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Approximating optimal combinatorial auctions for complements using restricted welfare maximization Pingzhong Tang and Tuomas Sandholm Computer Science Department Carnegie Mellon University

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High-level contributions New approach to mechanism design: “Social welfare with holes” – I.e., curtail the set of allocations based on agents’ reports (e.g., bids), and use welfare maximization within remaining set – Unlike maximum-in-range approach [Nisan and Ronen 07], where the allocation set is curtailed ex ante – Completely general (e.g., remove all but one allocation) – Trickier because not all report-based ways of curtailing are incentive compatible (paper contains an example) – We present the first (non-trivial) such curtailing that maintains incentive compatibility – Hopefully, a fruitful avenue going forward New, general form of reserve pricing for combinatorial auctions – Any efficient mechanism can be arbitrarily far from optimal revenue, while our reserves avoid this downside

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Background Optimal (i.e., expected revenue maximizing) auctions known for: – Single item [Myerson 81] – Multiple identical units [Maskin and Riley 89] – Multiple items with complementarities in a 1-dimensional setting [Levin 97] These are all based on virtual welfare maximization – Requires prior information – Complex and unintuitive – Inefficient Welfare maximizing allocation rule, but with reserve prices – Symmetric (1-item) setting: Identical to Myerson – Asymmetric (1-parameter) setting: 2-approximation [Hartline and Roughgarden EC-09]

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Our technical contributions We approximate Levin's optimal auction for complements using welfare maximization with a form of reserve pricing for combinatorial auctions – Reserve prices restrict allocations based on bids – In Levin's setting, we use a specific form: Monopoly reserves – We obtain a 2-approximation to optimal revenue And a 6-approximation using anonymous reserves Why are we doing this? – More efficient than Levin’s auction Any efficient mechanism has arbitrarily low revenue (e.g., in the paper) – Requires less info to verify correct execution of the mechanism (given the reserves) – Simpler, easier to understand – Better starting points for automated mechanism design (than, e.g., VCG)

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Myerson's setting Seller has 1 indivisible item for sale, which he values at 0 Set of bidders 1,…,n – Bidder i’s valuation, v i, is private knowledge – Distribution F i and regular density f i, according to which v i is drawn, are common knowledge – Quasi-linearity: u i = v i – payment i Two constraints: – (Ex interim) incentive compatibility – (Ex interim) individual rationality Objective: Maximize seller's expected revenue

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Myerson's solution Asymmetric case: f i ’s are different – Virtual valuation: – Allocation rule: Give the item to the bidder with the highest virtual valuation, if it is positive and retain the item otherwise – Payment rule: The lowest bid by i that would have won – Interpretation: Run second price auction on the virtual valuations, with reserve price 0 Symmetric case: f i = f j – Optimal auction is a 2 nd -price auction with monopoly reserve price – Monopoly reserve price: v ri such that

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More about asymmetric case… 2 nd -price auction with monopoly reserve prices (one per bidder) is a 2-approximation of Myerson's optimal auction [Hartline and Roughgarden EC-09] A key step: Myerson's Lemma [1981] – Lemma: For any truthful 1-item auction, expected payment from a bidder equals his expected virtual valuation

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What about asymmetric case? Hartline and Roughgarden EC-09: – Second price auction with monopoly reserve prices (one per bidder) is a 2-approximation of Myerson's optimal auction A key step: Myerson's Lemma [1981] – Lemma: For any truthful 1-item auction, expected payment from a bidder equals his expected virtual valuation – Proof of Myerson’s Lemma: Fixing others' reports, from bidder i’s perspective, Truthfulness implies a take-it-or-leave-it price (call it v 0 ) Expected payment from bidder i is Expected virtual valuation is By calculus, these two formulas are equal QED

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Levin's setting: Complements with 1-dimensional type Seller has 2 items for sale, which he values at 0 – All his results (and ours) extend to m items Set of bidders 1,…,n – Bidder i’s type θ i is private knowledge – Distribution F i and regular density f i, according to which θ i is drawn, are common knowledge – Bidder i 's valuation function is – Quasi-linearity: u i () = v i () - payment i Two constraints: – (Ex interim) incentive compatibility – (Ex interim) individual rationality Objective: Maximize seller's expected revenue v

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Levin's solution Virtual valuation: Allocation Rule: Maximize virtual social welfare, among all the positive virtual valuations Payment rule: θ1θ1 θ0θ0 Pay v i1 (θ 0 ), get item 1 Pay additional v i2 (θ 1 )+ v i3 (θ 1 ), get both items θ1θ1 θ0θ0 Pay v i2 (θ 0 ), get item 2 Pay additional v i1 (θ 1 )+ v i3 (θ 1 ), get both items Pay 0, get nothing Pay 0, get nothing Case I: Agent i wins item 1 first: Case II: Agent i wins item 2 first:

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Approximating Levin's auction Why difficult? – Multiple definitions of reserve prices in combinatorial settings One fake bidder, two fake bidders, bidder-specific... What are monopoly reserves in combinatorial settings? – Myerson's Lemma in this setting? – [Hartline and Roughgarden EC-09] approach doesn't apply

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Our allocation-curtailing approach applied to Levin's setting Idea – Preclude bidder-bundle pairs that have negative virtual valuations – Preclude bidder-bundle pairs where removing some item(s) from a bidder gives that bidder higher virtual value Theorem – Together with welfare-maximization allocation rule and Levin's payment rule, the preclusions above constitute an auction that is incentive compatible (in weakly dominant strategies), is individually rational, and 2-approximates Levin's revenue

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Desirable properties of our auction Incentive compatible, individually rational, 2-approximation – Important step for proving this is allocation monotonicity: Fixing others' reports, a bidder's set of allocated items is expanding in his report More efficient than Levin – Less restriction of the allocation space – Welfare maximizing in this less restricted space Requires less information, e.g., to verify correct execution – 5 numbers versus distribution function Easier to understand A bidder in his lowest type gets zero payoff For any allocation, a bidder's payment plus his virtual valuation is no less than his real valuation – We use this in 2-approximation proof

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Extending Myerson's Lemma to this setting Myerson's Lemma: Bidder’s expected payment equals his expected virtual value Our conditions: – 1. Truthful – 2. Allocation monotonic – 3. Lowest type gets zero payoff Our auction satisfies 1, 2 and 3 Levin's conditions: – a. Truthful – b. Revenue-maximizing – c. Utility functions satisfy the requirements of envelope theorem

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Proof of 2-approximation Let M be the social welfare maximizing mechanism under monopoly reserves (i.e, our auction) Step 1. By definition, M maximizes restricted social welfare Step 2. By Myerson’s lemma extended to this setting, expected revenue of M = expected sum of bidders’ virtual valuations in M Step 3. As we prove, in M, a bidder's payment plus his virtual valuation is no less than his real valuation Step 4. By Steps 2 and 3, 2 * [Expect revenue of M] ≥ social welfare of M Step 5. By Steps 1 and 4, 2 * [Expect revenue of M] ≥ social welfare of Levin Step 6. By individual rationality, social welfare of Levin ≥ revenue of Levin QED

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6-approximation of Levin’s optimal revenue using anonymous reserves Now, usual definition of reserve price: – Seller pretends to have valuation a for 1 st item, b for 2 nd item, and c for bundle Auction L: Levin's optimal auction on original set of bidders Auction D: Duplicate each bidder. Then apply welfare-maximizing allocation rule and Levin payment rule Step 1. Auction D 3-approximates Auction L Step 2. Let a, b, and c be random variables that simulate max i {v i1 }, max i {v i2 } and max i {v i1 + v i2 + v i3 }, respectively, in the original bidder set Step 3. Step 2 trivially yields a 2-approximation of D. Hence, a 6- approximation of L QED In contrast to 4-approximation for 1-item setting [Hartline & Roughgarden EC-09]

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Conclusions New general approach to mechanism design: Social welfare with holes New general form of reserve pricing under welfare maximization in combinatorial auctions Application of this idea to Levin's setting of 1-D complements: – 2-approximation to revenue 6-approximation with anonymous reserves – More efficient than Levin – Requires less info to verify correct execution (given reserves) – Easier to understand – Extended Myerson’s lemma to this setting

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Future work Characterizing truthful restrictions – 1-item setting: Equivalent to allocation monotonicity – Levin's setting: In our follow-on work we have found a necessary condition (e.g., can go from nothing to winning Item 1 to winning Item 2 to winning both) Plan to search for optimal auctions under this condition Application to other settings Application to automated mechanism design

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