Loading in 2 Seconds...
Loading in 2 Seconds...
Towards a Characterization of Truthful Combinatorial Auctions. Ron Lavi, Ahuva Mu’alem, Noam Nisan Hebrew University. Combinatorial Auctions. k indivisible nonidentical items for sale n bidders compete for subsets of these items
Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.
Ron Lavi, Ahuva Mu’alem, Noam Nisan
Hebrew University
unrestricted
domain
severely restricted domains

Multi Unit
Auctions (MUA)?

Combinatorial
Auctions (CA)?
Only affine maximizers
Many nonaffine maximizers exist
Quasilinear
Nonquasilinear
Preferences
vi
>i
Implementable SCFs
Affinemaximizers
Dictatorial
Impossibility result for unrestricted domains
Roberts (79)
GibbardSatterthwaite (70’s) Arrow (50’s)
Other implementations in restricted domains?
Singledimensional: Yes
CAs, MUAs, … : ???
“SinglePeaked”: Yes
“Saturated”: No
Wanted THM For CAs (and similar domains): Every implementable SCF is an affine maximizer.
Proved THM For CAs (and similar domains): Every playerdecisive, nondegenerate implementable SCF that satisfies IIA is an almost affine maximizer.
Dfn: fsatisfies IIA if:
f(v)=a and f(u)=b
Justifications:
Part 1: Truthful monotone
Part 2: SMON + technicalities almost affine maximizer
Observation: Affine maximization is as computationally hard as exact maximization.
Corollary 1: Any truthful unanimityrespecting CA that satisfies IIA and achieves a poly(n,k) approximation is not polytime.
Dfn: f is unanimityrespecting if, whenever all players singlemindedly desire bundles that together form a partition, this partition is chosen.
Corollary 2: No truthful polytime CA/MUA for two players, that must allocate all items, achieves better than 2approximation.
Describe main building blocks of proof:
Part I : Truthfulness, Monotonicity, and IIA.
Part II :Strong monotonicity affine maximization.
vi(f(vi, vi))– pi(vi, vi) > vi(f(wi , vi))– pi(wi, vi)
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 WMON if for any vi ,vi and ui:
Thm:
Comments:
Prop: If f is truthful then pi(v) = pi (a, vi ), 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 , vi ).
Proof (Truthfulness WMON):
f (vi , vi ) = a vi (a)  pi(a, vi ) > vi (b)  pi(b, vi ),
otherwise player i would declare ui instead of vi.
f (ui , vi ) = bui (b)  pi(b, vi ) > ui (a)  pi(a, vi ),
otherwise player i would declare vi instead of ui.
ui (b)ui (a) >vi (b) vi (a).
Dfn:f satisfies SMON if for any vi ,vi and ui:
f (vi , vi) = a and f (ui , vi) = b
implies ui (b)ui (a) > vi (b) vi (a).
Dfn: f satisfies IIA if:f(v)=a and f(u)=b
Lemma 1: WMON + IIA = SMON(for CAs, MUAs, and related restricted domains)
Lemma 2: WMON implies (w.l.o.g) SMON for CAs/MUAs among two players, where all goods must always be allocated.
Describe main building blocks of proof:
Part I : Truthfulness, Monotonicity, and IIA.
Part II :Strong monotonicity affine maximization.
Theorem: For CAs, MUAs, and related domains:
A is nondegenerate +
f satisfies SMON +
f is player decisive
f must be almost affine maximizer.
The proof essentially shows that every mechanism for CA that satisfies SMON operates as follows:
It is affine maximizer.
allocations
. . . .
a
b
. . . .
v1 =
x1
y1
. . . .
v2 =
x2
y2
players
.
.
. . . .
vn =
xn
yn
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
Antisymmetry:
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.
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:
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)