Pingzhong Tang and Tuomas Sandholm Computer Science Department Carnegie Mellon University

1. Approximating optimal combinatorialauctions for complements usingrestricted welfare maximization Pingzhong Tang and Tuomas Sandholm Computer Science Department Carnegie Mellon University

2. 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

3. Background • Optimal (i.e., expected revenue maximizing) auctions known for: • Single item [Myerson 81] • Multiple identical units [Maskinand 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]

4. 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)

5. Myerson's setting • Sellerhas 1 indivisible item for sale, which he values at 0 • Set of bidders 1,…,n • Bidder i’s valuation, vi , is private knowledge • Distribution Fiand regular density fi, according to which vi is drawn, are common knowledge • Quasi-linearity: ui= vi– paymenti • Two constraints: • (Ex interim) incentive compatibility • (Ex interim) individual rationality • Objective: Maximize seller's expected revenue

6. Myerson's solution • Asymmetric case: fi’sare 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: fi = fj • Optimal auction is a 2nd-price auction with monopoly reserve price • Monopoly reserve price: vri such that

7. More about asymmetric case… • 2nd-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

8. 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’stypeθiis private knowledge • Distribution Fi and regular density fi, according to which θiis drawn, are common knowledge • Bidder i 's valuation function is • Quasi-linearity: ui() = vi() - paymenti • Two constraints: • (Ex interim) incentive compatibility • (Ex interim) individual rationality • Objective: Maximize seller's expected revenue v

9. Levin's solution • Virtual valuation: • Allocation Rule: Maximize virtual social welfare, among all the positive virtual valuations • Payment rule: Case I: Agent i wins item 1 first: Pay 0, get nothing Pay vi1(θ0), get item 1 Pay additional vi2(θ1)+ vi3(θ1), get both items θ0 θ1 Case II: Agent i wins item 2 first: Pay 0, get nothing Pay vi2(θ0), get item 2 Pay additional vi1(θ1)+ vi3(θ1), get both items θ0 θ1

10. 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

11. 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

12. 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

13. 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

14. 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

15. 6-approximation of Levin’s optimal revenue using anonymous reserves • Now, usual definition of reserve price: • Seller pretends to have valuation a for 1stitem, b for 2nd 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 maxi{vi1}, maxi{vi2}and maxi{vi1+ vi2 + vi3}, 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]

16. 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

17. 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