on the sensitivity of incremental algorithms for combinatorial auctions

WECWIS, June 27, 2002

On the Sensitivity of Incremental On the Sensitivity of Incremental Algorithms for Combinatorial Algorithms for Combinatorial

AuctionsAuctions


AuctionsAuctions

Ryan Kastner, Christina Hsieh,

Miodrag Potkonjak, Majid Sarrafzadeh

[email protected]

Computer Science Department, UCLA

WECWIS

June 27, 2002



[email protected]


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


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


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


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



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


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)}


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


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


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


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


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


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


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


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



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)


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


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


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


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


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



AuctionsAuctions


AuctionsAuctions



[email protected]


WECWIS

June 27, 2002



[email protected]


WECWIS

June 27, 2002


Extra SlidesExtra SlidesExtra SlidesExtra Slides


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


ResultsResultsResultsResultsRegions


0.94

0.95

0.96

0.97

0.98

0.99

1

-10 0 10 20 30 40 50


Dif

fere

nce

in

Ob

ject

ive

Val

ue

5%

15%

30%


ResultsResultsResultsResultsPaths


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


Dif

fere

nce

in

Ob

ject

ive

Val

ue

5%

15%

30%

on the sensitivity of incremental algorithms for combinatorial auctions

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