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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

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## On the Sensitivity of Incremental Algorithms for Combinatorial Auctions

Ryan Kastner, Christina Hsieh,

kastner@cs.ucla.edu

Computer Science Department, UCLA

WECWIS

June 27, 2002

### 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

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

• 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

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

• 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

• 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

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

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

• 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

• 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

• 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

• 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

• 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

• 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

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 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

voluntary

dropouts

### Conclusions

• Main Idea: Incremental Combinatorial Auction

• Maximize valuation while maintaining solution

• Useful in many different contexts

• 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,

kastner@cs.ucla.edu

Computer Science Department, UCLA

WECWIS

June 27, 2002

### 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