Combinatorial Auctions
By: Shai Roitman e-mail: [email protected]
Auctions
• One to many mechanism
• Efficient Allocation of the items.
• Seller Auctions
• Buyer Auctions - Reversed Auctions
Known Auction Types
• Open Cry Auctions– English– Dutch
• Sealed Bid Auctions– First Price – Second Price
The equivalence of auctions
• True Valuations– English – Sealed Bid Second Price
• Winners Curse– Dutch – Sealed Bid First Price
Sealed Bid Auctions advantages
• Communication efficient
• The value of the bid can be kept private.
Items Value
• Private Value - An Item has a value to the bidder regardless of the value to the other bidders– Example: Consumer goods
• Public Value - The item has value in the context of other bidder estimations– Example: Stocks
Strategies for the Auctionsunder private value assumptions
• English Auction– Small increments until maximum price(true
value) reached.
• Second price Sealed Auction– Submit the evaluated value as the bid
• First price Sealed Auction & Dutch Auction– Need to evaluate others evaluation (may use
some distribution on the values of the other bidders) and use this evaluation for setting the bid.
– Winners Curse
• Complex analysis
Strategies for Auctions - continued
Multi Item Auctions - Multi Stage Auction
• Scenario– A set of items has to be sold
• Naive Solution– Hold auctions for each item or set of items one
at a time
Multi Item Auctions - Problems
• How to choose the order of the items to be sold?
• How to bundle several dependant items?
• If the items have dependencies multi stage auctions can lead to inefficient allocation
Combinatorial auction
• Items may be grouped as bundles.
• => Takes into considerations the dependencies between the items.
• => Greater economic efficiency
The Utility function
• Private - Public Value
• Super Additive - Supplemental items
• Sub Additive - Complementary items
• Monotonic - The more the better
• Convex - Diversity
Uses for combinatorial auctions
• FCC Radio spectrum
• Logistics
• Scheduling
• Any purchase of dependant multiple items.
Logistic explicit use case of combinatorial auctions
• Logistics.com - OptiBid(TM)– Trucking companies bid on bundles of lanes– Logistics.com - More than $5 billion in
transportation contracts been bid to date (January 2000) (Ford, Wal-Mart, K-Mart).
Incentive Issues - An example
• 3 bidders {1,2,3}• 2 items {x,y}• Bidder 1 values
– {x,y}=100 {x}={y}=0
• Bidder 2 values– {x,y}=0 {x}={y}=75
• Bidder 3 values– {x,y}=0 {x}={y}=40
Incentive Issues - An example - continued
• If bid truthfully - x->2 , y->3 (Revenue 115)• If Bidder 2 and Bidder 3 belief that the others
truthfully bid their values– Bidder 2 can shade his value of {x} and {y} to 65
and still get the same x->2 y->3 (Revenue 105)– Bidder 3 can shade his value of {x} and {y} to 30
and still get the same x->2 y->3 (Revenue 105)
• If Bidder 2 & Bidder 3 shade their value (65 & 30) then they will lose as {x,y}->1
• => Lost of economic efficiency
Incentive Issues - An example - continued
Threshold Problem
• a collections of bidders whose combined valuation for distinct portions of a subset of items exceed the bid submitted on that subset by some other bidder.
• Difficulty in coordination of their bids to outbid the single large bidder on that subset
Auction Scheme assumptions
• Independent private values for bidders
• values draw from a commonly known distribution
• risk neutral
Auction Design- An optimal mechanism
• Truth Revelation - revelation principle
• No Bidder is made worse off by participating
• Seller Maximum Expected Revenue
Efficiency
• If the allocation of objects to bidders chosen by the seller solves the following equations than the auction is efficient
NjMSjSy
NjjSy
MijSy
ts
jSySv
AuctionEfficient
Otherwise
jtoallocatedisSBundlejSy
objectsofBundlesMSS
objectsdistinctofSetM
BiddersofSetN
Let
MS
Si Nj
Nj MS
j
,1,0),(
1),(
1),(
..
),()(max
0
1),(
.
:
General CAP Formalization
NjMSjSy
NjjSy
MijSy
ts
jSySb
problemAuctionialCombinatorCAP
Otherwise
jtoallocatedisSBundlejSy
SbSb
SbundleforannouncedhasNjagentthatbidTheSb
objectsofBundlesMSS
objectsdistinctofSetM
BiddersofSetN
Let
MS
Si Nj
Nj MS
j
i
Nj
i
,1,0),(
1),(
1),(
..
),()(max
)(
0
1),(
)(max)(
.)(
.
:
Vickrey Clarke Groves (VCG) - part 1
*
,1,0),(
1),(
1),(
..
),()(max
.2
.1
yallocationoptimalthisCall
NjMSjSy
NjjSy
MijSy
ts
jSySvV
solvesthatallocationthechoosessellerThe
vreportsjAgent
MS
Si Nj
Nj MS
j
j
Vickrey Clarke Groves (VCG) - part 2
k
MS
Si kNj
kNj MS
jk
ysolutionthisCall
kNjMSjSy
kNjjSy
MijSy
ts
jSySvV
Nkeachfor
makemustbiddereachthatpaymentthecomputeTo
\,1,0),(
\1),(
1),(
..
),()(max
:.3
\
\
0)(
),()()(
.4
*
kp
kSySvVVkp
toequalismakeskbidderthatpaymentThe
MS
kk
Vickrey Clarke Groves (VCG) - part 3
Nk
k VVV
venueSellerRe
• If no agent has a significant effect on the average V is close to V^(-k) thus the revenue is close to the maximum revenue defined in the General CAP.
Problems in the VCG mechanism
• Solving the CAP problem is hard (NP-Hard)
• Using Approximate solutions => Not incentive compatible
• Payments in VCG are sensitive to the choice of the solution
General CAP Formalization
NjMSjSy
NjjSy
MijSy
ts
jSySb
problemAuctionialCombinatorCAP
Otherwise
jtoallocatedisSBundlejSy
SbSb
SbundleforannouncedhasNjagentthatbidTheSb
objectsofBundlesMSS
objectsdistinctofSetM
BiddersofSetN
Let
MS
Si Nj
Nj MS
j
i
Nj
i
,1,0),(
1),(
1),(
..
),()(max
)(
0
1),(
)(max)(
.)(
.
:
Multiple object in the CAP Formulation
NjMSjSy
NjjSy
itemstheinisiobjecttimesofnumberlMiljSy
ts
jSySb
problemAuctionialCombinatorCAP
Otherwise
jtoallocatedisSBundlejSy
SbSb
SbundleforannouncedhasNjagentthatbidTheSb
objectsofBundlesMSS
objectsdistinctofSetM
BiddersofSetN
Let
MS
Si Nj
Nj MS
j
i
Nj
i
,1,0),(
1),(
),(
..
),()(max
)(
0
1),(
)(max)(
.)(
.
:
The CAP (Combinatorial Auction Problem)
• Bidders must submit bid for every subset
• Transmitting the bid sets in a succinct manner
Restriction of conditions => solvable solution - an example
• Restriction– All bidders complement each other– all bidders are symmetric
• Solution– Auction all the items as one item in an optimal
single item auction
Cybernomics experiments
• Performed tests for additive values and valuations with synergies of small , medium or high intensity
• Results– Combinatorial multi round auctions always
superior in efficiency but lower in revenue– Slower convergence (finishing the auctions)
The CAP - continued
• partial solutions– Restriction on the way the bids are transmitted
• OR / OR* Trees
• Single mind restriction
– Sending an Oracle
• Problem of deciding the collection of bids to accept
The SPP Problem
• Given a set of M elements
• collection V of subsets with weights
• Find the largest weight collection of subsets that are pairwise disjoint.
The SPP Formalization
Vjx
Mixa
ts
xc
oblemSPPThe
otherwise
MielementcontainsVinsetjtheifa
otherwise
selectediscweightwithVinsetjtheifx
j
Vjjij
Vjjj
th
ij
jth
j
1,0
1
..
max
Pr
0
1
0
1
SPP Related Problem - Set Partitioning Problem (SPA)
Vjx
Mixa
ts
xc
oblemSPAThe
otherwise
MielementcontainsVinsetjthefa
otherwise
selectediscweightwithVinsetjtheifx
j
Vjjij
Vjjj
th
ij
jth
j
1,0
1
..
max
Pr
0
1
0
1
SPP Related Problem - Set Covering Problem
Vjx
Mixa
ts
xc
oblemSCPThe
otherwise
MielementcontainsVinsetjthefa
otherwise
selectediscweightwithVinsetjtheifx
j
Vjjij
Vjjj
th
ij
jth
j
1,0
1
..
min
Pr
0
1
0
1
What is the complexity of SPP?
• SPP Is a NP-Hard / Complete problem
• SPP Problem is exponential in |V| (V the number of subsets of M)!
• No Hope??
Effective solution to the CAP Problems
• Requirements– Number of distinct bids is not large– Underlying SPP problem can be solved
reasonable quick.
SPP Approximation
• There is no Polynomial algorithm that can deliver a worst case ration larger than n^(E-1) for any E>0
• There is a worst case ratio of O(n/(log n)^2) algorithm (Polynomial algorithm)
Other Approaches
• Decentralized Methods– Setting up a fictitious market determining an
allocation and prices– Choosing an allocation and bidders are required
to send improvements
Conclusions
• Combinatorial Auctions can lead to higher economic efficiency
• Practical Combinatorial Auctions are hard to implement with compliance to the truth revelation principle
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