On the sensitivity of incremental algorithms for combinatorial auctions
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On the Sensitivity of Incremental Algorithms for Combinatorial Auctions. Ryan Kastner , Christina Hsieh, Miodrag Potkonjak, Majid Sarrafzadeh [email protected] 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

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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

[email protected]

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

[email protected]

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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