Post on 09-Apr-2018
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Iterative Combinatorial Auctions:
Theory and Practice
David C. Parkes and Lyle H. UngarAAAI-00, pp. 74-81
Presented by Xi LIMay 2006
COMP670OHKUST
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Why iterative?• In Generalized Vickrey Auction (GVA)
– 2|G| bundles to evaluate– A large winner-determination (WD) problem
– Optimal
• In iBundle– Agents bid for less bundles
– Each iteration contains a smaller WD problem– Also optimal for reasonable agent biddingstrategy
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Outline
• iBundle – the model
• Optimality
• Computational Analysis
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Notation
Bid – each bid contains a buddle S and its
corresponding bid price
For each buddle S in round t , there are
- bid price , provided by agent i
- ask price , announced by the auctioneer
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Bidding Price
• ptbid,i(S) ≥ pt
ask(S) - ε , where ε > 0 is the
minimal increment of ask price.• Agent can also bid ε below the ask price,
but then it cannot bid a higher price infuture rounds.
• Agent will NOT bid a price that higher
than the value of the bundle,i.e. pt
bid,i(S) · vi(S)
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XOR Bids• Agents can place multiple bids for exclusive-or
bundles, e.g. S1 XOR S2, to indicate that an agentwants either all items in S1 or all items in S2 butnot both S1 and S2 (The allocation guarantees a
agent is satisfied by at most one bid (bundle) ornothing).
i.e.S
1: {apple, soap, mushroom}
XOR
S 2 : {apple, baseball}
XOR
S 3 : {dictionary, beer}
Agent
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The Auction - iBundle
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Approximate WD
People will not get lose his bid if he propose more money for fewer goods.
1
1
From Lehmann et al.(1999)
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Myopic Best-Response Bidding Strategy
• A myopic agent bids to maximize utility at
the current ask prices.• The myopic best-response strategy is to
submit an XOR bid for all bundles S thatmaximize utility ui(S)=vi(S)-p(S) at thecurrent prices. This maximizes the
probability of a successful bid for bid-monotonic WD algorithms.
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Dummy item• When agent i have un-conflict bids, add an
dummy item to each bids, so that all those bidswill not be compatible with each other ( so thatthe XOR bids don’t need special care)
B1: {1, 2, 3}
B2: {4, 5, 6}
add dummy item Xi B1: {1, 2, 3, Xi}
B2: {4, 5, 6, Xi}
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Outline
• iBundle – the model
• Optimality
• Computational Analysis
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Optimality of iBundle
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Proof of Theorem 1• IP Model
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Proof of Theorem 1• LP Relaxation
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Proof of Theorem 1• LP Dual
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Proof of Theorem 1
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Complementary-slackness conditions•
•
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Optimality
The constructed primal solution is integral, so VLP is feasible and
optimal
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Termination
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Outline• iBundle – the model
• Optimality
• Computational Analysis
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Computational Analysis• In iBundle, the worst case gives O(BV max / ε )
to converge• The value of ε determines
– The number of rounds to termination– The allocative efficiency
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Computational Results