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

Price hierarchy We consider several classes of pricing functions: Linear: pj for each jÎG, p(S) = ΣjÎSpj 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

Competitive equilibrium Let agent i’s surplus 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: πi(Si*, p) = maxS [vi(S) – pi(S), 0] (for all i) 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) Proof of Theorem: Chapter 8 of Combinatorial Auctions book. Proof technique: LP duality argument for suitably extended LP formulation of the CAP.

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 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)

Minimal CE prices Def. Minimal CE prices are CE prices where the seller’s 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

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 if: w(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 Bikhchandani and Ostroy 2002

Buyer-submodular Recall: Buyers are substitutes (BAS) if: w(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

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

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 Name Valuations Price structure Language Price update method Outcome KC Substitutes Non-anon items OR-items Greedy CE SAA Items GS XOR Minimal Min CE Aus Single VCG iBundle BSM Non-anon bundles General dVSV Clock-proxy Items (+proxy) RAD OR LP-based ???? AkBA Anon bundles iBEA MP Results assume truthful bidding

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 A set of items is overdemanded if demand sets unsatisfiable A set of bidders is undersupplied if some bidder not satisfied in allocation

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

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

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