1. Ascending Combinatorial Auctions Andrew Gilpin November 6, 2007

2. 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. 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. 4 More advantages of ascending CAs Distribution Transparency Dynamic exchange of information –With correlated values, can lead to increased revenue

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

6. 6 Notations and definitions Items: G = {1,…,m} Bidders: I = {1,…,n} Private values: v i (S) ≥ 0 –Free-disposal: v i (T) ≥ v i (S) for T  S –Normalization: v i ({}) = 0 Quasi-linear utility: u i (S, p) = v i (S) – p No budget constraints, seller has no value Efficient combinatorial allocation problem (CAP): max S Σ i v i (S i ) s.t. S i ∩ S j = {} for all i,j [CAP(I)] S* denotes efficient allocation CAP(I \ i) denotes CAP without bidder i

7. 7 Price hierarchy We consider several classes of pricing functions: 1.Linear: p j for each j  G, p(S) = Σ j  S p j 2.Non-linear: p(S) for each bundle S 3.Non-linear and non-anonymous: p i (S) for each bundle S and bidder i 3 generalizes 2 generalizes 1

8. 8 Competitive equilibrium Let π i (S i,p) = v i (S i ) – p i (S i ) Let Π S (S,p) = Σ i p i (S i ) Prices p and allocation S* are in competitive equilibrium (CE) if: 1.π i (S i *, p) = max S [v i (S) – p i (S), 0] (for all i) 2.Π S (S*, p) = max S Σ i p i (S i ) 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. p i = v i )

9. 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. 10 When do linear CE prices exist? D i (p) = {S : π i (S,p) ≥ max T π i (T,p), π i (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  D i (p’) s.t. {j  S : p j = p j ’}  T for all linear prices p’ ≥ p and S  D i (p), then v i 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

11. 11 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)

12. 12 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

13. 13 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) ≥  i  K [w(I) – w(I \ i)] for all K  I Thm BAS holds iff VCG payments are supported in minimal CE

14. 14 Buyer-submodular Recall: Buyers are substitutes (BAS) if: w(I) – w(I \ K) ≥  i  K [w(I) – w(I \ i)] for all K  I Slightly stronger version: Buyer- submodular (BSM): w(L) – w(L \ K) ≥  i  K [w(L) – w(L \ i)] for all K  L, L  I Some ascending CAs require the BSM condition to terminate in a minimal CE

15. 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. p i = v i ) Thm Let p be UCE with efficient allocation S*. The VCG payment to bidder i is: q i = p i (S i *) – [  I *(p) –  I\i *(p)] where  L *(p) = max S (p i (S i )) for bidders L  I, S feasible

16. 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. 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. 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. 19 Price-based ascending CAs Results assume truthful bidding NameValuationsPrice structureLanguagePrice update method Outcome KC SubstitutesNon-anon itemsOR-itemsGreedyCE SAASubstitutesItemsOR-itemsGreedyCE GSSubstitutesItemsXORMinimalMin CE AusSubstitutesItemsSingleGreedyVCG iBundleBSMNon-anon bundlesXORGreedyVCG GeneralMin CE dVSVBSMNon-anon bundlesXORMinimalVCG Clock-proxyBSMItems (+proxy)XORGreedyVCG GeneralMin CE RADGeneralItemsORLP-based???? AkBAGeneralAnon bundlesXORLP-based???? iBEAGeneralNon-anon bundlesXORGreedyVCG MPGeneralNon-anon bundlesXORMinimalVCG

20. 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. 21 Primal-dual auction design

22. 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  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

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

24. 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

25. 25 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.