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Approximation and Mechanism Design Jason D. Hartline Northwestern University May 15, 2010 Game theory has a great advantage in explicitly analyzing the consequences of trading rules that presumably are really common knowledge, it is

  1. Philosophy of Approximation What is the point of a 2-approximation? • Must make tradeoff between understanding and optimality. (1 + ǫ ) constant super-constant Performance great ok bad Understanding little lots little/none • Constant approximations identify salient features of model. Example: is X important in MD? competition? – no, if mech without X is constant approx – yes, otherwise. 7 A PPROXIMATION AND M ECHANISM D ESIGN – M AY 15, 2010

  2. Philosophy of Approximation What is the point of a 2-approximation? • Must make tradeoff between understanding and optimality. (1 + ǫ ) constant super-constant Performance great ok bad Understanding little lots little/none • Constant approximations identify salient features of model. Example: is X important in MD? competition? transfers? – no, if mech without X is constant approx – yes, otherwise. 7 A PPROXIMATION AND M ECHANISM D ESIGN – M AY 15, 2010

  3. Philosophy of Approximation What is the point of a 2-approximation? • Must make tradeoff between understanding and optimality. (1 + ǫ ) constant super-constant Performance great ok bad Understanding little lots little/none • Constant approximations identify salient features of model. Example: is X important in MD? competition? transfers? – no, if mech without X is constant approx – yes, otherwise. • Seller can always try ad hoc improvements on approximation. 7 A PPROXIMATION AND M ECHANISM D ESIGN – M AY 15, 2010

  4. Overview 1. Single-dimensional Bayesian settings. (e.g., single-item auctions) 2. Multi-dimensional Bayesian settings. (e.g., multi-item auctions) 3. Prior-free settings. 8 A PPROXIMATION AND M ECHANISM D ESIGN – M AY 15, 2010

  5. Part I: Approximation for single-dimensional Bayesian mechanism design (where agent preferences are given by a private value for service, zero for no service; preferences are drawn from a distribution)

  6. Example 2: Single-item auction Problem: Bayesian Single-item Auction Problem • a single item for sale, • n buyers, and • a dist. F = F 1 × · · · × F n from which the consumers’ values for the item are drawn. Goal: seller opt. auction for F . 10 A PPROXIMATION AND M ECHANISM D ESIGN – M AY 15, 2010

  7. Example 2: Single-item auction Problem: Bayesian Single-item Auction Problem • a single item for sale, • n buyers, and • a dist. F = F 1 × · · · × F n from which the consumers’ values for the item are drawn. Goal: seller opt. auction for F . Question: What is optimal auction? 10 A PPROXIMATION AND M ECHANISM D ESIGN – M AY 15, 2010

  8. Optimal Auction Design [Myerson ’81] 11 A PPROXIMATION AND M ECHANISM D ESIGN – M AY 15, 2010

  9. Optimal Auction Design [Myerson ’81] 1. Def: revenue curve : R i ( q ) = q · F − 1 (1 − q ) . 0 i 0 1 11 A PPROXIMATION AND M ECHANISM D ESIGN – M AY 15, 2010

  10. Optimal Auction Design [Myerson ’81] 1. Def: revenue curve : R i ( q ) = q · F − 1 (1 − q ) . 0 i 0 1 2. Def: virtual value : ϕ i ( v i ) = v i − 1 − F i ( v ) = marginal revenue. f i ( v i ) 11 A PPROXIMATION AND M ECHANISM D ESIGN – M AY 15, 2010

  11. Optimal Auction Design [Myerson ’81] 1. Def: revenue curve : R i ( q ) = q · F − 1 (1 − q ) . 0 i 0 1 2. Def: virtual value : ϕ i ( v i ) = v i − 1 − F i ( v ) = marginal revenue. f i ( v i ) 3. Def: virtual surplus : virtual value of winner(s). 11 A PPROXIMATION AND M ECHANISM D ESIGN – M AY 15, 2010

  12. Optimal Auction Design [Myerson ’81] 1. Def: revenue curve : R i ( q ) = q · F − 1 (1 − q ) . 0 i 0 1 2. Def: virtual value : ϕ i ( v i ) = v i − 1 − F i ( v ) = marginal revenue. f i ( v i ) 3. Def: virtual surplus : virtual value of winner(s). 4. Thm: E [ revenue ] = E [ virtual surplus ] . 11 A PPROXIMATION AND M ECHANISM D ESIGN – M AY 15, 2010

  13. Optimal Auction Design [Myerson ’81] 1. Def: revenue curve : R i ( q ) = q · F − 1 (1 − q ) . 0 i 0 1 2. Def: virtual value : ϕ i ( v i ) = v i − 1 − F i ( v ) = marginal revenue. f i ( v i ) 3. Def: virtual surplus : virtual value of winner(s). 4. Thm: E [ revenue ] = E [ virtual surplus ] . 5. Def: F i is regular iff revenue curve concave iff virtual values monotone. 0 0 1 11 A PPROXIMATION AND M ECHANISM D ESIGN – M AY 15, 2010

  14. Optimal Auction Design [Myerson ’81] 1. Def: revenue curve : R i ( q ) = q · F − 1 (1 − q ) . 0 i 0 1 2. Def: virtual value : ϕ i ( v i ) = v i − 1 − F i ( v ) = marginal revenue. f i ( v i ) 3. Def: virtual surplus : virtual value of winner(s). 4. Thm: E [ revenue ] = E [ virtual surplus ] . 5. Def: F i is regular iff revenue curve concave iff virtual values monotone. 0 0 1 6. Thm: for regular dists, optimal auction sells to bidder with highest positive virtual value. 11 A PPROXIMATION AND M ECHANISM D ESIGN – M AY 15, 2010

  15. Optimal Auction Design [Myerson ’81] 1. Def: revenue curve : R i ( q ) = q · F − 1 (1 − q ) . 0 i 0 1 2. Def: virtual value : ϕ i ( v i ) = v i − 1 − F i ( v ) = marginal revenue. f i ( v i ) 3. Def: virtual surplus : virtual value of winner(s). 4. Thm: E [ revenue ] = E [ virtual surplus ] . 5. Def: F i is regular iff revenue curve concave iff virtual values monotone. 0 0 1 6. Thm: for regular dists, optimal auction sells to bidder with highest positive virtual value. 7. Cor: for iid, regular dists, optimal auction is Vickrey with monopoly reserve price ϕ − 1 (0) . 11 A PPROXIMATION AND M ECHANISM D ESIGN – M AY 15, 2010

  16. Optimal Auctions Optimal Auctions: • iid, regular distributions : Vickrey with monopoly reserve price. • general : sell to bidder with highest positive virtual value. 12 A PPROXIMATION AND M ECHANISM D ESIGN – M AY 15, 2010

  17. Optimal Auctions Optimal Auctions: • iid, regular distributions : Vickrey with monopoly reserve price. • general : sell to bidder with highest positive virtual value. Discussion: • iid, regular case: seems unlikely in practice. • general case: nobody runs optimal auction (too complicated?). 12 A PPROXIMATION AND M ECHANISM D ESIGN – M AY 15, 2010

  18. Approximation with reserve prices Question: when is reserve pricing a good approximation? 13 A PPROXIMATION AND M ECHANISM D ESIGN – M AY 15, 2010

  19. Approximation with reserve prices Question: when is reserve pricing a good approximation? Thm: Vickrey with reserve = constant virtual price with Pr [ no sale ] = 1 / 2 is a 2-approximation. [Chawla, H, Malec, Sivan ’10] 13 A PPROXIMATION AND M ECHANISM D ESIGN – M AY 15, 2010

  20. Approximation with reserve prices Question: when is reserve pricing a good approximation? Thm: Vickrey with reserve = constant virtual price with Pr [ no sale ] = 1 / 2 is a 2-approximation. [Chawla, H, Malec, Sivan ’10] Proof: apply prophet inequality (tie-breaking by value) to virtual values. 13 A PPROXIMATION AND M ECHANISM D ESIGN – M AY 15, 2010

  21. Approximation with reserve prices Question: when is reserve pricing a good approximation? Thm: Vickrey with reserve = constant virtual price with Pr [ no sale ] = 1 / 2 is a 2-approximation. [Chawla, H, Malec, Sivan ’10] Proof: apply prophet inequality (tie-breaking by value) to virtual values. prophet inequality Vickrey with reserves prizes virtual values threshold t virtual price E [ max prize ] E [ optimal revenue ] E [ prize for t ] E [ Vickrey revenue ] 13 A PPROXIMATION AND M ECHANISM D ESIGN – M AY 15, 2010

  22. Approximation with reserve prices Question: when is reserve pricing a good approximation? Thm: Vickrey with reserve = constant virtual price with Pr [ no sale ] = 1 / 2 is a 2-approximation. [Chawla, H, Malec, Sivan ’10] Proof: apply prophet inequality (tie-breaking by value) to virtual values. prophet inequality Vickrey with reserves prizes virtual values threshold t virtual price E [ max prize ] E [ optimal revenue ] E [ prize for t ] E [ Vickrey revenue ] Discussion: • constant virtual price ⇒ bidder-specific reserves. • simple reserve prices natural, practical, and easy to find. • robust posted pricing with arbitrary tie-breaking works fine too. 13 A PPROXIMATION AND M ECHANISM D ESIGN – M AY 15, 2010

  23. Anonymous Reserves Question: for non-identical distributions, is anonymous reserve approximately optimal? (e.g., eBay) 14 A PPROXIMATION AND M ECHANISM D ESIGN – M AY 15, 2010

  24. Anonymous Reserves Question: for non-identical distributions, is anonymous reserve approximately optimal? (e.g., eBay) Thm: non-identical, regular distributions, Vickrey with anonymous reserve price is 3-approximation. [H, Roughgarden ’09] 14 A PPROXIMATION AND M ECHANISM D ESIGN – M AY 15, 2010

  25. Anonymous Reserves Question: for non-identical distributions, is anonymous reserve approximately optimal? (e.g., eBay) Thm: non-identical, regular distributions, Vickrey with anonymous reserve price is 3-approximation. [H, Roughgarden ’09] Proof: more complicated extension of prophet inequalities. 14 A PPROXIMATION AND M ECHANISM D ESIGN – M AY 15, 2010

  26. Anonymous Reserves Question: for non-identical distributions, is anonymous reserve approximately optimal? (e.g., eBay) Thm: non-identical, regular distributions, Vickrey with anonymous reserve price is 3-approximation. [H, Roughgarden ’09] Proof: more complicated extension of prophet inequalities. Discussion: • theorem is not tight, actual bound is in [2 , 3] . • justifies wide prevalence. • approximation good for platform design . 14 A PPROXIMATION AND M ECHANISM D ESIGN – M AY 15, 2010

  27. Extensions Beyond single-item auctions: general feasibility constraints . 15 A PPROXIMATION AND M ECHANISM D ESIGN – M AY 15, 2010

  28. Extensions Beyond single-item auctions: general feasibility constraints . Thm: for non-identical regular distributions, VCG with monopoly reserves is often a 2-approximation. [H, Roughgarden ’09] Thm: non-identical (possibly irregular) distributions, posted pricing mechanisms are often constant approximations. [Chawla, H, Malec, Sivan ’10] 15 A PPROXIMATION AND M ECHANISM D ESIGN – M AY 15, 2010

  29. Extensions Beyond single-item auctions: general feasibility constraints . Thm: for non-identical regular distributions, VCG with monopoly reserves is often a 2-approximation. [H, Roughgarden ’09] Thm: non-identical (possibly irregular) distributions, posted pricing mechanisms are often constant approximations. [Chawla, H, Malec, Sivan ’10] Proof technique: • optimal mechanism is a virtual surplus maximizer. • reserve-price mechanisms are virtual surplus approximators. 15 A PPROXIMATION AND M ECHANISM D ESIGN – M AY 15, 2010

  30. Extensions Beyond single-item auctions: general feasibility constraints . Thm: for non-identical regular distributions, VCG with monopoly reserves is often a 2-approximation. [H, Roughgarden ’09] Thm: non-identical (possibly irregular) distributions, posted pricing mechanisms are often constant approximations. [Chawla, H, Malec, Sivan ’10] Proof technique: • optimal mechanism is a virtual surplus maximizer. • reserve-price mechanisms are virtual surplus approximators. Basic Open Question: to what extent to simple mechanisms approxi- mate (well understood but complex) optimal ones? Challenges: non-downward-closed settings, negative virtual values. 15 A PPROXIMATION AND M ECHANISM D ESIGN – M AY 15, 2010

  31. Part II: Approximation for multi-dimensional Bayesian mechanism design (where agent preferences are given by values for each available service, zero for no service; preferences drawn from distribution)

  32. Example 3: unit-demand pricing Problem: Bayesian Unit-Demand Pricing • a single, unit-demand consumer. • n items for sale. • a dist. F = F 1 × · · · × F n from which the con- sumer’s values for each item are drawn. Goal: seller optimal item-pricing for F . 17 A PPROXIMATION AND M ECHANISM D ESIGN – M AY 15, 2010

  33. Example 3: unit-demand pricing Problem: Bayesian Unit-Demand Pricing • a single, unit-demand consumer. • n items for sale. • a dist. F = F 1 × · · · × F n from which the con- sumer’s values for each item are drawn. Goal: seller optimal item-pricing for F . Question: What is optimal pricing? 17 A PPROXIMATION AND M ECHANISM D ESIGN – M AY 15, 2010

  34. Optimal Pricing Optimal Pricing: consider distribution, feasibility constraints, incentive constraints, and solve! 18 A PPROXIMATION AND M ECHANISM D ESIGN – M AY 15, 2010

  35. Optimal Pricing Optimal Pricing: consider distribution, feasibility constraints, incentive constraints, and solve! Discussion: • little conceptual insight and • not generally tractable. 18 A PPROXIMATION AND M ECHANISM D ESIGN – M AY 15, 2010

  36. Analogy Challenge: approximate optimal but we do not understand it? 19 A PPROXIMATION AND M ECHANISM D ESIGN – M AY 15, 2010

  37. Analogy Challenge: approximate optimal but we do not understand it? Problem: Bayesian Unit-demand Problem: Bayesian Single-item Pricing (a.k.a., MD-PRICING) Auction (a.k.a., SD-AUCTION) • a single, unit-demand buyer, • a single item for sale, • n items for sale, and • n buyers, and • a dist. F from which the con- • a dist. F from which the con- sumer’s value for each item is sumers’ values for the item are drawn. drawn. Goal: seller opt. item-pricing for F . Goal: seller opt. auction for F . 19 A PPROXIMATION AND M ECHANISM D ESIGN – M AY 15, 2010

  38. Analogy Challenge: approximate optimal but we do not understand it? Problem: Bayesian Unit-demand Problem: Bayesian Single-item Pricing (a.k.a., MD-PRICING) Auction (a.k.a., SD-AUCTION) • a single, unit-demand buyer, • a single item for sale, • n items for sale, and • n buyers, and • a dist. F from which the con- • a dist. F from which the con- sumer’s value for each item is sumers’ values for the item are drawn. drawn. Goal: seller opt. item-pricing for F . Goal: seller opt. auction for F . Note: Same informational structure. 19 A PPROXIMATION AND M ECHANISM D ESIGN – M AY 15, 2010

  39. Analogy Challenge: approximate optimal but we do not understand it? Problem: Bayesian Unit-demand Problem: Bayesian Single-item Pricing (a.k.a., MD-PRICING) Auction (a.k.a., SD-AUCTION) • a single, unit-demand buyer, • a single item for sale, • n items for sale, and • n buyers, and • a dist. F from which the con- • a dist. F from which the con- sumer’s value for each item is sumers’ values for the item are drawn. drawn. Goal: seller opt. item-pricing for F . Goal: seller opt. auction for F . Note: Same informational structure. Thm: for any indep. distributions, MD-PRICING ≤ SD-AUCTION. 19 A PPROXIMATION AND M ECHANISM D ESIGN – M AY 15, 2010

  40. Analogy Challenge: approximate optimal but we do not understand it? Problem: Bayesian Unit-demand Problem: Bayesian Single-item Pricing (a.k.a., MD-PRICING) Auction (a.k.a., SD-AUCTION) • a single, unit-demand buyer, • a single item for sale, • n items for sale, and • n buyers, and • a dist. F from which the con- • a dist. F from which the con- sumer’s value for each item is sumers’ values for the item are drawn. drawn. Goal: seller opt. item-pricing for F . Goal: seller opt. auction for F . Note: Same informational structure. Thm: for any indep. distributions, MD-PRICING ≤ SD-AUCTION. Thm: a constant virtual price for MD-PRICING is 2-approx. [Chawla, H, Malec, Sivan ’10] 19 A PPROXIMATION AND M ECHANISM D ESIGN – M AY 15, 2010

  41. Analogy Challenge: approximate optimal but we do not understand it? Problem: Bayesian Unit-demand Problem: Bayesian Single-item Pricing (a.k.a., MD-PRICING) Auction (a.k.a., SD-AUCTION) • a single, unit-demand buyer, • a single item for sale, • n items for sale, and • n buyers, and • a dist. F from which the con- • a dist. F from which the con- sumer’s value for each item is sumers’ values for the item are drawn. drawn. Goal: seller opt. item-pricing for F . Goal: seller opt. auction for F . Note: Same informational structure. Thm: for any indep. distributions, MD-PRICING ≤ SD-AUCTION. Thm: a constant virtual price for MD-PRICING is 2-approx. [Chawla, H, Malec, Sivan ’10] Proof: prophet inequality. 19 A PPROXIMATION AND M ECHANISM D ESIGN – M AY 15, 2010

  42. Multi-item Auctions Sequential Posted Pricing: agents arrive in sequence, offer posted prices. 20 A PPROXIMATION AND M ECHANISM D ESIGN – M AY 15, 2010

  43. Multi-item Auctions Sequential Posted Pricing: agents arrive in sequence, offer posted prices. Thm: in many unit-demand settings, sequential posted pricings are a constant approximation to the optimal mechanism. [Chawla, H, Malec, Sivan ’10] 20 A PPROXIMATION AND M ECHANISM D ESIGN – M AY 15, 2010

  44. Multi-item Auctions Sequential Posted Pricing: agents arrive in sequence, offer posted prices. Thm: in many unit-demand settings, sequential posted pricings are a constant approximation to the optimal mechanism. [Chawla, H, Malec, Sivan ’10] Approach: 1. Analogy: “single-dimensional analog” (replace unit-demand agent with many single-dimensional agents) 20 A PPROXIMATION AND M ECHANISM D ESIGN – M AY 15, 2010

  45. Multi-item Auctions Sequential Posted Pricing: agents arrive in sequence, offer posted prices. Thm: in many unit-demand settings, sequential posted pricings are a constant approximation to the optimal mechanism. [Chawla, H, Malec, Sivan ’10] Approach: 1. Analogy: “single-dimensional analog” (replace unit-demand agent with many single-dimensional agents) 2. Upper bound: SD-AUCTION ≥ MD-PRICING (competition increases revenue) 20 A PPROXIMATION AND M ECHANISM D ESIGN – M AY 15, 2010

  46. Multi-item Auctions Sequential Posted Pricing: agents arrive in sequence, offer posted prices. Thm: in many unit-demand settings, sequential posted pricings are a constant approximation to the optimal mechanism. [Chawla, H, Malec, Sivan ’10] Approach: 1. Analogy: “single-dimensional analog” (replace unit-demand agent with many single-dimensional agents) 2. Upper bound: SD-AUCTION ≥ MD-PRICING (competition increases revenue) 3. Reduction: MD-PRICING ≥ SD-PRICING (pricings don’t use competition) 20 A PPROXIMATION AND M ECHANISM D ESIGN – M AY 15, 2010

  47. Multi-item Auctions Sequential Posted Pricing: agents arrive in sequence, offer posted prices. Thm: in many unit-demand settings, sequential posted pricings are a constant approximation to the optimal mechanism. [Chawla, H, Malec, Sivan ’10] Approach: 1. Analogy: “single-dimensional analog” (replace unit-demand agent with many single-dimensional agents) 2. Upper bound: SD-AUCTION ≥ MD-PRICING (competition increases revenue) 3. Reduction: MD-PRICING ≥ SD-PRICING (pricings don’t use competition) 4. Instantiation: SD-PRICING ≥ 1 β SD-AUCTION (virtual surplus approximation) 20 A PPROXIMATION AND M ECHANISM D ESIGN – M AY 15, 2010

  48. Sequential Posted Pricing Discussion Sequential Posted Pricing: agents arrive in sequence, offer posted prices. Thm: in many unit-demand settings, sequential posted pricings are a constant approximation to the optimal mechanism. [Chawla, H, Malec, Sivan ’10] 21 A PPROXIMATION AND M ECHANISM D ESIGN – M AY 15, 2010

  49. Sequential Posted Pricing Discussion Sequential Posted Pricing: agents arrive in sequence, offer posted prices. Thm: in many unit-demand settings, sequential posted pricings are a constant approximation to the optimal mechanism. [Chawla, H, Malec, Sivan ’10] Discussion: • robust to agent ordering, collusion, etc. • conclusive : competition not important for approximation. • practical : posted pricings widely prevalent. (e.g., eBay) • role of randomization is crucial. [Chawla, Malec, Sivan ’10] 21 A PPROXIMATION AND M ECHANISM D ESIGN – M AY 15, 2010

  50. Sequential Posted Pricing Discussion Sequential Posted Pricing: agents arrive in sequence, offer posted prices. Thm: in many unit-demand settings, sequential posted pricings are a constant approximation to the optimal mechanism. [Chawla, H, Malec, Sivan ’10] Discussion: • robust to agent ordering, collusion, etc. • conclusive : competition not important for approximation. • practical : posted pricings widely prevalent. (e.g., eBay) • role of randomization is crucial. [Chawla, Malec, Sivan ’10] Open Question: identify upper bounds beyond unit-demand settings that are • conceptually tractable and • approximable. 21 A PPROXIMATION AND M ECHANISM D ESIGN – M AY 15, 2010

  51. Part III: Approximation for prior-free mechanism design. (mechanisms should be good for any set of agent preferences, not just given distributional assumptions)

  52. The problem with priors Prior assumption: the mechanism designer knows the distribution of agent preferences. 23 A PPROXIMATION AND M ECHANISM D ESIGN – M AY 15, 2010

  53. The problem with priors Prior assumption: the mechanism designer knows the distribution of agent preferences. Where does prior come from: • historical data then using prior affects incentives of earlier transactions. (e.g. Coase Conjecture) • market analysis accuracy depends on market size, auctions are for small markets. 23 A PPROXIMATION AND M ECHANISM D ESIGN – M AY 15, 2010

  54. The problem with priors Prior assumption: the mechanism designer knows the distribution of agent preferences. Where does prior come from: • historical data then using prior affects incentives of earlier transactions. (e.g. Coase Conjecture) • market analysis accuracy depends on market size, auctions are for small markets. Question: can we design good auctions without knowledge of prior-distribution? 23 A PPROXIMATION AND M ECHANISM D ESIGN – M AY 15, 2010

  55. Resource augmentation Approach 1: “resource” augmentation. 24 A PPROXIMATION AND M ECHANISM D ESIGN – M AY 15, 2010

  56. Resource augmentation Approach 1: “resource” augmentation. Thm: for iid, regular, single-item auctions, the Vickrey auction on n + 1 bidders has more revenue than the optimal auction on n bidders. [Bulow, Klemperer ’96] 24 A PPROXIMATION AND M ECHANISM D ESIGN – M AY 15, 2010

  57. Resource augmentation Approach 1: “resource” augmentation. Thm: for iid, regular, single-item auctions, the Vickrey auction on n + 1 bidders has more revenue than the optimal auction on n bidders. [Bulow, Klemperer ’96] Discussion: [Dhangwatnotai, Roughgarden, Yan ’10] • “recruit one more bidder” is prior-free strategy. • “bicriteria” approximation result. • conclusive: competition more important than optimization. 24 A PPROXIMATION AND M ECHANISM D ESIGN – M AY 15, 2010

  58. Resource augmentation Approach 1: “resource” augmentation. Thm: for iid, regular, single-item auctions, the Vickrey auction on n + 1 bidders has more revenue than the optimal auction on n bidders. [Bulow, Klemperer ’96] Discussion: [Dhangwatnotai, Roughgarden, Yan ’10] • “recruit one more bidder” is prior-free strategy. • “bicriteria” approximation result. • conclusive: competition more important than optimization. • non-generic : e.g., for k -unit auctions, need k additional bidders. 24 A PPROXIMATION AND M ECHANISM D ESIGN – M AY 15, 2010

  59. Special Case: n = 1 Special Case: for regular distribution, the Vickrey revenue from two bidders is at least the optimal revenue from one bidder. 25 A PPROXIMATION AND M ECHANISM D ESIGN – M AY 15, 2010

  60. Special Case: n = 1 Special Case: for regular distribution, the Vickrey revenue from two bidders is at least the optimal revenue from one bidder. Geometric Proof: [Dhangwatnotai, Roughgarden, Yan ’10] 25 A PPROXIMATION AND M ECHANISM D ESIGN – M AY 15, 2010

  61. Special Case: n = 1 Special Case: for regular distribution, the Vickrey revenue from two bidders is at least the optimal revenue from one bidder. Geometric Proof: [Dhangwatnotai, Roughgarden, Yan ’10] • each bidder in Vickrey views other bid as “random reserve”. 25 A PPROXIMATION AND M ECHANISM D ESIGN – M AY 15, 2010

  62. Special Case: n = 1 Special Case: for regular distribution, the Vickrey revenue from two bidders is at least the optimal revenue from one bidder. Geometric Proof: [Dhangwatnotai, Roughgarden, Yan ’10] • each bidder in Vickrey views other bid as “random reserve”. • Vickrey revenue = 2 × random reserve revenue. 25 A PPROXIMATION AND M ECHANISM D ESIGN – M AY 15, 2010

  63. Special Case: n = 1 Special Case: for regular distribution, the Vickrey revenue from two bidders is at least the optimal revenue from one bidder. Geometric Proof: [Dhangwatnotai, Roughgarden, Yan ’10] • each bidder in Vickrey views other bid as “random reserve”. • Vickrey revenue = 2 × random reserve revenue. • random reserve revenue ≥ 1 2 × optimal reserve revenue: 25 A PPROXIMATION AND M ECHANISM D ESIGN – M AY 15, 2010

  64. Special Case: n = 1 Special Case: for regular distribution, the Vickrey revenue from two bidders is at least the optimal revenue from one bidder. Geometric Proof: [Dhangwatnotai, Roughgarden, Yan ’10] • each bidder in Vickrey views other bid as “random reserve”. • Vickrey revenue = 2 × random reserve revenue. • random reserve revenue ≥ 1 2 × optimal reserve revenue: R ( q ) 0 0 1 25 A PPROXIMATION AND M ECHANISM D ESIGN – M AY 15, 2010

  65. Special Case: n = 1 Special Case: for regular distribution, the Vickrey revenue from two bidders is at least the optimal revenue from one bidder. Geometric Proof: [Dhangwatnotai, Roughgarden, Yan ’10] • each bidder in Vickrey views other bid as “random reserve”. • Vickrey revenue = 2 × random reserve revenue. • random reserve revenue ≥ 1 2 × optimal reserve revenue: R ( q ) 0 0 1 25 A PPROXIMATION AND M ECHANISM D ESIGN – M AY 15, 2010

  66. Special Case: n = 1 Special Case: for regular distribution, the Vickrey revenue from two bidders is at least the optimal revenue from one bidder. Geometric Proof: [Dhangwatnotai, Roughgarden, Yan ’10] • each bidder in Vickrey views other bid as “random reserve”. • Vickrey revenue = 2 × random reserve revenue. • random reserve revenue ≥ 1 2 × optimal reserve revenue: R ( q ) 0 0 1 25 A PPROXIMATION AND M ECHANISM D ESIGN – M AY 15, 2010

  67. Special Case: n = 1 Special Case: for regular distribution, the Vickrey revenue from two bidders is at least the optimal revenue from one bidder. Geometric Proof: [Dhangwatnotai, Roughgarden, Yan ’10] • each bidder in Vickrey views other bid as “random reserve”. • Vickrey revenue = 2 × random reserve revenue. • random reserve revenue ≥ 1 2 × optimal reserve revenue: R ( q ) 0 0 1 25 A PPROXIMATION AND M ECHANISM D ESIGN – M AY 15, 2010

  68. Special Case: n = 1 Special Case: for regular distribution, the Vickrey revenue from two bidders is at least the optimal revenue from one bidder. Geometric Proof: [Dhangwatnotai, Roughgarden, Yan ’10] • each bidder in Vickrey views other bid as “random reserve”. • Vickrey revenue = 2 × random reserve revenue. • random reserve revenue ≥ 1 2 × optimal reserve revenue: R ( q ) 0 0 1 • So Vickrey with two bidders ≥ optimal revenue from one bidder. 25 A PPROXIMATION AND M ECHANISM D ESIGN – M AY 15, 2010

  69. Example 4: digital goods Question: how should a profit-maximizing seller sell a digital good ( n bidder, n copies of item)? 26 A PPROXIMATION AND M ECHANISM D ESIGN – M AY 15, 2010

  70. Example 4: digital goods Question: how should a profit-maximizing seller sell a digital good ( n bidder, n copies of item)? Bayesian Optimal Solution: if values are iid from known distribution, post the monopoly price ϕ − 1 (0) . [Myerson ’81] 26 A PPROXIMATION AND M ECHANISM D ESIGN – M AY 15, 2010

  71. Example 4: digital goods Question: how should a profit-maximizing seller sell a digital good ( n bidder, n copies of item)? Bayesian Optimal Solution: if values are iid from known distribution, post the monopoly price ϕ − 1 (0) . [Myerson ’81] Discussion: • optimal, • simple, but • not prior-free 26 A PPROXIMATION AND M ECHANISM D ESIGN – M AY 15, 2010

  72. Approximation via Single Sample Single-Sample Auction: (for digital goods) [Dhangwatnotai, Roughgarden, Yan ’10] 1. pick random agent i as sample. 2. offer all other agents price v i . 3. reject i . 27 A PPROXIMATION AND M ECHANISM D ESIGN – M AY 15, 2010

  73. Approximation via Single Sample Single-Sample Auction: (for digital goods) [Dhangwatnotai, Roughgarden, Yan ’10] 1. pick random agent i as sample. 2. offer all other agents price v i . 3. reject i . Thm: for iid, regular distributions, single sample auction on ( n + 1) -agents is 2-approx to optimal on n agents. [Dhangwatnotai, Roughgarden, Yan ’10] 27 A PPROXIMATION AND M ECHANISM D ESIGN – M AY 15, 2010

  74. Approximation via Single Sample Single-Sample Auction: (for digital goods) [Dhangwatnotai, Roughgarden, Yan ’10] 1. pick random agent i as sample. 2. offer all other agents price v i . 3. reject i . Thm: for iid, regular distributions, single sample auction on ( n + 1) -agents is 2-approx to optimal on n agents. [Dhangwatnotai, Roughgarden, Yan ’10] Proof: from geometric argument. 27 A PPROXIMATION AND M ECHANISM D ESIGN – M AY 15, 2010

  75. Approximation via Single Sample Single-Sample Auction: (for digital goods) [Dhangwatnotai, Roughgarden, Yan ’10] 1. pick random agent i as sample. 2. offer all other agents price v i . 3. reject i . Thm: for iid, regular distributions, single sample auction on ( n + 1) -agents is 2-approx to optimal on n agents. [Dhangwatnotai, Roughgarden, Yan ’10] Proof: from geometric argument. Discussion: • prior-free . • conclusive , don’t need precise distribution, only need single sample for approximation. more samples can improve approximation factor. • generic , applies to general settings. 27 A PPROXIMATION AND M ECHANISM D ESIGN – M AY 15, 2010

  76. Average-case vs Worst-case Note: prior-free auction cannot be optimal in every setting. 28 A PPROXIMATION AND M ECHANISM D ESIGN – M AY 15, 2010

  77. Average-case vs Worst-case Note: prior-free auction cannot be optimal in every setting. Average Case Approximation: ∃A , ∀ F ∈ IID , E v ∼ F [OPT F ( v )] E v ∼ F [ A ( v )] ≥ . β 28 A PPROXIMATION AND M ECHANISM D ESIGN – M AY 15, 2010

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