1 / 21

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

This article discusses the advantages and design dimensions of ascending combinatorial auctions (CAs). It covers the motivation behind CAs, transparency, dynamic exchange of information, and the various classes of pricing functions. The article also explores the existence of competitive equilibrium and minimal CE prices, as well as the conditions for linear and non-linear anonymous prices. Additionally, it analyzes the BAS and BSM conditions, universal CE prices, communicational complexity lower bounds, and different design dimensions of ascending CAs.

edithpenney
Download Presentation

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

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  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 • 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 • Def. 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-anonymous) 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. Design dimensions of 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 collusion risk • Bidding rules • Bid improvement rule / percentage improvement rule • Activity rules (to help de-motivate sniping) • Revealed preference rules • Termination conditions • Fixed vs. rolling • Bidding language • Proxy agents • Price update rules: myopic vs. planned ahead [Nguyen & Sandholm EC-14]

  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 straightforward bidding

  16. Price update methods • Greedy: Price is increased on some set of the over-demanded items/bundles • LP-based (Connection between auctions and optimization algorithms goes back at least to Danzig (1963). For an excellent modern presentation about this for CAs, see Mechanism Design: A Linear Programming Approach by Vohra, Cambridge Univ. Press, 2011.) • Subgradient algorithm (slower convergence, and convergence can require price adjustments to become infinitesimally small) • Primal-dual algorithm (faster converge)

  17. Subgradient-algorithm-based CA framework • Initialize prices (potentially on bundles and nonanynymous) to zero • Repeat • Each agent i choose a surplus-maximizing bundle Bi • If a bidder has zero surplus, he reports that he is inactive • Seller chooses revenue-maximizing allocation a* • If each active bidder k gets her most-preferred bundle Bk, stop • Otherwise, for each active bidder k, increase price pk(Bk) by Δ>0

  18. Primal-dual auction design

  19. Primal-dual-algorithm-based CAs • Algorithm: • 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 • The primal-dual approach also tells how much each price can be changed • Not all algorithms in this family are ascending • Can choose an ascending variant that works correctly --- by ensuring that a certain “overdemand property” is satisfied throughout the auction process. For this, it suffices to start from zero prices and use a “minimal” price update: • Prices are increased on the bids from a set of “minimally undersupplied bidders” in the provisional allocation

  20. Other CA designs used in practice • Clock-proxy auction [Ausubel, Cramton & Milgrom, Ch. 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 • In proxy round, bidders report values to their respective proxy agents. The proxy agents iteratively submit package bids on behalf of the bidders, selecting the best profit opportunity for a bidder given the bidder’s inputted values. The auctioneer then selects the provisionally winning bids that maximize revenues. This process continues until the proxy agents have no new bids to submit. • XOR bids; all bids remain active; revealed preference consistency requirement • 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 • Other core-selecting CAs [e.g., Day & Milgrom] • Constraint generation is used to make this computationally feasible • (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

  21. 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

More Related