Combinatorial Auctions Bidding and Allocation

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Combinatorial Auctions(Bidding and Allocation)

• Adapted from Noam Nisan

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The Setting

• Set of Products
• Each customer can bid 700 for AND
  1200 for OR 8 for
  6 for XOR 30 for 3
  for ANY 3

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Examples

• Classic
• (take-off right) AND (landing right)
• (frequency A) XOR (frequency B)
• E-commerce
• chair AND sofa -- of matching colors
• (machine A for 2 hours) AND (machine B for 1
  hour)
• XOR XOR

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Model

• We assumeEach bidder c has a valuation function
  ?c(S), for any set of products S, describing
  precisely the price c is willing to pay for S

No externalities?c depends solely on S

• ?c satisfies
• Free disposal S ? T ? ?c (S) ? ?c (T)
• May satisfy additionally
• Complementarity ?c (S?T) ? ?c (S) ?c (T)
• Substitutability ?c (S?T) ? ?c (S) ?c (T)

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Issues

• Consider only Sealed Bid Auctions
• Bidding languages and their expressiveness
• Allocation algorithms (maximizing total
  efficiency)
• Not deal with payment rules and bidders
  strategies

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How Does c Communicates ?c

• c sends his valuation ?c to auctioneer as
• a vector of numbers Problem Exponential size
• a computer program (applet) Problem requires
  exponential number of accesses by any auctioneer
  algorithm
• Using an Expressive, Efficient Bidding language

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Bidding LanguageRequirements

• Expressiveness
• Must be expressive enough to represent every
  possible valuation.
• Representation should not be too long
• Simplicity
• Easy for humans to understand
• Easy for auctioneer algorithms to handle

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AND, OR, and XOR bids

• left-sock, right-sock10
• blue-shirt8 XOR red-shirt7
• stamp-A6 OR stamp-B8

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General OR bids and XOR bids

• a,b7 OR d,e8 OR a,c4
• a0, a, b7, a, c4, a, b, c7, a, b,
  d, e15
• Can only express valuations with no
  substitutabilities.
• a,b7 XOR d,e8 XOR a,c4
• a0, a, b7, a, c4, a, b, c7, a, b,
  d, e8
• Can express any valuation
• Requires exponential size to represent
• a1 OR b1 OR OR z1

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OR of XORs example

• couch7 XOR chair5
• OR
• TV, VCR8 XOR Book3

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OR-of-XORs example 2

• Downward sloping symmetric valuationAny first
  item is valued at 9, the second at 7, and the
  third at 5.
• a9 XOR b9 XOR c9 XOR d9
• OR
• a7 XOR b7 XOR c7 XOR d7
• OR
• a5 XOR b5 XOR c5 XOR d5

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XOR of ORs example

• The Monochromatic valuationEven numbered items
  are red, and odd ones blue. Bidder wants to
  stick to one color, and values each item of that
  color at 1.
• a1 OR c1 OR e1 OR g1
• XOR
• b1 OR d1 OR f1 OR h1

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Bidding LanguageLimitations

• Theorem The downward sloping symmetric valuation
  with n items requires exponential size XOR-of-OR
  bids.
• Theorem The monochromatic valuation with n items
  requires exponential size OR-of-XOR bids.

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OR Bidding Language (Fujishima et al)

• Allow each bidder to introduce phantom items, and
  incorporate them in an OR bid.
• Example a,z7 OR b,z8 (z phantom)
• equivalent to (7 for a) XOR (8 for b)
• Lemma OR can simulate OR-of-XORs
• Lemma OR can simulate XOR-of-ORs

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Allocation

• A computational problem
• Input bids
• Outputs allocation of items to bidders
• Difficult computational problem (NP-complete)
• Existing approaches
• Very restricted bidding languages (Rothkopf
  et al)
• Search over allocation space (Fujishima etal,
  Sandholm)
• Fast heuristics (Fujishima
  etal, Lehman et al)

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Integer-Programming Formalization
Relaxation produces fractional allocations xj
specifies fraction of bid j obtained If were
lucky, the solution is 0,1

• n items m atomic bids
• Goal
• Maximize social efficiency
• subject to constraints

?0
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The Dual Linear Problem

• n items m atomic bids
• Goal
• Minimize Implicit Prices
• subject to constraints

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The meaning of the dual

• Intuition yi is the implicit price for item i
• Definition Allocation xj is supported by
  prices yi if
• Theorem There exists an allocation that is
  supported by prices iff the LP solution is 0,1

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When do we get 0,1 solutions?

• Theorem in each one of the cases below, the LP
  will produce optimal 0,1 results
• Hierarchical valuations
• 1-dimensional valuations
• Downward sloping symmetric valuation
• OR of XORs of singletons
• independent problems with 0,1 solutions
• problem with 0,1 solution low bids