Ascending Combinatorial Auctions

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Ascending Combinatorial Auctions

• Andrew Gilpin
• November 15, 2005

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Motivation for Ascending CAs

• Advanced clearing algorithms exist for clearing
  combinatorial auctions (CAs)
• Bidding problem huge and difficult
• Possible exponential communication cost
• Computational cost of value determination
• Even determining the value of a single bundle can
  be hard
• Clearing algorithms are useless without a simple
  bidding problem facing the bidders

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Advantages over sealed-bid

• Sealed-bid auctions do not allow for feedback and
  price discovery to guide the elicitation
• Ascending (or iterative) CAs
• Bidders submit multiple bids during an auction
• The auction provides feedback to the bidders,
  supporting adaptive and focused elicitation
• Efficient allocation possible without full value
  revelation, or even full value determination
• Efficiency in a sealed-bid auction requires full
  value revelation in every case

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More advantages of ascending CAs

• Distribution
• Transparency
• Dynamic exchange of information
• With correlated values, can lead to increased
  revenue

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Types of ascending CAs

• Price-based
• Decentralized protocols
• Proxied auctions
• Direct-elicitation approaches

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Notations and definitions

• Items G 1,,m
• Bidders I 1,,n
• Private values vi(S) 0
• Free-disposal vi(T) vi(S) for T Ê S
• Normalization vi() 0
• Quasi-linear utility ui(S, p) vi(S) p
• No budget constraints, seller has no value
• Efficient combinatorial allocation problem (CAP)
• maxS Si vi(Si) s.t. Si n Sj for all
  i,j CAP(I)
• S denotes efficient allocation
• CAP(I \ i) denotes CAP without bidder i

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Price hierarchy

• We consider several classes of pricing functions
• Linear pj for each jÎG, p(S) SjÎSpj
• Non-linear p(S) for each bundle S
• Non-linear and non-anonymous pi(S) for each
  bundle S and bidder i
• 3 generalizes 2 generalizes 1

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Competitive equilibrium

• Let pi(S,p) vi(S) pi(S)
• Let ?S(S,p) Si pi(Si)
• Prices p and allocation S are in competitive
  equilibrium (CE) if
• pi(Si, p) maxS vi(S) pi(S), 0 (for all i)
• ?S(S, p) maxS Si pi(Si) s.t. S feasible
• So, a CE (S,p) is such that S maximizes the
  payoff of every bidder and the seller, given the
  prices
• Allocation S is said to be supported by p in CE
• Theorem Allocation S is supported in CE iff S
  is efficient.
• CE prices always exist (e.g. pi vi)

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Existence of CE prices

• Some ascending CAs are designed to output a CE
• We just saw that non-linear, non-anonymous prices
  always exist
• But linear and non-linear anonymous prices do not
  always exist
• Under what conditions can the prices be
  guaranteed to exist?

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When do linear CE prices exist?

• Di(p) S pi(S,p) maxT pi(T,p), pi(S,p) 0
• This is bidder is demand set, i.e. the set of
  bundles that maximizes her payoff given prices
• Defn If there exists T Î Di(p) s.t. j Î S pj
  pj Í T for all linear prices p p and S Î
  Di(p), then vi satisfies the goods are
  substitutes condition
• Bidders continue to demand an item whose price
  does not change
• Special case Unit-demand valuations
• Theorem If valuations satisfy goods are
  substitutes, then linear CE prices exist

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When do non-linear prices exist?

• Non-linear prices exist if
• Valuations are supermodular
• Bidders are single-minded
• Bidders have safe valuations (each pair of
  bundles with positive value share at least one
  item)

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Minimal CE prices

• Restricts the set of feasible allocations
• Defn Minimal CE prices are CE prices where the
  sellers revenue is minimized
• For certain valuations, minimal CE prices
  correspond to VCG payments
• Thus, truthful bidding is ex post equilibrium
• Since minimal CE prices are a restriction of CE
  prices, a minimal CE allocation is efficient
• Minimal CE prices always provide upper bound on
  VCG payments

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Buyers are substitutes

• Let w(L) for L Í I denote the value of the
  efficient allocation for CAP(L)
• Defn A valuation v satisfies the buyers are
  substitutes (BAS) condition ifw(I) w(I \ K)
  SiÎK w(I) w(I \ i) for all K Ì I
• Thm BAS holds iff VCG payments are supported in
  minimal CE

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Buyer-submodular

• Recall Buyers are substitutes (BAS) ifw(I)
  w(I \ K) SiÎK w(I) w(I \ i) for all K Ì I
• Slightly stronger version Buyer-submodular
  (BSM)w(L) w(L \ K) SiÎK w(L) w(L \ i)
  for all K Ì L, L Í I
• Some ascending CAs require the BSM condition to
  terminate in a minimal CE

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Universal CE prices

• BAS does not hold in many practical cases
• By the previous theorem, VCG not reachable in
  minimal CE
• We can reach a stronger condition by further
  restricting the price equilibrium concept
• Defn Prices p are universal competitive
  equilibrium (UCE) prices if p are CE prices and
  p-i are CE prices for CAP(I \ i)
• UCE prices always exist (e.g. pi vi)
• Thm Let p be UCE with efficient allocation S.
  The VCG payment to bidder i is qi
  pi(Si) PI(p) PI\i(p)where
  PL(p) maxS (pi(Si)) for bidders L Í I, S
  feasible

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Communicational complexity lower-bounds

• Thm Any CA that implements an efficient
  allocation must compute CE prices
• Thm Any CA that implements the VCG outcome must
  compute UCE prices
• Ascending CAs are designed to run well on average
  (typical) instances
• Sealed-bid auctions always have the worst-case
  performance

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Designing ascending CAs

• Timing
• Continuous faster propagation of info, difficult
  winner determination
• Discrete runs according to planned schedule
• Feedback
• Prices, bids, provisional allocation
• Tradeoff between effective bid guidance and
  mitigating risk of collusion
• Bidding rules
• Bid improvement rule
• Percentage improvement rule
• Activity rules (to avoid sniping)
• Termination conditions
• Fixed vs. rolling
• Bidding language
• Proxy agents

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Price-based ascending CAs

• Each auction in this family has roughly the same
  structure
• In each round, announce prices and allocation
• Receive bids
• Update prices and allocation
• Stop if termination criterion met

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Price-based ascending CAs
Name Valuations Price structure Language Price update method Outcome
KC Substitutes Non-anon items OR-items Greedy CE
SAA Substitutes Items OR-items Greedy CE
GS Substitutes Items XOR Minimal Min CE
Aus Substitutes Items Single Greedy VCG
iBundle BSM Non-anon bundles XOR Greedy VCG
General Min CE
dVSV BSM Non-anon bundles XOR Minimal VCG
Clock-proxy BSM Items (proxy) XOR Greedy VCG
General Min CE
RAD General Items OR LP-based ????
AkBA General Anon bundles XOR LP-based ????
iBEA General Non-anon bundles XOR Greedy VCG
MP General Non-anon bundles XOR Minimal VCG

• Results assume truthful bidding

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Price update methods

• Greedy Price is increased on some set of the
  overdemanded items/bundles
• Minimal Price is increased on a minimal set of
  overdemanded items
• Or, on based on the bids from a set of minimally
  undersupplied bidders
• LP-based Prices adjusted based on optimal
  solution to an LP formulated to approximate CE
  prices
• A set of items is overdemanded if demand sets
  unsatisfiable
• A set of bidders is undersupplied if some bidder
  not satisfied in allocation

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Primal-dual auction design
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Primal-dual example iBundle

• Non-linear and anonymous prices
• XOR bidding
• Winning bids carried over from previous round
• A bidder is competitive if she has at least one
  bid above current ask price
• Prices are increased by e on bundles that receive
  a bid from a losing bidder
• Prices and provisional allocation provided as
  feedback
• Terminates when each competitive bidder wins a
  bundle
• Thm Terminates with allocation within 3minn,me
  of the efficient solution (under reasonable
  strategic assumptions)
• Proof uses LP duality and complementary-slackness

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Non-priced based approaches

• Decentralized
• Proxy auctions
• Direct-elicitation

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Open problems

• Design auction that makes appropriate tradeoff
  between cost of information revelation and market
  efficiency
• Design ex post truthful ascending CA that does
  not suffer from problems of VCG (collusion,
  low-revenue)
• Design auction that reaches VCG with general
  valuations, but without XOR bidding