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Single Parameter Combinatorial Auctions. Lei Wang Georgia Institute of Technology Joint work with Gagan Goel Chinmay Karande Google Georgia Tech. - PowerPoint PPT Presentation
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Single Parameter Combinatorial Auctions Lei Wang Georgia Institute of Technology Joint work with
Gagan Goel Chinmay Karande
Google Georgia Tech
Overview of Combinatorial Auction Setting
Mechanism Allocation: Payment: Truthfulness
Social welfare
Items: ,m E Agents: n A
Agent 's private valuation :ii F ( ), iF S S E
1
( )n
i ii
F S
1 ... nS S E
1,..., np p
reporting is dominantiF
Our Model and motivation
Motivation
might not be completely privateiF
Example: TV ad Auction
1: 00 2 : 00
2 : 00 3: 00
3: 00 4 : 00
4 : 00 5 : 00
5 : 00 6 : 00
private value :
viewers
v
x v x
( ) | ( ) |iF S v S
S
Our model and Motivation
Time slots
Advertiser
(S)
Our Model
Public function
Private value:
Valuations
Our Model and Motivation
: 2Ef
agent : ii v
agent 's valuation on : ( )ii S v f S
Single-parameter
Myerson’s Characterization of truthful mechanism Monotone allocation:
Payment is determined
Example: VCG mechanism
Approximation algorithm might not be monotone
, ' '
' ( ) ( ')
v S v S
v v f S f S
Our model and Motivation
1
max ( )n
i ii
v f S
Our result:
-approximate algorithm
log -approximate monotone algorithmn
log truthful mechanismn
Our Model and Motivation
Preliminary: Maximum In Range Mechanism
( )MIR v
: All allocations
1
max ( )n
A i ii
v f A
MIR
1 2( , ,..., )nv v v
OPT
( )approximation: min
( )v
MIR v
OPT v
Range:
Our conversion
Plan:
Choose a range R
Run MIR
Show:
-approxomation ALG log -monotone n
max ( , ) can be computed in polynomial timeA R SW v A
use ALG as a black box
( ) ( log ) max ( , )AOPT v n SW v A
Our conversion
1 2 ...... ,nv v v
1
( )n
i ii
OPT v f T
1 11
( ) ( )n
i
i i kki
v v f T
=area of the histogram
1( )
n
kkf T
1
1( )
n
kkf T
1( )f T
1v 2v nv
Fact: ( log ) max Rectangle Area under the curvev
( )h
1
1
1
0
0
0
1
( )
1( )
1( )
( ) log
v
v
v
h x dx
xh x dxx
h dxx
h v
Our conversion
v
1v 2v nv
1( )
n
kkf T
1
1( )
n
kkf T
1( )f T
Our conversion
1 11 1
(log ) max{ ( ),......, ( ),......, ( )}n i
n k i ik k
OPT n v f T v f T v f T
..........
1
( )i
kk
f T
1T 2T iT
1
( ) ALG( ,1)i
kk
f T i
Our conversion
Our conversion
Construction of our range
For each ,1 :i i n !ni
i
Step1:Split
Step 2: Allocate
Step1:Split
1=( ,..., )=ALG[ ,1]iiA S S i
Our conversion
1S 2S 3S
Step 2: Allocate
Our conversion
1S
2S3S
3S
3S
3S
3S
2S
2S
2S
2S
1S
1S
1S
1S
!ni
i
Range
(1)list of bundles ALG( ,1) for all size of "serviced agents";
(2)all possible allocations of bundles on the list.
i
1
!n
i
ni
i
Our conversion
Properties
max ( , ) can be computed in polynomial timeA R SW v A
( )log
max ( , )A R
OPT vn
SW v A
Our conversion
Proof
max ( , ) can be computed in polynomial timeA R SW v A
1 2 ...... nv v v
.......... ..........
1MAX( ): ( ,..., )iiA S S
1v 2v iv nv
1 2Suppose ( ) ( ) ...... ( )if S f S f S
1S 2S iS
iterate over the size
Proof: max Rectangle max ( , )A R SW v A
OPT (log ) max Rectanglen
OPT ( log ) max ( , )A Rn SW v A
( )log
max ( , )A R
OPT vn
SW v A
Conclusion
Conversion
-approxomation
log -approxomation+monotonen
Future direction Randomized mechanism
Randomized maximum in range
Randomized rounding
Truthfulness v.s. Approximability Huge clash in non-Bayesian setting
On the hardness of being truthful
C.Papadimitriou and Y.Singer FOCS’08 No clash in Bayesian setting
Bayesian algorithmic mechanism design
J.Hartline and B.Lucier STOC’10 Towards Optimal Bayesian Algorithmic Mechanism Desi
gn X.Bei and Z. Huang SODA’11 Is there any clash for single-parameter?
Thank you!
谢谢
