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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

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
slide15
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

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

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
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