Ascending Combinatorial Auctions = a restricted form of preference elicitation in CAs

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Ascending Combinatorial Auctions a restricted
form of preference elicitation in CAs

• Tuomas Sandholm

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Advantages of ascending CAs

• Same motivation as other multiagent preference
  elicitation methods
• Transparency
• Dynamic exchange of information
• With correlated values, can lead to increased
  revenue

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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 agent is surplus pi(Si,p) vi(Si) pi(Si)
• 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 do they exist?

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When do linear CE prices exist?

• Theorem If each agents valuation function
  satisfies goods are substitutes, then linear CE
  prices exist
• Special cases
• Unit-demand valuations
• Additive valuations
• Downward-sloping valuations

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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 cases
• Unit-demand valuations
• Additive valuations
• Downward-sloping valuations
• Theorem If valuations satisfy goods are
  substitutes, then linear CE prices exist

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When do non-linear anonymous prices exist?

• Non-linear anonymous prices exist if
• valuations are supermodular, i.e., increasing
  returns, or
• bidders are single-minded, or
• bidders have safe valuations (each pair of
  bundles with positive value share at least one
  item)

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Minimal CE prices

• Def. 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)
• Def. 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
• Then, 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 (non-linear, non-anonymos) always
  exist (e.g. pi vi)
• Minimal CE prices are universal iff BAS holds
• VCG outcome and payments determinable from UCE
  prices
• 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

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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
  over-demanded items/bundles
• Minimal Price is increased on a minimal set of
  over-demanded items
• Or, on the bids from a set of minimally
  undersupplied bidders
• LP (primal-dual)-based
• Formulate CA as an LP with integral optima. Dual
  should allow convergence to UCE prices (or
  minimal CE prices in the case of BAS)
• Use bidding language that is expressive for
  straightforward bidding, and formulate a WDP to
  compute feasible primal solution that minimizes
  violation of complementary slackness conditions
  as represented by bids
• Terminate when provisional allocation and ask
  prices satisfy complementary slackness conditions
  (and thus represent a CE), and also satisfy any
  additional conditions needed to compute VCG
  payments (e.g., UCE conditions or minimal CE
  conditions under BAS)
• Otherwise, adjust prices to make progress toward
  an optimal dual solution that satisfies these
  conditions

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Primal-dual auction design
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Primal-dual example iBundle(2)

• Non-linear, 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
• In general, could use primal-dual LP algorithms
  to jump the prices to the next vertex instead
  of incrementing them just a bit.
• 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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Other CA designs used in practice

• Clock-proxy auction Chapter 5 of CA book
• Run a parallel clock auctions for the items until
  no item is over-demanded. Then run a
  last-and-final proxy round
• Combines the simple and transparent price
  discovery of the clock auction with the
  efficiency of the proxy auction
• Linear pricing maintained as long as possible,
  but is abandoned in the proxy round to improve
  efficiency and enhance revenue
• Revealed preference consistency requirement
• Other core-selecting CAs e.g., Day Milgrom
• (actually select a core for revealed valuations,
  assuming bidders act truthfully)
• But bidders are not generally motivated to bid
  truthfully
• If bidders use envy-reducing strategies, then
  these converge to an envy-free fixed point, and
  those points have revenue same or greater than
  VCG Othman Sandholm AAAI-10
• Can be supported by envy-quotes
• Constraint generation is used to make this
  computationally feasible

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Open problems

• Design ex post truthful ascending CA that does
  not suffer from problems of VCG (collusion,
  low-revenue)
• See two technical preference elicitation problems
  in our JMLR-04 paper

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Open problems

• Design auction that makes appropriate tradeoff
  between cost of information revelation and market
  efficiency
• Design auction that reaches VCG with general
  valuations, but without XOR bidding

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Recommended reading

1. Iterative Combinatorial Auctions. David Parkes.
   Chapter 2 of Combinatorial Auctions book.
2. Ascending Auctions. Liad Blumrosen. Section 11.7
   of AGT book.