Ascending Combinatorial Auctions a restricted form of preference
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: 1. Linear: pj for each jÎG, p(S) = ΣjÎSpj 2. Non-linear: p(S) for each bundle S 3. 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: 1. πi(Si*, p) = max. S [vi(S) – pi(S), 0] (for all i) 2. ΠS(S*, p) = max. S 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)
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 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 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
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: Buyersubmodular (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 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) – PIi*(p)] where PL*(p) = max. S ∑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 Substitutes Items OR-items Greedy CE GS Substitutes Items XOR Minimal Min CE Aus Substitutes Items Single Greedy VCG i. Bundle BSM Non-anon bundles XOR Greedy VCG General Min CE d. VSV BSM Non-anon bundles XOR Minimal VCG Clock-proxy BSM Items (+proxy) XOR Greedy VCG General Min CE RAD General Items OR LP-based ? ? Ak. BA General Anon bundles XOR LP-based ? ? i. BEA General Non-anon bundles XOR Greedy VCG MP General Non-anon bundles XOR Minimal VCG 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
Primal-dual example: i. Bundle(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 3 min{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 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)
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.
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