Combinatorial Auctions
By: Shai Roitman
e-mail: shairoi@cs.huji.ac.il
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 Auctions
under private value assumptions
• English Auction
– Small increments until maximum price(true
value) reached.
• Second price Sealed Auction
– Submit the evaluated value as the bid
Strategies for Auctions continued
• 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
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
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•
•
•
•
Private - Public Value
Super Additive - Supplemental items
Sub Additive - Complementary items
Monotonic - The more the better
Convex - Diversity
Uses for combinatorial auctions
•
•
•
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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)
Incentive Issues - An example continued
• If Bidder 2 & Bidder 3 shade their value (65
& 30) then they will lose as {x,y}->1
• => Lost of economic efficiency
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
Let :
N  Set of Bidders
M  Set of distinct objects
S .S  M  Bundles of objects
1
y(S , j)  
0
Efficient Auction
max 
v
j
Bundle S is allocated to j 

Otherwise

(S ) y(S , j)
jN S  M
s.t.
  y ( S , j ) 1
i  M
 y(S , j)  1
j  N
iS jN
S M
y ( S , j )  0,1S  M , j  N
General CAP Formalization
Let :
N  Set of Bidders
M  Set of distinct objects
S .S  M  Bundles of objects
b i ( S )  The bid that agent j  N has announced for bundle S .
b( S )  max b i ( S )
jN
Bundle S is allocated to j 
1
y(S , j)  

0
Otherwise


CAP (Combinatorial Auction problem )
max 
b
j
(S ) y(S , j)
jN S  M
s.t.
  y ( S , j ) 1
i  M
 y(S , j)  1
j  N
iS jN
S M
y ( S , j )  0,1S  M , j  N
Vickrey Clarke Groves (VCG) part 1
1. Agent j reports v j
2.The seller chooses the allocation that solves
V  max 
v
j
(S ) y(S , j)
jN S  M
s.t.
  y ( S , j ) 1
i  M
 y(S , j)  1
j  N
iS jN
S M
y ( S , j )  0,1S  M , j  N
Call this optimal allocation y *
Vickrey Clarke Groves (VCG) part 2
3.To compute the payment that each bidder must make :
for each k  N
V  k  max
 v
j
(S ) y(S , j)
jN \ k S  M
s.t.
  y ( S , j ) 1
i  M
 y(S , j)  1
j  N \ k
iS jN \ k
S M
y ( S , j )  0,1S  M , j  N \ k
Call this solution y k
4.The payment that bidder k makes is equal to


p (k )  V  k  V   v k ( S ) y * ( S , k )
S M


p(k )  0
Vickrey Clarke Groves (VCG) part 3
Seller Re venue

V   V k  V

kN
• 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
Let :
N  Set of Bidders
M  Set of distinct objects
S .S  M  Bundles of objects
b i ( S )  The bid that agent j  N has announced for bundle S .
b( S )  max b i ( S )
jN
Bundle S is allocated to j 
1
y(S , j)  

0
Otherwise


CAP (Combinatorial Auction problem )
max 
b
j
(S ) y(S , j)
jN S  M
s.t.
  y ( S , j ) 1
i  M
 y(S , j)  1
j  N
iS jN
S M
y ( S , j )  0,1S  M , j  N
Multiple object in the CAP
Formulation
Let :
N  Set of Bidders
M  Set of distinct objects
S .S  M  Bundles of objects
b i ( S )  The bid that agent j  N has announced for bundle S .
b( S )  max b i ( S )
jN
Bundle S is allocated to j 
1
y(S , j)  

Otherwise
0

CAP (Combinatorial Auction problem )
max
 b
j
(S ) y(S , j)
jN S  M
s.t.
  y(S , j)  l
i  M l  number of times object i is in the items
 y(S , j)  1
j  N
iS jN
S M
y ( S , j )  0,1S  M , j  N
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
1 if the j th set in V with weight c j is selected 
xj  

0
otherwise


1 if the j th set in V contains element i  M 
aij  

0
otherwise


The SPP Pr oblem
max  c j x j
jV
s.t.
a
jV
ij
x j  1 i  M
x j  0,1 j  V
SPP Related Problem - Set
Partitioning Problem (SPA)
1 if the j th set in V with weight c j is selected 
xj  

0
otherwise


1 f the j th set in V contains element i  M 
aij  

0
otherwise


The SPA Pr oblem
max  c j x j
jV
s.t.
a
jV
ij
x j  1 i  M
x j  0,1 j  V
SPP Related Problem - Set
Covering Problem
1 if the j th set in V with weight c j is selected 
xj  

0
otherwise


1 f the j th set in V contains element i  M 
aij  

0
otherwise


The SCP Pr oblem
min
c
jV
j
xj
s.t.
a
jV
ij
x j  1 i  M
x j  0,1 j  V
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^(E1) 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