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

**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: vi(S) = value that i assigns to acquiring the set S • viis non-decreasing (“free disposal”) • vi() = 0 • Objective: Find a partition (S1…Sn) of {1..k} that maximizes the social welfare: ivi(Si)

**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).

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

**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?

**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 vi(a), sum-squares; tradeoffs, … • What other truthful mechanisms are there?

**Abstraction: Social Choice Function** • A set of possible alternatives, A. • For CAs: A = {S1..Sn that are a partition of 1..k} • Each player has a valuation vi Vi,vi : A R • For CAs: Vi = {vi that satisfy 1, 2, 3}(1) depends only on Si (2) monotone (3) vi() = 0 • Truthful implementation: adding payments s.t. bidders will maximize their utility by revealing their true vi

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

**Comparison with the non-quasi-linear case** Quasi-linear Non-quasi-linear Preferences vi >i Implementable SCFs Affine-maximizers Dictatorial Impossibility result for unrestricted domains Roberts (79) Gibbard-Satterthwaite (70’s) Arrow (50’s) Other implementations in restricted domains? Single-dimensional: Yes CAs, MUAs, … : ??? “Single-Peaked”: Yes “Saturated”: No

**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.

**Independence of Irrelevant Alternatives** Dfn: fsatisfies IIA if: f(v)=a and f(u)=b Justifications: • We needed itin 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.

**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

**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

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

**Truthful Implementation of Social Choice Functions** • A mechanism is m = (f, p1,p2, , pn), where f isa SCF, and pi : V R is the payment function of player i. • Dfn:Truthful Implementation in dominant strategies [rational players tell the truth]: vi, v-i, wi : vi(f(vi, v-i))– pi(vi, v-i) > vi(f(wi , v-i))– pi(wi, v-i) • Not all SCFs can be implemented. If there exists an implementation it is essentially unique.

**Weak Monotonicity** If the result changes from a to b then i’s value for b increased at least as his value for a. Dfn:f satisfies W-MON if for any vi ,v-i and ui: 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.

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

**Strong Monotonicity and IIA** Dfn:f satisfies S-MON if for any vi ,v-i and ui: f (vi , v-i) = a and f (ui , v-i) = b implies ui (b)-ui (a) > vi (b) -vi (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!

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

**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 vi(S)>M for all i and non-empty bundles S, f is affine maximizer for v. f must be almost affine maximizer.

**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.

**The order induced by a S.C.F** allocations . . . . a b . . . . v1 = x1 y1 . . . . v2 = x2 y2 players . . . . . . vn = xn yn

**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. Player 1 gets all goods . . . . a b c . . . . v1 = x1 y1 1 . . . . v2 = x2 y2 0 . . . . . . vn = xn yn 0 x@a e1@c

**Some properties of ' > '** Anti-symmetry: x@a> y@b ¬ (y @b > x @a). Comparability to e1@c: Eitherx@a > ( ·e1)@c orx@a< ( ·e1)@c ( for > x1). Weak transitivity: x@a> ( ·e1)@c > y@b ¬ (y@b> x@a). Remark:for unrestricted domains '> 'is full order.

**The measure of x@a** Dfn: Themeasure of x@a is defined as m( x@a ) = inf { R | x@a< ( ·e1)@c }. Claim (measure preserves ‘>’) : If m( x@a ) < m( y@b ) then ¬ [x@a > y@b]. Corollary: f chooses alternative with highest measure. Left to show:

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

**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 ?