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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

Ryan Kastner, Christina Hsieh,

Miodrag Potkonjak, Majid Sarrafzadeh

Computer Science Department, UCLA

WECWIS

June 27, 2002

Outline Combinatorial Auctions

• 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 Combinatorial Auctions

• 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 Combinatorial Auctions

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 Combinatorial Auctions

• 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: Combinatorial AuctionsTask 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 Combinatorial Auctions

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 Combinatorial Auctions

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 Combinatorial Auctions

• 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 Combinatorial Auctions

• 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 Combinatorial Auctions

• 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 Combinatorial Auctions

• 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 Combinatorial Auctions

• 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

ILP for Incremental Winner Determination Combinatorial Auctions

• 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 Combinatorial Auctions

x

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 Combinatorial Auctions

• 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 Combinatorial Auctions

voluntary

dropouts

Results – 0% Involuntary Dropout Combinatorial Auctions

Conclusions Combinatorial Auctions

• 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 Auctions

Ryan Kastner, Christina Hsieh,

Miodrag Potkonjak, Majid Sarrafzadeh

Computer Science Department, UCLA

WECWIS

June 27, 2002

Extra Slides Combinatorial Auctions

Benchmarks Combinatorial Auctions

• 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

Results Combinatorial Auctions

Results Combinatorial Auctions