Transcript Document

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.
• 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?
[NR]
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), sumsquares; tradeoffs, …
• What other truthful mechanisms are there?
Abstraction: Social Choice Function
f : V1  . . . Vn  A
• 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)
f (v)  arg max aA { i wi vi (a )   a }
• Roberts ’79 : If Vi = R|A| (unrestricted domain) then only affine
maximizers can be implemented!
• For single dimensional domains (Vi = R), many non-affinemaximizers 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
severely restricted domains
|
Many non-affine
maximizers exist
|
Multi Unit
Auctions (MUA)
[BGN]
unrestricted
domain
|
|
?
Combinatorial
Auctions (CA)
?
Only affine
maximizers
Comparison with the non-quasi-linear case
Quasi-linear
Non-quasi-linear
Preferences
vi
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
>i
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 playerdecisive, 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
i : vi (a )  vi (b)  ui (a )  ui (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.
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 is
a 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
Dfn: f satisfies W-MON if for any vi , v-i and ui:
f ( vi , vi )  a and f (ui , vi )  b
implies ui (b)  vi (b)  ui ( a )  vi ( a )
If the result changes
from a to b then i’s
value for b increased at
least as his value for a.
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 ) = 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).
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
i : vi (a )  vi (b)  ui (a )  ui (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 must be almost
f satisfies S-MON +
affine maximizer.
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.
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
players
allocations
a
b
. . . .
v1 =
x1
y1
. . . .
v2 =
x2
y2
. . . .
.
.
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.
a
b
. . . .
c
v1 =
x1
y1
. . . .
1
v2 =
x2
y2
. . . .
0
.
.
vn =
xn
x@a
yn
. . . .
0
e1@c
Player 1
gets all
goods
Some properties of ' > '
Anti-symmetry:
x@a > y@b  ¬ (y @b > x @a).
Comparability to e1@c:
Either x@a > ( ·e1)@c or x@a < ( ·e1)@c
Weak transitivity:
x@a > ( ·e1)@c > y@b  ¬ (y@b > x@a).
Remark: for unrestricted domains ' > ' is full order.
( for  > x1 ).
The measure of x@a
Dfn: The measure of x@a is defined as
m( x@a ) = inf {  R | x@a < ( ·e1)@c }.
x @ a  (  e1 ) @ c
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:
m( x @ a)  in1 wi  xi   a

Measure is affine
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.
m(· @a)
m(· @ ci)
m((( +)·ei)@ci)
m((·ei)@ci)
m((x+·ei)@a)
m(x@a)
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.
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 ?