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

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

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

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

Competitive equilibrium Let agent i’s surplus 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 )

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? …

When do linear CE prices exist? Theorem If each agent’s valuation function satisfies “goods are substitutes”, then linear CE prices exist Special cases –Unit-demand valuations –Additive valuations –Downward-sloping valuations

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

When do non-linear anonymous prices exist? Non-linear anonymous prices exist if 1.valuations are supermodular, i.e., increasing returns, or 2.bidders are single-minded, or 3.bidders have safe valuations (each pair of bundles with positive value share at least one item)

Minimal CE prices Def. 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

Buyers are substitutes Let w(L) for L  I denote the value of the efficient allocation for CAP(L) Def. 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

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

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 always exist (e.g. p i = v i ) 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: 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

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

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

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

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

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

Primal-dual auction design

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  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 3min{n,m}  of the efficient solution (under reasonable strategic assumptions) –Proof uses LP duality and complementary-slackness

Non-priced based approaches Decentralized Proxy auctions Direct-elicitation

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 draft] –Can be supported by envy-quotes –Constraint generation is used to make this computationally feasible

An open problem Design ex post truthful ascending CA that does not suffer from problems of VCG (collusion, low-revenue)

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

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.