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

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

2. 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

3. 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

4. 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) = maxSΣi 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)

5. 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? …

6. 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

7. 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)

8. 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

9. 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

10. 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

11. 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. 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)]wherePL*(p) = maxS ∑pi(Si) for bidders L Í I, S feasible

12. 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

13. 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

14. 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

15. Price-based ascending CAs Results assume truthful bidding

16. 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

17. 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

18. 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

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

20. Recommended reading • Iterative Combinatorial Auctions. David Parkes. Chapter 2 of Combinatorial Auctions book. • Ascending Auctions. Liad Blumrosen. Section 11.7 of AGT book.