WECWIS, June 27, 2002 On the Sensitivity of On the Sensitivity of Incremental Algorithms for Incremental Algorithms for Combinatorial Auctions Combinatorial Auctions Ryan Kastner, Christina Hsieh, Miodrag Potkonjak, Majid Sarrafzadeh [email protected]Computer Science Department, UCLA WECWIS June 27, 2002
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WECWIS, June 27, 2002 On the Sensitivity of Incremental Algorithms for Combinatorial Auctions Ryan Kastner, Christina Hsieh, Miodrag Potkonjak, Majid Sarrafzadeh.
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WECWIS, June 27, 2002
On the Sensitivity of Incremental On the Sensitivity of Incremental Algorithms for Combinatorial Algorithms for Combinatorial
AuctionsAuctions
On the Sensitivity of Incremental On the Sensitivity of Incremental Algorithms for Combinatorial Algorithms for Combinatorial
Incremental Algorithms for CA Uses of Incremental CA ILP for Incremental Winner Determination
Results Conclusions
WECWIS, June 27, 2002
Combinatorial AuctionsCombinatorial AuctionsCombinatorial AuctionsCombinatorial Auctions 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
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
$$$
Maximize Maximize Objects Objects MMBids Bids BB
$9
$6
WECWIS, June 27, 2002
Winner Determination ProblemWinner Determination ProblemWinner Determination ProblemWinner 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)
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 solutionWe focus on the ILP solutionWe focus on the ILP solutionWe focus on the ILP solution
WECWIS, June 27, 2002
Winner Determination via ILPWinner Determination via ILPWinner Determination via ILPWinner Determination via ILP
1
0jxLet
if bid j is selected as a winner
otherwise
1
0ijc
otherwise
if item i is in bid j
B
iiixv
1
max s.t. ,11
B
jjijxc Mi ,,2,1
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
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
WECWIS, June 27, 2002
Supply Chains and CAsSupply Chains and CAs Supply Chains and CAsSupply 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
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
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|
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 determinationUse ILP to solve incremental winner determination
WECWIS, June 27, 2002
ILP for Incremental Winner DeterminationILP for Incremental Winner DeterminationILP for Incremental Winner DeterminationILP 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
Introduce a new decision variable zi corresponding to each winning bid b S0 that corresponds to b also being a winning bid in S1
1
0izLet
if bid i is not selected as a winner in S1
if bid i is selected as a winner in S1
For each bid bi S0
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
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
WECWIS, June 27, 2002
ILP for Incremental Winner DeterminationILP for Incremental Winner DeterminationILP for Incremental Winner DeterminationILP for Incremental Winner Determination
New objective function Maximize valuation of the winners Maintain winners from original (unperturbed) solution S0
New objective function Maximize valuation of the winners Maintain winners from original (unperturbed) solution S0
OW
iii
B
iii zwxv
11
max
s.t. Mi ,,2,1 ,11
B
jjijxc
OWbizx iii 1
Original constraint : every item won at most one time
Original constraint : every item won at most one time
New constraint : relates original winners to new winners
New constraint : relates original winners to new winners
wi – propensity for keeping bid as a winner (user assigned)
wi – propensity for keeping bid as a winner (user assigned)