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
• kastner_at_cs.ucla.edu
• Computer Science Department, UCLA
• WECWIS
• June 27, 2002

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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

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Combinatorial Auctions

• Given a set of distinct objects M and set of bids
  B where B is a tuple S ? v s.t. S ? powerSetM
  and v is a positive real number, determine a set
  of bids W (W ? B) s.t. ? wv is maximized

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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
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Winner Determination via ILP

• Let xj be a decision variable that determines if
  bid j is selected as a winner
• Let cij be a decision variable relating item i to
  bid j
• Let vi be the valuation of bid j

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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
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Modeling Supply Chains Task Dependency Graph

• Goods labeled as circles
• Producers/consumers labeled 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
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Supply Chains and CA

• 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

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Supply Chain Perturbation

• 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

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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

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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

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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

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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
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ILP for Incremental Winner Determination

• Introduce a new decision variable zi
  corresponding to each winning bid b ? S0 that
  corresponds to b also being a winning bid in S1

For each bid bi ? S0
if bid i is selected as a winner in S1
Let
if bid i is not selected as a winner in S1

• Other other variables similar to ILP for winner
  determination
• Let xj be a decision variable that determines if
  bid j is selected as a winner
• Let cij be a decision variable relating item i to
  bid j
• Let vi be the valuation of bid j

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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

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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
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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

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Results
voluntary dropouts
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Results 0 Involuntary Dropout
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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

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On the Sensitivity of Incremental Algorithms for
Combinatorial Auctions

• Ryan Kastner, Christina Hsieh,
• Miodrag Potkonjak, Majid Sarrafzadeh
• kastner_at_cs.ucla.edu
• Computer Science Department, UCLA
• WECWIS
• June 27, 2002

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Extra Slides
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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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Results
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Results