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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,

Miodrag Potkonjak, Majid Sarrafzadeh

Computer Science Department, UCLA

WECWIS

June 27, 2002

- 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

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

- 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

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

- 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

- 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

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

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

- 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

- 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

- 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

- 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

- 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

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

- 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

- 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

- 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