Ascending Combinatorial Auctions

Andrew Gilpin

November 6, 2007

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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 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 Σi vi(Si) s.t. Si ∩ 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:

1. Linear: pj for each jG, p(S) = ΣjSpj

2. Non-linear: p(S) for each bundle S

3. 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 πi(Si,p) = vi(Si) – pi(Si)• Let ΠS(S,p) = Σi pi(Si)• Prices p and allocation S* are in competitive

equilibrium (CE) if:1. πi(Si*, p) = maxS [vi(S) – pi(S), 0] (for all i)2. ΠS(S*, p) = maxS Σi 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 : πi(S,p) ≥ maxT πi(T,p), πi(S,p) ≥ 0}• This is bidder i’s demand set, i.e. the set of bundles that

maximizes her payoff given prices• Defn If there exists T Di(p’) s.t. {j S : pj = pj’} T for all

linear prices p’ ≥ p and S 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:1. Valuations are supermodular

2. Bidders are single-minded

3. 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 seller’s revenue is minimized and allocation is efficient

• 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 I denote the value of the efficient allocation for CAP(L)

• Defn A valuation v satisfies the buyers are substitutes (BAS) condition if:w(I) – w(I \ K) ≥ iK [w(I) – w(I \ i)] for all K I

• Thm BAS holds iff VCG payments are supported in minimal CE

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Buyer-submodular

• Recall: Buyers are substitutes (BAS) if:w(I) – w(I \ K) ≥ iK [w(I) – w(I \ i)] for all K I

• Slightly stronger version: Buyer-submodular (BSM):w(L) – w(L \ K) ≥ iK [w(L) – w(L \ i)] for all K L, L 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*) – [I*(p) – I\i*(p)]where L*(p) = maxS (pi(Si)) for bidders L 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

Results assume truthful bidding

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

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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 unsatisfiableA 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 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 3min{n,m} of the

efficient solution (under reasonable strategic assumptions)– Proof uses LP duality and complementary-slackness

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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

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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.