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)  Goal: find a partition S 1,…,S n such that the total social welfare  v i (S i ) 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)

Truthful Approximations?  A 2 approximation algorithm exists [Feige], and a matching lower bound is also known [Dobzinski-Nisan-Schapira].  A deterministic O(m ½ )-truthful approximation algorithm exists [Dobzinski-Nisan-Schapira].  Our Goal: lower bounds on the power of polynomial time truthful mechanisms

VCG (applied to combinatorial auctions)  A truthful mechanism for combinatorial auctions (VCG):  Find the optimal allocation (O 1,…,O n ). Assign the bidders items accordingly.  Pay each bidder i:  j≠i v j (O j ).  Proof (of truthfulness):  The utility of a bidder is the welfare of the allocation: e.g., Bidder 1’s utility is v 1 (O 1 )+  j>1 v j (O j ) =  j v j (O j ) = 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.  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? A O(m ½ )–Truthful Approximation Algorithm No Probably Not

A General Setting  A set of alternatives A.  n players, for each player i valuation v i : A  R.  A social choice function:  i v i  A.  We want to find payments (if such exist) such that the social choice function is implemented truthfully.

 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. Is There Anything Beyond VCG? MIR are the only truthful mechanisms Single parameter domains Very rich domains ??? E.g., combinatorial auctions Many truthful Mechanisms  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.

A Roadmap for Proving Hardness A Truthful Mechanism Affine Maximizer MIR AlgorithmThe Power of Efficient MIR Algorithms LMN Nisan-Ronen a m 1/6 lower bound for CAs with subadditive bidders using MIR algorithms. Conjecture: Every mechanism for “rich enough” domain must be affine maximizer. A way to set lower bounds on the only technique we have

An  (m 1/6 ) 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.

The Cover Number  Lemma: Let A be an MIR algorithm with range R. If cover(R) = |R| < e m/400, then A provides an approximation ratio no better than  Proof: Using the probabilistic method.  Fix an allocation T=(T 1,T 2 ) from the range.  Construct an instance with additive bidders: v(S) =  j  S v({j})  For each item j, set with probability ½ v 1 ({j})=1 and v 2 ({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.

The Intersection Number  A set of allocations D={(A 1,B 1 ),…,(A d,B d )} is called intersection set if each A i intersects with every B j, except B i, and each B i intersects with every A j, except A i.  Let intersect(R) be the size of the largest intersection set in R.

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.

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.

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=a 1 …a d, Bob holds b=b 1 …b d. Is there some t with a t =b t =1? Requires d bits of communication.  The Reduction:  Let {(A 1,B 1 ),…,(A d,B d )} be the maximal intersection set of the alg. For each index i with a i =1, set v A (S)=2 for all A i  S. Otherwise v A (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.