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

– vi is non-decreasing (“free disposal”)

– vi () = 0

• Objective: Find a partition (S1…Sn) of {1..k} that

maximizes the social welfare: i vi (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

AVVf n ...: 1

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]

})({maxarg)( i aiiAa avwvf

severely restricted domains

|

|

Multi Unit

Auctions (MUA)?

|

Combinatorial

Auctions (CA)?

unrestricted domain

|

Only affine maximizers

Many non-affine maximizers exist

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

viPreferences

Non-quasi-linearQuasi-linear

>i

Our ResultWanted 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: 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.

)()()()(: buaubvavi iiii

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 ImplicationsObservation: 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 MonotonicityDfn: 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.

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),(and),(

avaubvbu

bvufavvf

iiii

iiii

If the result changes

from a to b then i’s value for b increased at least as his value for a.

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 ) = b ui (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).

Proof: Truthfulness W-MON

Strong Monotonicity and IIADfn: 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!

)()()()(: buaubvavi iiii

Rest of Talk

Describe main building blocks of proof:

Part I : Truthfulness, Monotonicity, and IIA.

Part II :Strong monotonicity affine maximization.

Main TheoremTheorem: 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

. . . .x1 y1

a b

v1 =

. . . .x2 y2v2 =

. . . .xn ynvn =

.

.pla

yers

allocations

. . . .

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.

. . . .x1 y1 1

a b c

v1 =

. . . .x2 y2 0v2 =

. . . .xn yn 0vn =

.

.

Player 1 gets all goods

x@a e1@c

. . . .

Anti-symmetry:

x@a > y@b ¬ (y @b > x @a).

Comparability to e1@c:

Either x@a > ( ·e1)@c or x@a < ( ·e1)@c ( for > x1 ).

Weak transitivity:

x@a > ( ·e1)@c > y@b ¬ (y@b > x@a).

Remark: for unrestricted domains ' > ' is full order.

Some properties of ' > '

The measure of x@a Dfn: The measure 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:

ceax @)(@ 1 ceax @)(@ 1

)@( 1 ani ii xwaxm

Measure is affineClaim: 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(·· @a) m(·· @ ci)

m(x@a)

m((x+·ei)@a)

m((·ei)@ci)

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 ?