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On the Sensitivity of Incremental Algorithms for Combinatorial Auctions. Ryan Kastner , Christina Hsieh, Miodrag Potkonjak, Majid Sarrafzadeh kastner@cs.ucla.edu Computer Science Department, UCLA WECWIS June 27, 2002. Outline. Basics Combinatorial Auctions (CA)

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On the sensitivity of incremental algorithms for combinatorial auctions

On the Sensitivity of Incremental Algorithms for Combinatorial Auctions

Ryan Kastner, Christina Hsieh,

Miodrag Potkonjak, Majid Sarrafzadeh

kastner@cs.ucla.edu

Computer Science Department, UCLA

WECWIS

June 27, 2002


Outline

Outline

  • 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 auctions

Combinatorial Auctions

Maximize

Bids B

Objects M

$$$

$9

$6

  • 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


Winner determination problem

Winner 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)

We focus on the ILP solution


Winner determination via ilp

Winner Determination via ILP

otherwise

Let

if bid j is selected as a winner

otherwise

if item i is in bid j

s.t.

  • Let xjbe a decision variable that determines if bid j is selected as a winner

  • Let cijbe a decision variable relating item i to bid j

  • Let vibe the valuation of bid j


Supply chains and cas

Supply 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

Large, dynamic supply chains require automated negotiation/formation


Modeling supply chains task dependency graph

Modeling Supply Chains: Task Dependency Graph

  • Goodslabeled as circles

  • Producers/consumerslabeled 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)}

A3

$5

G1

G3

A1

$4

C1

$12.27

A4

$9

G2

G4

A2

$3

C2

$21.68

A5

$5


Supply chains and ca

Supply Chains and CA

Efficient Allocation

A3

$5

A3

$5

G1

G1

G3

G3

A1

$4

A1

$4

C1

$12.27

C1

$12.27

A4

$9

A4

$9

G2

G2

G4

G4

A2

$3

A2

$3

C2

$21.68

C2

$21.68

A5

$5

A5

$5

  • “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


Supply chain perturbation

Supply Chain Perturbation

A3

$5

G1

G3

A1

$4

C1

$12.27

A4

$20

G2

G4

A2

$3

C2

$21.68

Perturbation: A4 changes cost from $9 to $20

Perturbation: A4 changes cost from $9 to $20

A5

$5

A3

$5

G1

G1

G3

A1

$4

A1

$4

C1

$12.27

A4

$9

A4

$9

G2

G2

G4

G4

A2

$3

A2

$3

C2

$21.68

C2

$21.68

A5

$5

  • 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


Incremental algorithms

Incremental 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


Perturbations for ca

Perturbations 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

$9

$5

$7

$5

$5


Uses for incremental ca

Uses 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


Incremental winner determination

Incremental 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|

Use ILP to solve incremental winner determination


Ilp for incremental winner determination

ILP for Incremental Winner Determination

  • Introduce a new decision variable zicorresponding to each winning bid b S0that corresponds to b also being a winning bid in S1

For each bid bi S0

if bid iis selected as a winner in S1

Let

if bid iis not selected as a winner in S1

  • Other other variables similar to ILP for winner determination

    • Let xjbe a decision variable that determines if bid j is selected as a winner

    • Let cijbe a decision variable relating item i to bid j

    • Let vibe the valuation of bid j


On the sensitivity of incremental algorithms for combinatorial auctions

ILP for Incremental Winner Determination

  • New objective function

    • Maximize valuation of the winners

    • Maintain winners from original (unperturbed) solution S0

  • wi– propensity for keeping bid as a winner (user assigned)

  • Original constraint : every item won at most one time

s.t.

  • New constraint : relates original winners to new winners


Experimental flow

Experimental Flow

x

Add

perturbation

(randomly

remove x%

of winning

bids)

CATS

Winner

determination

ILP solver

S0

# bids

I0

# goods

Incremental

winner

determination

ILP solver

% involuntary

dropouts

I1

Winner

determination

ILP solver

incremental S1

objective value

optimal S1

objective value


Benchmarks

Benchmarks

  • 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


Results

Results

voluntary

dropouts


Results 0 involuntary dropout

Results – 0% Involuntary Dropout


Conclusions

Conclusions

  • 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


On the sensitivity of incremental algorithms for combinatorial auctions1

On the Sensitivity of Incremental Algorithms for Combinatorial Auctions

Ryan Kastner, Christina Hsieh,

Miodrag Potkonjak, Majid Sarrafzadeh

kastner@cs.ucla.edu

Computer Science Department, UCLA

WECWIS

June 27, 2002


Extra slides

Extra Slides


Benchmarks1

Benchmarks

  • 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


Results1

Results


Results2

Results


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