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WECWIS, June 27, 2002
On the Sensitivity of Incremental On the Sensitivity of Incremental Algorithms for Combinatorial Algorithms for Combinatorial
AuctionsAuctions
On the Sensitivity of Incremental On the Sensitivity of Incremental Algorithms for Combinatorial Algorithms for Combinatorial
AuctionsAuctions
Ryan Kastner, Christina Hsieh,
Miodrag Potkonjak, Majid Sarrafzadeh
kastner@cs.ucla.edu
Computer Science Department, UCLA
WECWIS
June 27, 2002
Ryan Kastner, Christina Hsieh,
Miodrag Potkonjak, Majid Sarrafzadeh
kastner@cs.ucla.edu
Computer Science Department, UCLA
WECWIS
June 27, 2002
WECWIS, June 27, 2002
OutlineOutlineOutlineOutline
Basics Combinatorial Auctions (CA) Integer Linear Programming (ILP) for Winner Determination
Motivating Example: Supply Chains Incremental Algorithms
Incremental Algorithms for CA Uses of Incremental CA ILP for Incremental Winner Determination
Results Conclusions
Basics Combinatorial Auctions (CA) Integer Linear Programming (ILP) for Winner Determination
Motivating Example: Supply Chains Incremental Algorithms
Incremental Algorithms for CA Uses of Incremental CA ILP for Incremental Winner Determination
Results Conclusions
WECWIS, June 27, 2002
Combinatorial AuctionsCombinatorial AuctionsCombinatorial AuctionsCombinatorial Auctions Given a set of distinct objects M and set of bids B where
B is a tuple S v s.t. S powerSet{M} and v is a positive real number, determine a set of bids W (W B) s.t. w·v is maximized
Given a set of distinct objects M and set of bids B where B is a tuple S v s.t. S powerSet{M} and v is a positive real number, determine a set of bids W (W B) s.t. w·v is maximized
$$$
Maximize Maximize Objects Objects MMBids Bids BB
$9
$6
WECWIS, June 27, 2002
Winner Determination ProblemWinner Determination ProblemWinner Determination ProblemWinner Determination Problem
Informal Definition: Auctioneer must figure out who to give the items to in order to make the most money
NP-Hard need heuristics to quickly solve large instances
Many exact methods to solve winner determination problem
Dynamic Programming – Rothkopf et al. Optimized Search – Sandholm CASS, VSA, CA-MUS – Layton-Brown et al. Integer Linear Program (ILP)
Informal Definition: Auctioneer must figure out who to give the items to in order to make the most money
NP-Hard need heuristics to quickly solve large instances
Many exact methods to solve winner determination problem
Dynamic Programming – Rothkopf et al. Optimized Search – Sandholm CASS, VSA, CA-MUS – Layton-Brown et al. Integer Linear Program (ILP)
We focus on the ILP solutionWe focus on the ILP solutionWe focus on the ILP solutionWe focus on the ILP solution
WECWIS, June 27, 2002
Winner Determination via ILPWinner Determination via ILPWinner Determination via ILPWinner Determination via ILP
1
0jxLet
if bid j is selected as a winner
otherwise
1
0ijc
otherwise
if item i is in bid j
B
iiixv
1
max s.t. ,11
B
jjijxc Mi ,,2,1
Let xj be a decision variable that determines if bid j is selected as a winner
Let cij be a decision variable relating item i to bid j
Let vi be the valuation of bid j
Let xj be a decision variable that determines if bid j is selected as a winner
Let cij be a decision variable relating item i to bid j
Let vi be the valuation of bid j
WECWIS, June 27, 2002
Supply Chains and CAsSupply Chains and CAs Supply Chains and CAsSupply Chains and CAs
Trend: Supply chains becoming large and dynamic More complementary companies – larger supply chains Specialization becoming prevalent – deeper supply chains Market changes rapidly – need quick reformation Automated negotiation – CA for supply chains
Supply Chain formation/negotiation through CA Welsh et al. give an CA approach to solving supply chain
problem Model supply chain through task dependency network
Trend: Supply chains becoming large and dynamic More complementary companies – larger supply chains Specialization becoming prevalent – deeper supply chains Market changes rapidly – need quick reformation Automated negotiation – CA for supply chains
Supply Chain formation/negotiation through CA Welsh et al. give an CA approach to solving supply chain
problem Model supply chain through task dependency network
Large, dynamic supply chains require automated negotiation/formation
Large, dynamic supply chains require automated negotiation/formation
WECWIS, June 27, 2002
Modeling Supply Chains: Modeling Supply Chains: Task Dependency GraphTask Dependency GraphModeling Supply Chains: Modeling Supply Chains: Task Dependency GraphTask Dependency Graph
A1
$4
A2
$3
G1
G2
A4
$9
A3
$5
A5
$5
G3
G4
C1
$12.27
C2
$21.68
Goods labeled as circles Producers/consumers labeled as rectangles Arrows indicate the goods needed to produce another
good Bids are the number of goods needed/produced and the
price to produce e.g. bid(A4) = {$9,(G1,1),(G2,1),(G4,1)}
Goods labeled as circles Producers/consumers labeled as rectangles Arrows indicate the goods needed to produce another
good Bids are the number of goods needed/produced and the
price to produce e.g. bid(A4) = {$9,(G1,1),(G2,1),(G4,1)}
WECWIS, June 27, 2002
Supply Chains and CASupply Chains and CASupply Chains and CASupply Chains and CA “Winning” bidders (companies) are included in supply
chain CA guarantees an optimal supply chain formation
Allocation of goods is efficient – producers get all input goods they need
Maximizes the value of the supply chain – the goods that are produced are done so in the least expensive possible manner
“Winning” bidders (companies) are included in supply chain
CA guarantees an optimal supply chain formation Allocation of goods is efficient – producers get all input goods
they need Maximizes the value of the supply chain – the goods that are
produced are done so in the least expensive possible manner
A3
$5
A5
$5
G3C1
$12.27
A1
$4
A2
$3
G1
G2
A4
$9
G4C2
$21.68
A3
$5
A5
$5
G3C1
$12.27
A1
$4
A2
$3
G1
G2
A4
$9
G4C2
$21.68
Efficient AllocationEfficient Allocation
WECWIS, June 27, 2002
Supply Chain PerturbationSupply Chain PerturbationSupply Chain PerturbationSupply Chain Perturbation What happens when there is a change in the supply
chain? Want to keep current producer/consumer relationships intact Want to maximize the efficiency of supply chain
Not always possible to maintain previous relationships when supply chain changes
What happens when there is a change in the supply chain?
Want to keep current producer/consumer relationships intact Want to maximize the efficiency of supply chain
Not always possible to maintain previous relationships when supply chain changes
Perturbation: Perturbation: A4 changes cost from $9 to $20A4 changes cost from $9 to $20
A1
$4
A2
$3
G1
G2
A4
$9
G4C2
$21.68
A3
$5
A5
$5
G3C1
$12.27
A1
$4
A2
$3
G1
G2
A4
$9
G4C2
$21.68
A1
$4
A2
$3
G1
G2
A4
$20
A3
$5
A5
$5
G3
G4
C1
$12.27
C2
$21.68
Perturbation: Perturbation: A4 changes cost from $9 to $20A4 changes cost from $9 to $20
WECWIS, June 27, 2002
Incremental AlgorithmsIncremental AlgorithmsIncremental AlgorithmsIncremental Algorithms
An original instance I0 of a problem is solved by a full algorithm to give solution S0
Perturbed instances, I1,I2,,In are generated one by one in sequence
Each instance is solved by an incremental algorithm which uses Si-1 as a starting point find solution Si
An original instance I0 of a problem is solved by a full algorithm to give solution S0
Perturbed instances, I1,I2,,In are generated one by one in sequence
Each instance is solved by an incremental algorithm which uses Si-1 as a starting point find solution Si
WECWIS, June 27, 2002
Perturbations for CAPerturbations for CAPerturbations for CAPerturbations for CA
A bidder retracts their bid. This removes the bid from consideration
A bidder changes the valuation of their bid
A bidder prefers a different set of items
A new bidder enters the bidding process
A bidder retracts their bid. This removes the bid from consideration
A bidder changes the valuation of their bid
A bidder prefers a different set of items
A new bidder enters the bidding process
$9
$5 $7
$5 $5
WECWIS, June 27, 2002
Uses for Incremental CAUses for Incremental CAUses for Incremental CAUses for Incremental CA
Supply chain reformation/adjustment
Iterative Combinatorial Auctions Progressive combinatorial auction – bidding done in rounds Different protocols governing various aspects
Stopping conditions, price reporting, rules to withdrawal bids Types of Iterative CA
AkBA – Wurman and Wellman iBundle – Parkes and Unger Generalized Vickrey Auction – Varian and MacKie-Mason
Aid development of heuristics for large instances of CA
Supply chain reformation/adjustment
Iterative Combinatorial Auctions Progressive combinatorial auction – bidding done in rounds Different protocols governing various aspects
Stopping conditions, price reporting, rules to withdrawal bids Types of Iterative CA
AkBA – Wurman and Wellman iBundle – Parkes and Unger Generalized Vickrey Auction – Varian and MacKie-Mason
Aid development of heuristics for large instances of CA
WECWIS, June 27, 2002
Incremental Winner DeterminationIncremental Winner DeterminationIncremental Winner DeterminationIncremental Winner Determination
Given an original instance I0 of a problem solved by a full algorithm to give solution S0
S0 is the set of winners which we call the original winners OW Determined through ILP – exact solution
I0 is perturbed to give a new instance I1
We wish to find a solution S1 to the instance I1 while: Maximizing the valuation of the bids in the solution S1
Maintaining the original winners from solution S0 i.e. maximize |S0 S1|
Given an original instance I0 of a problem solved by a full algorithm to give solution S0
S0 is the set of winners which we call the original winners OW Determined through ILP – exact solution
I0 is perturbed to give a new instance I1
We wish to find a solution S1 to the instance I1 while: Maximizing the valuation of the bids in the solution S1
Maintaining the original winners from solution S0 i.e. maximize |S0 S1|
Use ILP to solve incremental winner determinationUse ILP to solve incremental winner determination
WECWIS, June 27, 2002
ILP for Incremental Winner DeterminationILP for Incremental Winner DeterminationILP for Incremental Winner DeterminationILP for Incremental Winner Determination
Introduce a new decision variable zi corresponding to each winning bid b S0 that corresponds to b also being a winning bid in S1
Introduce a new decision variable zi corresponding to each winning bid b S0 that corresponds to b also being a winning bid in S1
1
0izLet
if bid i is not selected as a winner in S1
if bid i is selected as a winner in S1
For each bid bi S0
Other other variables similar to ILP for winner determination Let xj be a decision variable that determines if bid j is selected as a
winner Let cij be a decision variable relating item i to bid j
Let vi be the valuation of bid j
Other other variables similar to ILP for winner determination Let xj be a decision variable that determines if bid j is selected as a
winner Let cij be a decision variable relating item i to bid j
Let vi be the valuation of bid j
WECWIS, June 27, 2002
ILP for Incremental Winner DeterminationILP for Incremental Winner DeterminationILP for Incremental Winner DeterminationILP for Incremental Winner Determination
New objective function Maximize valuation of the winners Maintain winners from original (unperturbed) solution S0
New objective function Maximize valuation of the winners Maintain winners from original (unperturbed) solution S0
OW
iii
B
iii zwxv
11
max
s.t. Mi ,,2,1 ,11
B
jjijxc
OWbizx iii 1
Original constraint : every item won at most one time
Original constraint : every item won at most one time
New constraint : relates original winners to new winners
New constraint : relates original winners to new winners
wi – propensity for keeping bid as a winner (user assigned)
wi – propensity for keeping bid as a winner (user assigned)
WECWIS, June 27, 2002
Experimental FlowExperimental FlowExperimental FlowExperimental Flow
# bids# bids
# goods# goodsCATS
Winner
determination
ILP solver
II00 SS00
Add
perturbation
(randomly
remove x%
of winning
bids)
xx
Winner
determination
ILP solver
II11
optimal Soptimal S11
objective valueobjective value
Incremental
winner
determination
ILP solver
incremental Sincremental S11
objective valueobjective value
% involuntary% involuntary
dropoutsdropouts
WECWIS, June 27, 2002
BenchmarksBenchmarksBenchmarksBenchmarks
Combinatorial Auction Test Suite (CATS) – Leyton-Brown et al.
We focused on three specific distributions Matching – correspondence of time slices on multiple
resources e.g. airport takeoff/landing rights Regions – adjacency in two dimensional space e.g.
drilling rights Paths – purchase of connection between two points
e.g. truck routes
Combinatorial Auction Test Suite (CATS) – Leyton-Brown et al.
We focused on three specific distributions Matching – correspondence of time slices on multiple
resources e.g. airport takeoff/landing rights Regions – adjacency in two dimensional space e.g.
drilling rights Paths – purchase of connection between two points
e.g. truck routes
WECWIS, June 27, 2002
ResultsResultsResultsResultsMatching
% Involuntary Dropouts vs. Difference in Objective Value
0.96
0.965
0.97
0.975
0.98
0.985
0.99
0.995
1
-2 0 2 4 6 8 10 12 14 16
% Involuntary Dropouts
Dif
fere
nce
in
Ob
ject
ive
Val
ue
5%
10%
15%
30%
voluntary
dropouts
WECWIS, June 27, 2002
Results – 0% Involuntary DropoutResults – 0% Involuntary DropoutResults – 0% Involuntary DropoutResults – 0% Involuntary DropoutMatching (15% Voluntary Dropout)
Difference in Objective Value
0.86
0.88
0.9
0.92
0.94
0.96
0.98
1
0 5000 10000 15000 20000 25000
# of Bids
Dif
fere
nce
in
Ob
ject
ive
Val
ue
WECWIS, June 27, 2002
ConclusionsConclusionsConclusionsConclusions Main Idea: Incremental Combinatorial Auction
Maximize valuation while maintaining solution Useful in many different contexts
Supply chain reformation/adjustment Iterative Combinatorial Auctions
Studied incremental tradeoff through incremental CA ILP formulation
Increased perturbation leads to worse solution Large instances can be solved near-optimally while maintaining
solution Future work
Incremental CA algorithms Fault tolerant CA solutions
Main Idea: Incremental Combinatorial Auction Maximize valuation while maintaining solution
Useful in many different contexts Supply chain reformation/adjustment Iterative Combinatorial Auctions
Studied incremental tradeoff through incremental CA ILP formulation
Increased perturbation leads to worse solution Large instances can be solved near-optimally while maintaining
solution Future work
Incremental CA algorithms Fault tolerant CA solutions
WECWIS, June 27, 2002
On the Sensitivity of Incremental On the Sensitivity of Incremental Algorithms for Combinatorial Algorithms for Combinatorial
AuctionsAuctions
On the Sensitivity of Incremental On the Sensitivity of Incremental Algorithms for Combinatorial Algorithms for Combinatorial
AuctionsAuctions
Ryan Kastner, Christina Hsieh,
Miodrag Potkonjak, Majid Sarrafzadeh
kastner@cs.ucla.edu
Computer Science Department, UCLA
WECWIS
June 27, 2002
Ryan Kastner, Christina Hsieh,
Miodrag Potkonjak, Majid Sarrafzadeh
kastner@cs.ucla.edu
Computer Science Department, UCLA
WECWIS
June 27, 2002
WECWIS, June 27, 2002
Extra SlidesExtra SlidesExtra SlidesExtra Slides
WECWIS, June 27, 2002
BenchmarksBenchmarksBenchmarksBenchmarks
Matching 35 instances ~[25 – 20000] bids ~[50 – 3600] goods
Paths 21 instances ~[100 – 20000] bids ~[30 – 2800] goods
Regions 18 instances ~[100 – 10000] bids ~[40 – 2000] goods
Matching 35 instances ~[25 – 20000] bids ~[50 – 3600] goods
Paths 21 instances ~[100 – 20000] bids ~[30 – 2800] goods
Regions 18 instances ~[100 – 10000] bids ~[40 – 2000] goods
WECWIS, June 27, 2002
ResultsResultsResultsResultsRegions
% Involuntary Dropouts vs. Difference in Objective Value
0.94
0.95
0.96
0.97
0.98
0.99
1
-10 0 10 20 30 40 50
% Involuntary Dropouts
Dif
fere
nce
in
Ob
ject
ive
Val
ue
5%
15%
30%
WECWIS, June 27, 2002
ResultsResultsResultsResultsPaths
% Involuntary Dropouts vs. Difference in Objective Value
0.989
0.99
0.991
0.992
0.993
0.994
0.995
0.996
0.997
0.998
0.999
1
-0.5 0 0.5 1 1.5 2 2.5
% Involuntary Dropouts
Dif
fere
nce
in
Ob
ject
ive
Val
ue
5%
15%
30%
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