Truthful Mechanisms for Combinatorial Auctions with Subadditive Bidders

1. Truthful Mechanisms for Combinatorial Auctions with Subadditive Bidders Speaker: Shahar Dobzinski Based on joint works with Noam Nisan & Michael Schapira

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]. • What about truthful approximations? • The private information of each bidder is his valuation.

4. Outline • A deterministic VCG-based O(m½)-approximation mechanism • An W(m1/6) lower bound on VCG-based mechanisms. • A randomized almost-logarithmic approximation mechanism.

5. Reminder: Maximal in Range Algorithms • VCG: Allocate Oi to bidder i. Bidder i gets a payment of Sk≠ivk(Ok). • (O1,…,On) is the optimal solution. • Still truthful if we limit the range. • Range := { A=(A1,…,An) |v1,…,vn: A(v1,…,vn)=A } • The Algorithm[Dobzinski-Nisan-Schapira]: • Choose the best allocation where either: • One bidder gets all items OR • Each bidder gets at most one item. • Clearly, the algorithm is maximal-in-range and can be implemented in polynomial time.

6. Proof of the Approximation Ratio Theorem: If all valuations are subadditive, the algorithm provides an O(m1/2)-approximation. Proof: Let OPT=(L1,..,Ll,S1,...,Sk), where for each Li, |Li|>m1/2, and for each Si, |Si|≤m1/2. |OPT|= Sivi(Li) + Sivi(Si) • Case 2:Sivi(Si) ≥ Sivi(Li) • (“small” bundles contribute most of the optimal social welfare) • Sivi(Si) ≥ |OPT|/2 Claim: Let v be a subadditive valuation and S a bundle. Then there exists an item jS s.t. v({j}) ≥ v(S)/|S|. Proof: immediate from subadditivity. • Thus, for each bidder i that was assigned a small bundle, there is an item ciSi, such that: vi({ci}) > vi(Si) / m1/2. Allocate ci to bidder i. • Case 1: Sivi(Li) > Sivi(Si) • (“large” bundles contribute most of the optimal social welfare) • Sivi(Li) > |OPT|/2 • At most m1/2 bidders get at least m1/2 items in OPT. •  There is a bidder i s.t.: vi(M) ≥ vi(Li) ≥ |OPT|/2m1/2.

7. Outline • A deterministic VCG-based O(m½)-approximation mechanism • An W(m1/6) lower bound for VCG-based mechanisms. • A randomized almost-logarithmic approximation mechanism.

8. About the Lower Bound • Why lower bounds on VCG-Based mechanisms (a.k.a. maximal-in-range algorithms)? • Conjectured characterization: All mechanisms that give a good approximation ratio for combinatorial auctions with subadditive bidders are maximal in their range. • Even if the conjecture is false, still the only technique that we currently know.

9. An W(m1/6) lower bound on VCG-based mechanisms [Dobzinski-Nisan] • We define two complexity: • 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. • 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 • Intuitively, the size of the range • But we don’t want to count “degenerate allocations”… • A set of allocations C covers a set of allocations R if for each allocation S in R there is an allocation T in C such that TiCi for i={1,2}. • cover(R) is the size of the smallest set C that covers R. • Observation: An MIR on range C provides a better approximation ratio than on R.

11. The Cover Number • Lemma: Let A be an MIR algorithm with range R. If cover(R) < em/400, then A provides an approximation ratio of at most 1.99. • Proof: Using the probabilistic method. • Fix an allocation T=(T1,T2) from the minimal cover C. • 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.

12. The Intersection Number • A set of allocations D is called an intersection set if for each (A1,A2)≠(B1,B2)D we have that A1 intersects B2 and A2 intersects B1. • Let intersect(R) be the size of the largest intersection set in R.

13. 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 t bits of communication. • Given a disjointness instance, construct a combinatorial auction with subadditive bidders: • Let {(A1,B1),…,(Ad,Bd)} be the intersection set. Set vA(S)=2 if there is an index i s.t. ai=1 and 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.

14. 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 VCG-based algorithm provides an approximation ratio better than 2.

15. Summary • A deterministic VCG-based O(m½)-approximation mechanism • An W(m1/6) lower bound on VCG-based mechanisms. • A randomized almost-logarithmic approximation mechanism.

16. Open Questions • Deterministic mechanisms\lower bounds for combinatorial auctions with general valuations? • Is the gap between randomized and deterministic mechanisms essential?

17. Randomness and Mechanism Design • Randomization might help in mechanism design settings. • Two notions of randomization: • “The universal sense”: a distribution over deterministic mechanisms (stronger) • “In expectation”: truthful behavior maximizes the expectation of the profit (weaker) • Risk-averse bidders might benefit from untruthful behavior. • The outcomes of the random coins must be kept secret.

18. Results • Feige shows a randomized O(logm/loglogm)-truthful in expectation mechanism. • We show that there exists an O(logm*loglogm) truthful in the universal sense mechanism.

19. The Framework • Two cases: • Case 1: There is a dominant bidder. • A bidder with v(M) > OPT/(100log m loglog m) (denote the denominator by c) • We can simply allocate all items to this bidder. • Case 2: There is no dominant bidder. • In this case we can use random sampling: partition the bidders into two sets, acquire statistics from one set, and use it to get an approximate solution with the other set. • How to put the two cases together? • Flipping a coin works, but with probability of only ½. • Next we will see how to increase the probability of success to 1-e.

20. The Mechanism A second price auction with a reserve price of OPT/c I have an estimate of OPT SECPRICE group • Partition the bidders into 3 sets: • STAT with probability e/2, SECPRICE with probability 1-e, and FIXED with probability e/2. • First case: there is a dominant bidder. Statistics Group

21. The Mechanism • Second case: there is no dominant bidder. A second price auction with a reserve price of OPT/c FIXED group I have a (good) estimate of OPT Statistics Group

22. Case 2: No “Dominant” Bidder • Assumption: For all bidders vi(OPTi) < OPT / c • In the FIXED group: a fixed-price auction where each item has a price of p (depends on the statistics group) Everything costs p Take your most profitable bundle My price is 2*p Too Expensive! I paid p

23. Still Missing… • Why does the fixed price auction (with a “good price”) provides a good approximation ratio? • Can we find this “good price” using the statistics group?

24. A Combinatorial Property of Subadditive Valuations • Lemma: Let v be a subadditive valuation and S a bundle of items. Then we can assign each item in S a price in {0,p} such that: • For each TS: v(T) > SjT|T|*p • |S|*p > v(S)/(100*logm)