1. Computation and Incentives in Combinatorial Public Projects Michael Schapira Yale University and UC Berkeley Joint work with Dave Buchfuhrer and Yaron Singer

2. Take Home Messages Combinatorial Public Projects are cool! More suitable arena for exploring truthful computation? Should we rethink AMD solution concept?

3. Designing Algorithms for Environments With Selfish Agents computational efficiency incentive- compatibility When can these coexist? [Nisan-Ronen]

4. Paradigmatic Problem: Combinatorial Auctions A set of m items on sale {1,…m}. n bidders {1,…,n}. Each bidder i has valuation function v i : 2 [m] → R ≥0. –normalized, non-decreasing. Goal: find a partition of the items between the bidders S 1,…,S n such that the social welfare  i v i (S i ) is maximized

5. What Do We Want? Quality of the solution: As close to the optimum as possible. Computationally tractable: Polynomial running time (in n and m). Truthful: Motivate (via payments) bidders to report their true values. –The utility of each agent is u i = v i (S) – p i –Solution concepts: dominant strategies, ex- post Nash.

6. State of the Art “It is probably fair to summarize that most computational issues have been resolved, while most strategic questions have remained open… despite much work and some mild progress… The basic question of how well can computationally-efficient incentive-compatible combinatorial auctions … perform remains as open as in the beginning of the decade, and gets my (biased) AGT open problem of the decade award.” Noam Nisan

7. Why is This Happening? We do not understand truthfulness. –Roberts’ Theorem... Combinatorial auctions are complex –Too much noise… (combinatorics) Other approach: find “minimal” environments where computation and incentives clash. –and then go back to combinatorial auctions.

8. Combinatorial Public Projects Problem (CPPP) [Papadimitriou-S-Singer] Set of n agents; Set of m resources; Each agent i has a valuation function: v i : 2 [m] → R ≥0 –normalized, non-decreasing. Goal: Given a parameter k, choose a set of resources S* of size k which maximizes the social welfare: S* = argmax i  i v i (S) S [m], |S|=k

9. Complement-Free Hierarchy [Lehmann-Lehmann-Nisan] Fractionally- Subadditive (“XOS”) Complement-Free (Subadditive) Submodular Gross Substitute Unit-Demand (“XS”) Multi-Unit- Demand (“OXS”) Capped Additive (“Budget-Additive”) Coverage Questions: 1.Where does CPPP cease to be tractable? (VCG!) 2.Where does CPPP cease to be approximable?

10. Complement-Free Hierarchy: Tractability Fractionally- Subadditive (“XOS”) Complement-Free (Subadditive) Submodular Gross Substitute Unit-Demand (“XS”) Multi-Unit- Demand (“OXS”) Capped Additive (“Budget-Additive”) Coverage CPPP combinatorial auctions even for n=1

11. Complement-Free Hierarchy: Approximability Fractionally- Subadditive (“XOS”) Complement-Free (Subadditive) Submodular Gross Substitute Unit-Demand (“XS”) Multi-Unit- Demand (“OXS”) Capped Additive (“Budget-Additive”) Coverage CPPP combinatorial auctions

12. Complement-Free Hierarchy: Area of Interest Submodular Gross Substitute Unit-Demand (“XS”) Multi-Unit- Demand (“OXS”) Capped Additive (“Budget-Additive”) Coverage Unit-Demand (“XS”) Coverage even for n=1

13. Two Simple Environments CPPP with unit-demand agents –Each agent only wants one resource! CPPP with one coverage valuation

14. 2-{0,1}-Unit-Demand resources 0 0 0 1 1 Each user only wants (value 1) at most two resources and does not want (value 0) all others. user

15. Combinatorial auctions with such valuations are trivial. –matching CPPP with such valuations is NP-hard. –Vertex Cover –But approximable –(Solvable for constant n’s) The perfect starting point. –What about truthful computation? 2-{0,1}-Unit-Demand

16. all sets of resources of size k Maximal-In-Range Mechanisms (= VCG-Based) Definition: A is MIR if there is some R A {|S | = k| S [m]} s.t. A(v 1,…v n ) = argmax S in R v 1 (S)+…+v n (S) * We shall refer to R A as A’s range. RARA A

17. Thm [S-Singer] : There exists a computationally-efficient MIR mechanism for CPPP with complement- free valuations with appx ratio 1/ √m. Thm: No computationally-efficient MIR mechanism for CPPP with 2-{0,1}- unit-demand valuations has appx ratio better than 1/√m –unless SAT is in P/poly. 2-{0,1}-Unit-Demand

18. What about general truthful mechanisms? Thm: There exists a computationally- efficient MIR mechanism for CPPP with 2-{0,1}-unit-demand valuations that has appx ratio ½. –Simply choose the k most demanded resources. 2-{0,1}-Unit-Demand

19. What about truthful mechanisms for CPPP with unit-demand valuations? –Characterization? No techniques? –VCG –random sampling –LP Open Question

20. Truthfulness With One Player? The interests of the player and mechanism are aligned (value = social welfare) player mechanism What do you want? I want this! Take it!

21. CPPP With a Coverage Valuation Defn: A valuation v is a coverage valuation if there is –a universe U –m subsets of U, T 1,…,T m –and  >0 such that for every set of resources S: v(S) =  |U j in S T j |

22. Computational Perspective: A 1-1/e approximation ratio is achievable (not truthful!)  A tight lower bound exists [Feige]. Strategic Perspective: A truthful solution is trivially achievable via VCG payments (but NP-hard to obtain) What about achieving both simultaneously ? CPPP With a Coverage Valuation

23. Thm: No computationally-efficient and truthful mechanism for CPPP with one coverage valuation has appx ratio better than 1/√m –Unless SAT is in P/poly. –Tight Strengthens and simplifies a recent result in [Papadimitriou-S-Singer] –For n=2 –For submodular valuations. Hardness of Truthfulness With One Player?

24. 3 Challenges Complexity theory mechanism design combinatorics (hardness of truthful mechanisms) (characterization of truthful mechanisms) (structure of truthful mechanisms)

25. Characterization Lemma (informal): Every truthful mechanism for CPPP with one coverage valuation is MIR. –True for all one-player mechanism design environments Inapproximability Lemma: No computationally-efficient MIR mechanism for CPPP with one coverage valuation has appx ratio better than 1/√m –unless SAT is in P/poly. The Proof: Overview

26. If a computationally-efficient MIR mechanism A has appx ratio better than 1/√m then |R A | ≥ 2  m (for some constant  >0). –probabilistic construction. So, a MIR mechanism A that has appx ratio better than 1/√m optimizes over exponentially many outcomes. Proof of Inapproximability Lemma (sketch)

27. All sets of resources of size k Computational Hardness –CPPP with one coverage valuation is NP-hard. –So, optimizing over the set of all possible outcomes is hard. –What about optimizing over a set of outcomes of exponential size?  Intuition: also hard! RARA

28. The VC Dimension universe 1 x 3 x 5 1 2 3 4 5 1 2 x x 5 x x x 4 x collection of subsets R shattered set 1 2 3 4 5

29. Lower Bounding the VC Dimension The Sauer-Shelah Lemma: Let R be a collection of subsets of a universe U. Then, there exists a subset E of U such that: –E is shattered by R. –|E| ≥  ( log(|R|)/log(|U|) ). R A is a collection of subsets of the universe of resources.

30. The Reduction We know that |R A | ≥ 2  m (for some constant . Hence, there is a set of resources of size m  (for some constant  >0) that is shattered by R A. We can now show that the MIR mechanism A solves exactly a smaller (but not too small!) CPPP with one coverage valuation! RARA

31. Truthfulness With One Player? Somewhat Strange… Do we need to rethink the framework? player mechanism What do you want? I don’t know!

32. Positive Results for CPPP Submodular Gross Substitute Unit-Demand (“XS”) Multi-Unit- Demand (“OXS”) Capped Additive (“Budget-Additive”) Coverage FPTAS for constant n’s optimal algorithm for n=2

33. Take Home Messages Combinatorial Public Projects are cool! More suitable arena for exploring truthful computation? Should we rethink AMD solution concept?

34. Back to Combinatorial Auctions… [Mossel-Papadimitriou-S-Singer] A set of m items on sale {1,…m}. n bidders {1,…,n}. Each bidder i has valuation function v i : 2 [m] → R ≥0. –normalized, non-decreasing. Goal: find a partition of the items between the bidders S 1,…,S n such that social welfare  i v i (S i ) is maximized

35. What About Combinatorial Auctions? Complexity theory mechanism design combinatorics embedding hard problems in partial ranges Truthful=MIR (VC dimension) consider only MIR generalize the VC dimension to handle partitions of a universe.

36. The Case of 2 Bidders Not trivial even for n=2! The trivial MIR mechanism: allocate the bundle of all items to the highest bidder. –½ appx. ratio. Is this the best we can do (with MIR)? –Yes! [Buchfuhrer et al.] –extends to general n’s.

37. Intuition 1 2 3 4 5 5 items MIR algorithm A 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 RARA A is (implicitly) optimally solving a 2-item auction 2 bidders

38. Intuition We wish to prove the existence of a subset of items E that is “shattered” by A’s range (R A ). –“Embed” a small NP-hard auction in E. –Not too small! (|E| ≥ m  ) VC dimension –We need to bound the VC dimension of collections of partitions! –Of independent interest.

39. VC Dimension of Partitions We want to prove an analogue of the Sauer-Shelah Lemma for the case of partitions of a universe. –That do not necessarily cover the universe. Problem: The size of the collection of partitions does not tell us much. Recent advances [Mossel-Papadimitriou-S-Singer, Buchfuhrer-Umans, Dughmi-Fu-Kleinberg]

40. Directions for Future Research Understanding truthful computation in the context of CPPP with unit-demand valuations. Implications for combinatorial auctions. Many open questions regarding the approximability of CPPP. Truthfulness in single-player environments?

41. Thank You