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# Limitations of VCG-Based Mechanisms - PowerPoint PPT Presentation

Limitations of VCG-Based Mechanisms. Shahar Dobzinski Joint work with Noam Nisan. Combinatorial Auctions. m items, n bidders, each bidder i has a valuation function v i :2 M ->R + . Common assumptions: Normalization : v i (  )=0 Monotonicity : S T  v i (T) ≥ v i (S)

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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: 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)

• 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.

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

• 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.

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.

• 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.

• 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.