj(Oj) = Sjvj(Oj) = OPT

- VCG is truthful iff the algorithm is maximal-in-range [Nisan-Ronen]
- MIR: limit the range and fully optimize over the restricted range.

- The Algorithm[Dobzinski-Nisan-Schapira]:
- Choose the maximum-value allocation where either:
- One bidder gets all items OR
- Each bidder gets at most one item.

- Choose the maximum-value allocation where either:
- The algorithm is MIR (and can be made truthful using VCG payments).
- Is there a (substantially) better MIR polynomial time algorithm?
- Are there other types of truthful mechanisms?

No

Probably Not

- A set of alternatives A.
- n players, for each player i valuation vi: A R.
- A social choice function: Pivi A.
- We want to find payments (if such exist) such that the social choice function is implemented truthfully.

Many truthful Mechanisms

MIR are the only truthful mechanisms

???

- Roberts theorem (informal): if the domain of valuations is unrestricted then MIR mechanisms are the only truthful mechanisms.
- Lavi, Mu’alem, and Nisan (informal): For rich enough domains (e.g., combinatorial auctions) and some technical (?) conditions, MIR mechanisms might be the only truthful mechanisms that give a good approximation ratio.

E.g., combinatorial auctions

Very rich domains

Single parameter domains

- Single parameter domains: the private information of each player consists of one number.
- Monotone algorithm: a player that wins and raises his bid is still a winner.
- An algorithm is truthful iff it is monotone.

LMN

A Truthful Mechanism

Affine Maximizer

Conjecture: Every mechanism for “rich enough” domain must be affine maximizer.

A way to set lower bounds on the only technique we have

Nisan-Ronen

MIR Algorithm

The Power of Efficient MIR Algorithms

a m1/6 lower bound for CAs with subadditive bidders using MIR algorithms.

- Two complexity measures:
- Cover Number: (approximately) the range size
- must be “large” in order to obtain a good approximation ratio.

- Intersection Number: a lower bound on the communication complexity (the # of queries to the black boxes).
- We therefore want it to be “small” (polynomial).

- Cover Number: (approximately) the range size
- Lemma (informal): If the cover number is large then the intersection number must be large too.
- From now on, only 2 bidders, thus a lower bound of 2.

- Lemma: Let A be an MIR algorithm with range R. If cover(R) = |R| < em/400, then A provides an approximation ratio no better than 1.99.
- Proof: Using the probabilistic method.
- Fix an allocation T=(T1,T2) from the range.
- Construct an instance with additive bidders: v(S) = SjS v({j})
- For each item j, set with probability ½ v1({j})=1 and v2({j})=0 (or vice versa with probability ½ ).
- The optimal welfare in this instance is m, but each item j contributes 1 to the welfare provided by T only if we hit the corresponding bundle in T (with probability 1/2).
- The expected welfare that T provides is m/2, and we can get a better welfare only with exponential small probability.

- A set of allocations D={(A1,B1),…,(Ad,Bd)} is called intersection set if each Ai intersects with every Bj, except Bi, and each Bi intersects with every Aj, except Ai.
- Let intersect(R) be the size of the largest intersection set in R.

- In order to obtain an approximation ratio better than 2, the cover number must be exponentially large.
- If the MIR algorithm runs in polynomial time then the intersection number must be polynomial too.
- Lemma (informal): If the cover number is exponentially large then the intersection number is exponentially large too.
- Corollary: No polynomial time MIR algorithm provides an approximation ratio better than 2.

- MIR as an algorithmic technique
- Arora’s PTAS for Euclidean TSP, multi-unit auctions, …
- Improve the m/(log m)½-approximation algorithm for combinatorial auctions with general bidders

- “Real” hardness of truthful approximation results.

- Lemma: Let A be an MIR algorithm with range R. Let intersect(R)=d. Then, the communication complexity of A is at least d.
- Proof:
- Reduction from disjointness: Alice holds a=a1…ad, Bob holds b=b1…bd. Is there some t with at=bt=1? Requires d bits of communication.
- The Reduction:
- Let {(A1,B1),…,(Ad,Bd)} be the maximal intersection set of the alg.
For each index i with ai=1, set vA(S)=2 for all Ai S. Otherwise vA(S)=1. Similar valuation for Bob.

- The valuations are subadditive.

- Let {(A1,B1),…,(Ad,Bd)} be the maximal intersection set of the alg.
- A common 1 bit optimal welfare of 4. Our algorithm is maximal in range, and the optimal allocation is in the range, so our algorithm always return the optimal solution. But this requires d bits of communication.