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Ascending Combinatorial Auctions

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Advanced clearing algorithms exist for clearing combinatorial auctions (CAs) ... Linear: pj for each j G, p(S) = Sj Spj. Non-linear: p(S) for each bundle S ... – PowerPoint PPT presentation

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Title: Ascending Combinatorial Auctions


1
Ascending Combinatorial Auctions
  • Andrew Gilpin
  • November 15, 2005

2
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

3
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

4
More advantages of ascending CAs
  • Distribution
  • Transparency
  • Dynamic exchange of information
  • With correlated values, can lead to increased
    revenue

5
Types of ascending CAs
  • Price-based
  • Decentralized protocols
  • Proxied auctions
  • Direct-elicitation approaches

6
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

7
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

8
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)

9
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?

10
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

11
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)

12
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

13
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

14
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

15
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

16
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

17
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

18
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

19
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

20
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

21
Primal-dual auction design
22
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

23
Non-priced based approaches
  • Decentralized
  • Proxy auctions
  • Direct-elicitation

24
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
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