Limitations of VCG-Based Mechanisms

1. Limitations of VCG-Based Mechanisms Shahar Dobzinski Joint work with Noam Nisan

2. Combinatorial Auctions • m items, n bidders, each bidder i has a valuation function vi:2M->R+. Common assumptions: • Normalization: vi()=0 • Monotonicity: ST  vi(T) ≥ vi(S) • Goal: find a partition S1,…,Sn such that the total social welfareSvi(Si) is maximized. • Algorithms must run in time polynomial in n and m. • In this talk the valuations are subadditive: for every S,T  M: v(S)+v(T) ≥ v(ST) (but all of our results also hold for submodular valuations)

3. Truthful Approximations? • 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

4. VCG (applied to combinatorial auctions) • 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.

5. A O(m½)–Truthful Approximation Algorithm • 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. • 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

6. A General Setting • 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.

7. Is There Anything Beyond VCG? 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.

8. A Roadmap for Proving Hardness 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.

9. An W(m1/6) Lower Bound on MIR Mechanisms • 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). • 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.

10. The Cover Number • 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) = SjS 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.

11. The Intersection Number • 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.

12. Putting it Together • 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.

13. Open Questions • 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.

14. The Intersection Number • 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. • 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.