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]
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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.
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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!
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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 ?
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