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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

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towards a characterization of truthful combinatorial auctions

Towards a Characterization of Truthful Combinatorial Auctions

Ron Lavi, Ahuva Mu’alem, Noam Nisan

Hebrew University

combinatorial auctions
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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 ) = 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).

slide18

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 talk1
Rest of Talk

Describe main building blocks of proof:

Part I : Truthfulness, Monotonicity, and IIA.

Part II :Strong monotonicity affine maximization.

main theorem
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
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
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 f1
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
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
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
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
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 ?