1. Towards a Characterization of Truthful Combinatorial Auctions Ron Lavi, Ahuva Mu’alem, Noam Nisan Hebrew University

2. Combinatorial Auctions k indivisible non-identical items for sale n bidders compete for subsets of these items Each bidder i has a valuation for each set of items: v i (S) = value that i assigns to acquiring the set S –v i is non-decreasing (“free disposal”) –v i (  ) = 0 Objective: Find a partition (S 1 …S n ) of {1..k} that maximizes the social welfare:  i v i (S i )

3. Motivation Abstracts complex resource allocation problems in systems with distributed ownership (e.g. scheduling, allocation of network resources). Real Applications (e.g. the FCC spectrum auction).

4. Main Issues Complexity : Computing Optimal Allocation is NP. –Handle it by approximation algorithms or by allocation heuristics that perform well in practice. Strategic : Valuations v i are private information. –Study rational bidders that aim to maximize v i (S i ) – price –Wlog: concentrate on Truthful Auctions –We can apply the classic positive result of mechanism design: VCG mechanisms.

5. The Clash: Complexity - Incentives VCG payments ensure truthfulness only if optimal allocation is chosen – but this is NP-complete! Problem is near universal: VCG will work with no other “reasonable” allocation algorithm. [NR] Main Open Problem: Are there any truthful polynomial time mechanisms? –Can poly-time truthful mechanisms give good approximations? –Can poly-time truthful mechanisms be reasonable heuristics?

6. A broader question VCG is the only known general method to design truthful mechanisms. Many times, VCG is not suitable for us: –Computing the exact optimal welfare may be computationally hard. –Desire different goals than welfare maximization: Rawls-like max-min; max  i log v i (a), sum- squares; tradeoffs, … What other truthful mechanisms are there?

7. Abstraction: Social Choice Function A set of possible alternatives, A. –For CAs: A = {S 1..S n that are a partition of 1..k} Each player has a valuation v i  V i, v i : A  R –For CAs: V i = {v i that satisfy 1, 2, 3} (1) depends only on S i (2) monotone (3) v i (  ) = 0 Truthful implementation: adding payments s.t. bidders will maximize their utility by revealing their true v i

8. What SCFs can be implemented ?? Affine maximizers (or weighted-VCG): (can always be implemented) Roberts ’79 : If V i = R |A| (unrestricted domain) then only affine maximizers can be implemented! For single dimensional domains (V i = R), many non-affine- maximizers are known. [LOS, MN, AT,.....] OPEN: Are there any implementable non-affine maximizers for multi-dimensional domains V i  R |A| ? Only one known example - for multi-unit CAs [BGN] severely restricted domains | | Multi Unit Auctions (MUA) ? | Combinatorial Auctions (CA) ? unrestricted domain | Only affine maximizers Many non-affine maximizers exist

9. Comparison with the non-quasi-linear case “Single-Peaked”: Yes “Saturated”: No Single-dimensional: Yes CAs, MUAs, … : ??? Other implementations in restricted domains? Gibbard-Satterthwaite (70’s)  Arrow (50’s) Roberts (79)Impossibility result for unrestricted domains DictatorialAffine-maximizersImplementable SCFs vivi Preferences Non-quasi-linearQuasi-linear >i>i

10. Our Result Wanted THM For CAs (and similar domains) : Every implementable SCF is an affine maximizer. –False as is. Proved THM For CAs (and similar domains) : Every player- decisive, non-degenerate implementable SCF that satisfies IIA is an almost affine maximizer. –IIA condition can be dropped for 2-player auctions that always allocate all items.

11. Independence of Irrelevant Alternatives Dfn: f satisfies IIA if: f(v)=a and f(u)=b Justifications: –We needed it in the proof. –Similar justifications as for Arrow’s IIA. –Condition is w.l.o.g for unrestricted domains and for 2-player auctions that always allocate all items.

12. Proof Structure Part 1: Truthful  monotone –Every implementable SCF is W-MON –WMON is also a sufficient condition (for many domains) –W-MON + IIA = SMON –IIA requirement can be dropped in some domains Part 2: SMON + technicalities  almost affine maximizer –An SMON SCF induces an order-like structure –This structure implies a way to “measure” alternatives –This measure implies affine maximization of the SCF

13. Computational Implications Observation: Affine maximization is as computationally hard as exact maximization. Corollary 1: Any truthful unanimity-respecting CA that satisfies IIA and achieves a poly(n,k) approximation is not poly-time. Dfn: f is unanimity-respecting if, whenever all players single-mindedly desire bundles that together form a partition, this partition is chosen. Corollary 2: No truthful poly-time CA/MUA for two players, that must allocate all items, achieves better than 2-approximation. For MUA, without truthfulness, an FPAS exists. A simple truthful 2-approximation exists

14. Rest of Talk Describe main building blocks of proof: Part I : Truthfulness, Monotonicity, and IIA. Part II : Strong monotonicity  affine maximization.

15. Truthful Implementation of Social Choice Functions A mechanism is m = (f, p 1, p 2, , p n ), where f is a SCF, and p i : V  R is the payment function of player i. Dfn: Truthful Implementation in dominant strategies [rational players tell the truth] :  v i, v -i, w i : v i (f(v i, v -i )) – p i (v i, v -i ) > v i (f(w i, v -i )) – p i (w i, v -i ) Not all SCFs can be implemented. If there exists an implementation it is essentially unique.

16. Weak Monotonicity Dfn: f satisfies W-MON if for any v i, v -i and u i : Thm: Truthfulness  W-MON. W-MON  Truthfulness ( for CA, MUA, and related domains ). Comments: Generalizes monotonicity for single dimensional domains. Equivalent to Roberts’ PAD for unrestricted domains, but makes sense also in restricted domains. Many other natural monotonicity conditions don’t work. If the result changes from a to b then i’s value for b increased at least as his value for a.

17. Prop: If f is truthful then p i (v) = p i (a, v -i ), where f(v) = a. proof: Otherwise, if p i (v) depends on v i, then player i would untruthfully declare the v’ i that minimizes p i (v’ i, v -i ). Proof (Truthfulness  W-MON): f (v i, v -i ) = a  v i (a) - p i (a, v -i ) > v i (b) - p i (b, v -i ), otherwise player i would declare u i instead of v i. f (u i, v -i ) = b  u i (b) - p i (b, v -i ) > u i (a) - p i (a, v -i ), otherwise player i would declare v i instead of u i.  u i (b) - u i (a) > v i (b) - v i (a). Proof: Truthfulness  W-MON

18. Strong Monotonicity and IIA Dfn: f satisfies S-MON if for any v i, v -i and u i : f (v i, v -i ) = a and f (u i, v -i ) = b implies u i (b) - u i (a) > v i (b) - v i (a). Dfn: f satisfies IIA if: f(v)=a and f(u)=b Lemma 1: W-MON + IIA = S-MON (for CAs, MUAs, and related restricted domains) Lemma 2: W-MON implies (w.l.o.g) S-MON for CAs/MUAs among two players, where all goods must always be allocated. –But not in general!

19. Rest of Talk Describe main building blocks of proof: Part I : Truthfulness, Monotonicity, and IIA. Part II : Strong monotonicity  affine maximization.

20. Main Theorem Theorem: For CAs, MUAs, and related domains: A is non-degenerate + f satisfies S-MON + f is player decisive A is “non-degenerate” if there is an allocation where player 1 and player i receive a non-empty bundle (for any i>1). f is “player decisive” if any player can always receive all the goods by bidding high enough on them. f is “almost affine maximizer” if it is affine maximizer for all large enough valuations: there exists a constant M s.t. for any type v with v i (S)>M for all i and non-empty bundles S, f is affine maximizer for v. f must be almost affine maximizer.

21. Proof idea The proof essentially shows that every mechanism for CA that satisfies S-MON operates as follows: –It has a measure function - attaching a value to every alternative and choosing the one with the highest measure. (Inspired by the min-function model of Archer and Tardos). –This measure function must be affine -- it is the weighted sum of valuations for the alternative. It is affine maximizer.

22. The order induced by a S.C.F.. x1x1 y1y1 ab v 1 =.. x2x2 y2y2 v 2 =.. xnxn ynyn v n =.... players allocations..

23. The order induced by a S.C.F Definition: x@a > y@b [“x at a” is larger than “y at b”] if there exists v with: f(v)=a, v(a)=x, v(b)=y... x1x1 y1y1 1 abc v 1 =.. x2x2 y2y2 0 v 2 =.. xnxn ynyn 0 v n =.... Player 1 gets all goods x@ae 1 @c..

24. Anti-symmetry: x @ a > y @ b  ¬ (y @ b > x @ a). Comparability to e 1 @ c: Either x @ a > (  ·e 1 ) @ c or x @ a x 1 ). Weak transitivity: x @ a > (  ·e 1 ) @ c > y @ b  ¬ (y @ b > x @ a). Remark: for unrestricted domains ' > ' is full order. Some properties of ' > '

25. The measure of x@a Dfn: The measure of x@a is defined as m( x @ a ) = inf {   R | x @ a < (  ·e 1 ) @ c }. Claim (measure preserves ‘>’) : If m( x @ a ) y @ b]. Corollary: f chooses alternative with highest measure. Left to show:

26. Measure is affine Claim: For any a and large enough  : m((x +  ·e i )@a) - m(x@a) = m(((  +  ) ·e i )@c i ) - m((  ·e i )@c i ), where c i is the allocation in which i gets all goods. Notice: This difference does not depend on x, or on a. Cor1: m((x +  ·e i )@a) - m(x@a) = h i (  ). (*) Cor2: measure is affine Proof: Any monotone function that has (*) is affine. m(((  +  )·e i )@c i ) · m(· @a) · m(· @ c i ) m(x@a) m((x+  ·e i )@a) m((  ·e i )@c i )

27. Summary We investigated the problem of characterizing truthful mechanisms for Combinatorial Auctions. We have seen the impact of two monotonicity types: –The weak one: characterizes truthfulness. –The strong one: implies affine maximization. –The difference between them is similar to Arrow’s IIA condition, and is w.l.o.g for some special cases. Corollary: truthfulness + IIA (+ minor technicalities) almost affine maximization computational hardness Main open question: Is IIA really necessary ?