A Universally-Truthful Approximation Scheme for Multi-unit Auction

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A Universally-Truthful Approximation Scheme for Multi-unit Auction

  1. 1. Introduction Tools The Mechanism Summary A Universally-Truthful Approximation Scheme for Multi-unit Auction Author : Berthold Vöcking Presenter : Thatchaphol Saranurak Seminar Algorithmic Game Theory Saarland University 6 Dec 2011
  2. 2. Introduction Tools The Mechanism Summary Outline Introduction Problem 3 Kinds of Truthfulness Related Work Tools ∆-Perturbed Maximizer Consensus Function with Drop-outs The Mechanism Social Choice Function Feasibility Approximation Ratio Universal Truthfulness Summary
  3. 3. Introduction Tools The Mechanism Summary Outline Introduction Problem 3 Kinds of Truthfulness Related Work Tools ∆-Perturbed Maximizer Consensus Function with Drop-outs The Mechanism Social Choice Function Feasibility Approximation Ratio Universal Truthfulness Summary
  4. 4. Introduction Tools The Mechanism Summary Multi-unit Auction • m identical items • n bidders • Bidders bid : Valuation function vi • vi : {0, 1, ..., m} → R 0 • how much i is willing to pay for each amount • V = {v | v(0) = 0 and v is non-decreasing } • Bidders get : Allocation s • s = (s1 , s2 , ..., sn ) ∈ {0, ..., m}n : how much each gets • feasible set A = {s | ∑n si i=1 m} • vi (s) = vi (si ) • Objective : maximize social welfare ∑i vi (s) = v(s)
  5. 5. Introduction Tools The Mechanism Summary Multi-unit Auction • m identical items • n bidders • Bidders bid : Valuation function vi • vi : {0, 1, ..., m} → R 0 • how much i is willing to pay for each amount • V = {v | v(0) = 0 and v is non-decreasing } • Bidders get : Allocation s • s = (s1 , s2 , ..., sn ) ∈ {0, ..., m}n : how much each gets • feasible set A = {s | ∑n si i=1 m} • vi (s) = vi (si ) • Objective : maximize social welfare ∑i vi (s) = v(s)
  6. 6. Introduction Tools The Mechanism Summary Multi-unit Auction • m identical items • n bidders • Bidders bid : Valuation function vi • vi : {0, 1, ..., m} → R 0 • how much i is willing to pay for each amount • V = {v | v(0) = 0 and v is non-decreasing } • Bidders get : Allocation s • s = (s1 , s2 , ..., sn ) ∈ {0, ..., m}n : how much each gets • feasible set A = {s | ∑n si i=1 m} • vi (s) = vi (si ) • Objective : maximize social welfare ∑i vi (s) = v(s)
  7. 7. Introduction Tools The Mechanism Summary Multi-unit Auction • m identical items • n bidders • Bidders bid : Valuation function vi • vi : {0, 1, ..., m} → R 0 • how much i is willing to pay for each amount • V = {v | v(0) = 0 and v is non-decreasing } • Bidders get : Allocation s • s = (s1 , s2 , ..., sn ) ∈ {0, ..., m}n : how much each gets • feasible set A = {s | ∑n si i=1 m} • vi (s) = vi (si ) • Objective : maximize social welfare ∑i vi (s) = v(s)
  8. 8. Introduction Tools The Mechanism Summary Multi-unit Auction • m identical items • n bidders • Bidders bid : Valuation function vi • vi : {0, 1, ..., m} → R 0 • how much i is willing to pay for each amount • V = {v | v(0) = 0 and v is non-decreasing } • Bidders get : Allocation s • s = (s1 , s2 , ..., sn ) ∈ {0, ..., m}n : how much each gets • feasible set A = {s | ∑n si i=1 m} • vi (s) = vi (si ) • Objective : maximize social welfare ∑i vi (s) = v(s)
  9. 9. Introduction Tools The Mechanism Summary Multi-unit Auction • m identical items • n bidders • Bidders bid : Valuation function vi • vi : {0, 1, ..., m} → R 0 • how much i is willing to pay for each amount • V = {v | v(0) = 0 and v is non-decreasing } • Bidders get : Allocation s • s = (s1 , s2 , ..., sn ) ∈ {0, ..., m}n : how much each gets • feasible set A = {s | ∑n si i=1 m} • vi (s) = vi (si ) • Objective : maximize social welfare ∑i vi (s) = v(s)
  10. 10. Introduction Tools The Mechanism Summary Multi-unit Auction • m identical items • n bidders • Bidders bid : Valuation function vi • vi : {0, 1, ..., m} → R 0 • how much i is willing to pay for each amount • V = {v | v(0) = 0 and v is non-decreasing } • Bidders get : Allocation s • s = (s1 , s2 , ..., sn ) ∈ {0, ..., m}n : how much each gets • feasible set A = {s | ∑n si i=1 m} • vi (s) = vi (si ) • Objective : maximize social welfare ∑i vi (s) = v(s)
  11. 11. Introduction Tools The Mechanism Summary Mechanism • Mechanism (f , p) • social choice function f : V n → A • payment scheme p = (p1 , p2 , ..., pn ) • pi : V n → R • k-approximation mechanism • social welfaresocial welfare optimum from mechanism ≥k • Polynomial time mechanism • poly(n, log m)
  12. 12. Introduction Tools The Mechanism Summary Mechanism • Mechanism (f , p) • social choice function f : V n → A • payment scheme p = (p1 , p2 , ..., pn ) • pi : V n → R • k-approximation mechanism • social welfaresocial welfare optimum from mechanism ≥k • Polynomial time mechanism • poly(n, log m)
  13. 13. Introduction Tools The Mechanism Summary Mechanism • Mechanism (f , p) • social choice function f : V n → A • payment scheme p = (p1 , p2 , ..., pn ) • pi : V n → R • k-approximation mechanism • social welfaresocial welfare optimum from mechanism ≥k • Polynomial time mechanism • poly(n, log m)
  14. 14. Introduction Tools The Mechanism Summary Mechanism • Mechanism (f , p) • social choice function f : V n → A • payment scheme p = (p1 , p2 , ..., pn ) • pi : V n → R • k-approximation mechanism • social welfaresocial welfare optimum from mechanism ≥k • Polynomial time mechanism • poly(n, log m)
  15. 15. Introduction Tools The Mechanism Summary Mechanism • Mechanism (f , p) • social choice function f : V n → A • payment scheme p = (p1 , p2 , ..., pn ) • pi : V n → R • k-approximation mechanism • social welfaresocial welfare optimum from mechanism ≥k • Polynomial time mechanism • poly(n, log m)
  16. 16. Introduction Tools The Mechanism Summary Utility of Bidder • let s = f (vi , v−i ) • v−i : all valuation functions, except is valuation • Utility of bidder i : vi (s) − pi (vi , v−i )
  17. 17. Introduction Tools The Mechanism Summary Utility of Bidder • let s = f (vi , v−i ) • v−i : all valuation functions, except is valuation • Utility of bidder i : vi (s) − pi (vi , v−i )
  18. 18. Introduction Tools The Mechanism Summary Outline Introduction Problem 3 Kinds of Truthfulness Related Work Tools ∆-Perturbed Maximizer Consensus Function with Drop-outs The Mechanism Social Choice Function Feasibility Approximation Ratio Universal Truthfulness Summary
  19. 19. Introduction Tools The Mechanism Summary Deterministic Mechanism Denition Deterministically truthful • Utility vi (s) − pi is maximized when i bids the true vi
  20. 20. Introduction Tools The Mechanism Summary Randomized Mechanism • Randomized mechanism • Probability distribution over deterministic mechanisms Denition Universally truthful • Each mechanism in distribution is truthful Denition Truthful in expectation • Expected utility is maximized when i bids the true vi • Truthful only if bidders dont know outcome of random bits
  21. 21. Introduction Tools The Mechanism Summary Randomized Mechanism • Randomized mechanism • Probability distribution over deterministic mechanisms Denition Universally truthful • Each mechanism in distribution is truthful Denition Truthful in expectation • Expected utility is maximized when i bids the true vi • Truthful only if bidders dont know outcome of random bits
  22. 22. Introduction Tools The Mechanism Summary Randomized Mechanism • Randomized mechanism • Probability distribution over deterministic mechanisms Denition Universally truthful • Each mechanism in distribution is truthful Denition Truthful in expectation • Expected utility is maximized when i bids the true vi • Truthful only if bidders dont know outcome of random bits
  23. 23. Introduction Tools The Mechanism Summary Randomized Mechanism • Randomized mechanism • Probability distribution over deterministic mechanisms Denition Universally truthful • Each mechanism in distribution is truthful Denition Truthful in expectation • Expected utility is maximized when i bids the true vi • Truthful only if bidders dont know outcome of random bits
  24. 24. Introduction Tools The Mechanism Summary Outline Introduction Problem 3 Kinds of Truthfulness Related Work Tools ∆-Perturbed Maximizer Consensus Function with Drop-outs The Mechanism Social Choice Function Feasibility Approximation Ratio Universal Truthfulness Summary
  25. 25. Introduction Tools The Mechanism Summary Compare Dierent Power of Truthfulness-es Truthfulness Mechanism Deterministically truthful 2-approx, poly-time Universally truthful ? Truthful in expectation(*) FPTAS • PTAS • for xed ε 0, (1 − ε )-approximation • run in poly(n, log m) • may run in exp(1/ε ). • FPTAS • PTAS, but run in poly(1/ε ) • (*) suggested : no poly-time universally truthful mechanism has approximation ratio better than 2
  26. 26. Introduction Tools The Mechanism Summary Compare Dierent Power of Truthfulness-es Truthfulness Mechanism Deterministically truthful 2-approx, poly-time Universally truthful ? Truthful in expectation(*) FPTAS • PTAS • for xed ε 0, (1 − ε )-approximation • run in poly(n, log m) • may run in exp(1/ε ). • FPTAS • PTAS, but run in poly(1/ε ) • (*) suggested : no poly-time universally truthful mechanism has approximation ratio better than 2
  27. 27. Introduction Tools The Mechanism Summary Compare Dierent Power of Truthfulness-es Truthfulness Mechanism Deterministically truthful 2-approx, poly-time Universally truthful ? Truthful in expectation(*) FPTAS • PTAS • for xed ε 0, (1 − ε )-approximation • run in poly(n, log m) • may run in exp(1/ε ). • FPTAS • PTAS, but run in poly(1/ε ) • (*) suggested : no poly-time universally truthful mechanism has approximation ratio better than 2
  28. 28. Introduction Tools The Mechanism Summary Compare Dierent Power of Truthfulness-es Truthfulness Mechanism Deterministically truthful 2-approx, poly-time Universally truthful ? Truthful in expectation(*) FPTAS • PTAS • for xed ε 0, (1 − ε )-approximation • run in poly(n, log m) • may run in exp(1/ε ). • FPTAS • PTAS, but run in poly(1/ε ) • (*) suggested : no poly-time universally truthful mechanism has approximation ratio better than 2
  29. 29. Introduction Tools The Mechanism Summary This Paper : Universally Truthful Mechanism has PTAS Truthfulness Mechanism Deterministically truthful 2-approx, poly-time Universally truthful PTAS Truthful in expectation FPTAS
  30. 30. Introduction Tools The Mechanism Summary Outline Introduction Problem 3 Kinds of Truthfulness Related Work Tools ∆-Perturbed Maximizer Consensus Function with Drop-outs The Mechanism Social Choice Function Feasibility Approximation Ratio Universal Truthfulness Summary
  31. 31. Introduction Tools The Mechanism Summary Multiple-choice Knapsack Problem • n classes of objects • m objects, for each class • each object k has weight wk and prot pk • select 1 object from each class • sum of weight ≤ m • maximize sum of prot
  32. 32. Introduction Tools The Mechanism Summary Multiple-choice Knapsack Problem • n classes of objects • m objects, for each class • each object k has weight wk and prot pk • select 1 object from each class • sum of weight ≤ m • maximize sum of prot
  33. 33. Introduction Tools The Mechanism Summary Reduce Maximizing Social Welfare → Knapsack Problem • Dene (i, j) be object for allocating j items to bidder i • w(i,j) = j • p(i,j) = vi (j) • Solve Knapsack = Optimize social welfare • But still cannot solve in poly(n, log m) • #object is n × m not poly(n, log m) • Knapsack is NP-Hard!
  34. 34. Introduction Tools The Mechanism Summary Reduce Maximizing Social Welfare → Knapsack Problem • Dene (i, j) be object for allocating j items to bidder i • w(i,j) = j • p(i,j) = vi (j) • Solve Knapsack = Optimize social welfare • But still cannot solve in poly(n, log m) • #object is n × m not poly(n, log m) • Knapsack is NP-Hard!
  35. 35. Introduction Tools The Mechanism Summary Reduce Maximizing Social Welfare → Knapsack Problem • Dene (i, j) be object for allocating j items to bidder i • w(i,j) = j • p(i,j) = vi (j) • Solve Knapsack = Optimize social welfare • But still cannot solve in poly(n, log m) • #object is n × m not poly(n, log m) • Knapsack is NP-Hard!
  36. 36. Introduction Tools The Mechanism Summary Reduce Maximizing Social Welfare → Knapsack Problem • Dene (i, j) be object for allocating j items to bidder i • w(i,j) = j • p(i,j) = vi (j) • Solve Knapsack = Optimize social welfare • But still cannot solve in poly(n, log m) • #object is n × m not poly(n, log m) • Knapsack is NP-Hard!
  37. 37. Introduction Tools The Mechanism Summary Reduce Maximizing Social Welfare → Knapsack Problem • Dene (i, j) be object for allocating j items to bidder i • w(i,j) = j • p(i,j) = vi (j) • Solve Knapsack = Optimize social welfare • But still cannot solve in poly(n, log m) • #object is n × m not poly(n, log m) • Knapsack is NP-Hard!
  38. 38. Introduction Tools The Mechanism Summary Perturbed Valuation Function • Dene • ∆ 0 is some constant j • xi ∼ [0, 1] • q(k) : {0, ..., m} → Z is number of factor 2 of k • 96 = 25 × 3 so q(96) = 5 • q(k) ≤ log m • exception : q(0) = log m + 1 • Perturbed valuation function j vi (j) = v(j) + (2q(j) + xi )∆
  39. 39. Introduction Tools The Mechanism Summary Perturbed Valuation Function • Dene • ∆ 0 is some constant j • xi ∼ [0, 1] • q(k) : {0, ..., m} → Z is number of factor 2 of k • 96 = 25 × 3 so q(96) = 5 • q(k) ≤ log m • exception : q(0) = log m + 1 • Perturbed valuation function j vi (j) = v(j) + (2q(j) + xi )∆
  40. 40. Introduction Tools The Mechanism Summary Perturbed Valuation Function • Dene • ∆ 0 is some constant j • xi ∼ [0, 1] • q(k) : {0, ..., m} → Z is number of factor 2 of k • 96 = 25 × 3 so q(96) = 5 • q(k) ≤ log m • exception : q(0) = log m + 1 • Perturbed valuation function j vi (j) = v(j) + (2q(j) + xi )∆
  41. 41. Introduction Tools The Mechanism Summary Perturbed Valuation Function • Dene • ∆ 0 is some constant j • xi ∼ [0, 1] • q(k) : {0, ..., m} → Z is number of factor 2 of k • 96 = 25 × 3 so q(96) = 5 • q(k) ≤ log m • exception : q(0) = log m + 1 • Perturbed valuation function j vi (j) = v(j) + (2q(j) + xi )∆
  42. 42. Introduction Tools The Mechanism Summary Perturbed Valuation Function • Dene • ∆ 0 is some constant j • xi ∼ [0, 1] • q(k) : {0, ..., m} → Z is number of factor 2 of k • 96 = 25 × 3 so q(96) = 5 • q(k) ≤ log m • exception : q(0) = log m + 1 • Perturbed valuation function j vi (j) = v(j) + (2q(j) + xi )∆
  43. 43. Introduction Tools The Mechanism Summary Perturbed Valuation Function • Dene • ∆ 0 is some constant j • xi ∼ [0, 1] • q(k) : {0, ..., m} → Z is number of factor 2 of k • 96 = 25 × 3 so q(96) = 5 • q(k) ≤ log m • exception : q(0) = log m + 1 • Perturbed valuation function j vi (j) = v(j) + (2q(j) + xi )∆
  44. 44. Introduction Tools The Mechanism Summary Perturbed Valuation Function • Dene • ∆ 0 is some constant j • xi ∼ [0, 1] • q(k) : {0, ..., m} → Z is number of factor 2 of k • 96 = 25 × 3 so q(96) = 5 • q(k) ≤ log m • exception : q(0) = log m + 1 • Perturbed valuation function j vi (j) = v(j) + (2q(j) + xi )∆
  45. 45. Introduction Tools The Mechanism Summary Poly-time Knapsack • Set prot of object p(i,j) = vi (j) instead of vi (j) • Claim: Expected running time of Knapsack is poly(n, log m, P/∆) • P is second largest number of max bid vi (m) • set ∆ be proportion to P ⇒ run in poly-time
  46. 46. Introduction Tools The Mechanism Summary Poly-time Knapsack • Set prot of object p(i,j) = vi (j) instead of vi (j) • Claim: Expected running time of Knapsack is poly(n, log m, P/∆) • P is second largest number of max bid vi (m) • set ∆ be proportion to P ⇒ run in poly-time
  47. 47. Introduction Tools The Mechanism Summary Poly-time Knapsack • Set prot of object p(i,j) = vi (j) instead of vi (j) • Claim: Expected running time of Knapsack is poly(n, log m, P/∆) • P is second largest number of max bid vi (m) • set ∆ be proportion to P ⇒ run in poly-time
  48. 48. Introduction Tools The Mechanism Summary Informal Explanation of Achieving Poly-time Knapsack j vi (j) = v(j) + (2q(j) + xi )∆ • term q(·) : can focus on only poly(n, log m, P/∆) objects • Nice proof but have to skip. • term x : can run Knapsack in poly(#object) time in expectation • How ?
  49. 49. Introduction Tools The Mechanism Summary Informal Explanation of Achieving Poly-time Knapsack j vi (j) = v(j) + (2q(j) + xi )∆ • term q(·) : can focus on only poly(n, log m, P/∆) objects • Nice proof but have to skip. • term x : can run Knapsack in poly(#object) time in expectation • How ?
  50. 50. Introduction Tools The Mechanism Summary Informal Explanation of Achieving Poly-time Knapsack j vi (j) = v(j) + (2q(j) + xi )∆ • term q(·) : can focus on only poly(n, log m, P/∆) objects • Nice proof but have to skip. • term x : can run Knapsack in poly(#object) time in expectation • How ?
  51. 51. Introduction Tools The Mechanism Summary Use Randomness to Bound Running Time (Smoothed Analysis) • Hard instances of Knapsack are isolated • Random variable averages running time of hard instances with easy instances (gures from: Smoothed Analysis Homepage)
  52. 52. Introduction Tools The Mechanism Summary Use Randomness to Bound Running Time (Smoothed Analysis) • Hard instances of Knapsack are isolated • Random variable averages running time of hard instances with easy instances (gures from: Smoothed Analysis Homepage)
  53. 53. Introduction Tools The Mechanism Summary Use Randomness to Bound Running Time (Smoothed Analysis) • Hard instances of Knapsack are isolated • Random variable averages running time of hard instances with easy instances (gures from: Smoothed Analysis Homepage)
  54. 54. Introduction Tools The Mechanism Summary ∆-Perturbed Maximizer • ∆-perturbed maximizer = solving Knapsack with perturbed valuation function • Additive error • Reject low bidding
  55. 55. Introduction Tools The Mechanism Summary ∆-Perturbed Maximizer • ∆-perturbed maximizer = solving Knapsack with perturbed valuation function • Additive error • Reject low bidding
  56. 56. Introduction Tools The Mechanism Summary Additive Error • ∆-perturbed maximizer maximizes ∑n vi (s) i=0 • n Social welfare is less than ∑i=0 vi (s) n n 0 ≤ ∑ vi (s) − ∑ vi (s) ≤ (2 log m + 3)∆n i=0 i=0 j • vi (j) = vi (j) + (2q(j) + xi )∆ • q(j) ≤ log m + 1 j • xi ≤ 1 • vi (j) − vi (j) ≤ (2 log m + 2 + 1)∆
  57. 57. Introduction Tools The Mechanism Summary Additive Error • ∆-perturbed maximizer maximizes ∑n vi (s) i=0 • n Social welfare is less than ∑i=0 vi (s) n n 0 ≤ ∑ vi (s) − ∑ vi (s) ≤ (2 log m + 3)∆n i=0 i=0 j • vi (j) = vi (j) + (2q(j) + xi )∆ • q(j) ≤ log m + 1 j • xi ≤ 1 • vi (j) − vi (j) ≤ (2 log m + 2 + 1)∆
  58. 58. Introduction Tools The Mechanism Summary Additive Error • ∆-perturbed maximizer maximizes ∑n vi (s) i=0 • n Social welfare is less than ∑i=0 vi (s) n n 0 ≤ ∑ vi (s) − ∑ vi (s) ≤ (2 log m + 3)∆n i=0 i=0 j • vi (j) = vi (j) + (2q(j) + xi )∆ • q(j) ≤ log m + 1 j • xi ≤ 1 • vi (j) − vi (j) ≤ (2 log m + 2 + 1)∆
  59. 59. Introduction Tools The Mechanism Summary Additive Error • ∆-perturbed maximizer maximizes ∑n vi (s) i=0 • n Social welfare is less than ∑i=0 vi (s) n n 0 ≤ ∑ vi (s) − ∑ vi (s) ≤ (2 log m + 3)∆n i=0 i=0 j • vi (j) = vi (j) + (2q(j) + xi )∆ • q(j) ≤ log m + 1 j • xi ≤ 1 • vi (j) − vi (j) ≤ (2 log m + 2 + 1)∆
  60. 60. Introduction Tools The Mechanism Summary Additive Error • ∆-perturbed maximizer maximizes ∑n vi (s) i=0 • n Social welfare is less than ∑i=0 vi (s) n n 0 ≤ ∑ vi (s) − ∑ vi (s) ≤ (2 log m + 3)∆n i=0 i=0 j • vi (j) = vi (j) + (2q(j) + xi )∆ • q(j) ≤ log m + 1 j • xi ≤ 1 • vi (j) − vi (j) ≤ (2 log m + 2 + 1)∆
  61. 61. Introduction Tools The Mechanism Summary Additive Error • ∆-perturbed maximizer maximizes ∑n vi (s) i=0 • n Social welfare is less than ∑i=0 vi (s) n n 0 ≤ ∑ vi (s) − ∑ vi (s) ≤ (2 log m + 3)∆n i=0 i=0 j • vi (j) = vi (j) + (2q(j) + xi )∆ • q(j) ≤ log m + 1 j • xi ≤ 1 • vi (j) − vi (j) ≤ (2 log m + 2 + 1)∆
  62. 62. Introduction Tools The Mechanism Summary Reject Low Bidding • Max bid of bidder i :vi (m) • Claim: if vi (m) ∆ → bidder i get nothing si = 0 j vi (j) = v(j) + (2q(j) + xi )∆ • vi (j) vi (m) ∆ • q(j) + 1 ≤ q(0) and q is multiplied by ∆ • vi is maximized when j = 0
  63. 63. Introduction Tools The Mechanism Summary Reject Low Bidding • Max bid of bidder i :vi (m) • Claim: if vi (m) ∆ → bidder i get nothing si = 0 j vi (j) = v(j) + (2q(j) + xi )∆ • vi (j) vi (m) ∆ • q(j) + 1 ≤ q(0) and q is multiplied by ∆ • vi is maximized when j = 0
  64. 64. Introduction Tools The Mechanism Summary Reject Low Bidding • Max bid of bidder i :vi (m) • Claim: if vi (m) ∆ → bidder i get nothing si = 0 j vi (j) = v(j) + (2q(j) + xi )∆ • vi (j) vi (m) ∆ • q(j) + 1 ≤ q(0) and q is multiplied by ∆ • vi is maximized when j = 0
  65. 65. Introduction Tools The Mechanism Summary Reject Low Bidding • Max bid of bidder i :vi (m) • Claim: if vi (m) ∆ → bidder i get nothing si = 0 j vi (j) = v(j) + (2q(j) + xi )∆ • vi (j) vi (m) ∆ • q(j) + 1 ≤ q(0) and q is multiplied by ∆ • vi is maximized when j = 0
  66. 66. Introduction Tools The Mechanism Summary Reject Low Bidding • Max bid of bidder i :vi (m) • Claim: if vi (m) ∆ → bidder i get nothing si = 0 j vi (j) = v(j) + (2q(j) + xi )∆ • vi (j) vi (m) ∆ • q(j) + 1 ≤ q(0) and q is multiplied by ∆ • vi is maximized when j = 0
  67. 67. Introduction Tools The Mechanism Summary Reject Low Bidding • Max bid of bidder i :vi (m) • Claim: if vi (m) ∆ → bidder i get nothing si = 0 j vi (j) = v(j) + (2q(j) + xi )∆ • vi (j) vi (m) ∆ • q(j) + 1 ≤ q(0) and q is multiplied by ∆ • vi is maximized when j = 0
  68. 68. Introduction Tools The Mechanism Summary Summary of ∆-Perturbed Maximizer • Maximizes social welfare • in poly-time • (2 log m + 3)∆n additive error • reject low bid
  69. 69. Introduction Tools The Mechanism Summary Summary of ∆-Perturbed Maximizer • Maximizes social welfare • in poly-time • (2 log m + 3)∆n additive error • reject low bid
  70. 70. Introduction Tools The Mechanism Summary Summary of ∆-Perturbed Maximizer • Maximizes social welfare • in poly-time • (2 log m + 3)∆n additive error • reject low bid
  71. 71. Introduction Tools The Mechanism Summary Outline Introduction Problem 3 Kinds of Truthfulness Related Work Tools ∆-Perturbed Maximizer Consensus Function with Drop-outs The Mechanism Social Choice Function Feasibility Approximation Ratio Universal Truthfulness Summary
  72. 72. Introduction Tools The Mechanism Summary Want Function l Such That 1. Drop out : fail to compute with low prob Pr[l(a) = ⊥] = ε 2. Bound : if l(a) = ⊥ l(a) ≤ a 3. Separation : if l(a1 ) l(a2 ) = ⊥ a1 l(a2 ) − 1
  73. 73. Introduction Tools The Mechanism Summary Want Function l Such That 1. Drop out : fail to compute with low prob Pr[l(a) = ⊥] = ε 2. Bound : if l(a) = ⊥ l(a) ≤ a 3. Separation : if l(a1 ) l(a2 ) = ⊥ a1 l(a2 ) − 1
  74. 74. Introduction Tools The Mechanism Summary Want Function l Such That 1. Drop out : fail to compute with low prob Pr[l(a) = ⊥] = ε 2. Bound : if l(a) = ⊥ l(a) ≤ a 3. Separation : if l(a1 ) l(a2 ) = ⊥ a1 l(a2 ) − 1
  75. 75. Introduction Tools The Mechanism Summary 2.Bound • τ ∼ [0, 1] 1 • xτ (i) = (i + τ ) ε ; i∈Z = () ( + 1) . =⊥ 1 1/ • let k be largest integer s.t. xτ (k) ≤ a xτ (k) ; d 1 • lτ (a) = ⊥ ;d≤1 • So lτ (a) ≤a
  76. 76. Introduction Tools The Mechanism Summary 2.Bound • τ ∼ [0, 1] 1 • xτ (i) = (i + τ ) ε ; i∈Z = () ( + 1) . =⊥ 1 1/ • let k be largest integer s.t. xτ (k) ≤ a xτ (k) ; d 1 • lτ (a) = ⊥ ;d≤1 • So lτ (a) ≤a
  77. 77. Introduction Tools The Mechanism Summary 2.Bound • τ ∼ [0, 1] 1 • xτ (i) = (i + τ ) ε ; i∈Z = () ( + 1) . =⊥ 1 1/ • let k be largest integer s.t. xτ (k) ≤ a xτ (k) ; d 1 • lτ (a) = ⊥ ;d≤1 • So lτ (a) ≤a
  78. 78. Introduction Tools The Mechanism Summary 2.Bound • τ ∼ [0, 1] 1 • xτ (i) = (i + τ ) ε ; i∈Z = () ( + 1) . =⊥ 1 1/ • let k be largest integer s.t. xτ (k) ≤ a xτ (k) ; d 1 • lτ (a) = ⊥ ;d≤1 • So lτ (a) ≤a
  79. 79. Introduction Tools The Mechanism Summary 3.Separation (2 ) 1 1 2 1 1/ lτ (a1 ) lτ (a2 ) ⇒ a1 lτ (a2 ) − 1
  80. 80. Introduction Tools The Mechanism Summary 1.Drop out • τ ∼ [0, 1] 1 • xτ (i) = (i + τ ) ε ; i∈Z 1 • Can view xτ (k + 1) as random number picked ∼ (a, a + ε ] • Drop out if it is picked ∼ (a, a + 1] () ( + 1) 1 1/ • 1 Pr[lτ (a) = ⊥] = =ε 1/ε
  81. 81. Introduction Tools The Mechanism Summary 1.Drop out • τ ∼ [0, 1] 1 • xτ (i) = (i + τ ) ε ; i∈Z 1 • Can view xτ (k + 1) as random number picked ∼ (a, a + ε ] • Drop out if it is picked ∼ (a, a + 1] () ( + 1) 1 1/ • 1 Pr[lτ (a) = ⊥] = =ε 1/ε
  82. 82. Introduction Tools The Mechanism Summary 1.Drop out • τ ∼ [0, 1] 1 • xτ (i) = (i + τ ) ε ; i∈Z 1 • Can view xτ (k + 1) as random number picked ∼ (a, a + ε ] • Drop out if it is picked ∼ (a, a + 1] () ( + 1) 1 1/ • 1 Pr[lτ (a) = ⊥] = =ε 1/ε
  83. 83. Introduction Tools The Mechanism Summary 1.Drop out • τ ∼ [0, 1] 1 • xτ (i) = (i + τ ) ε ; i∈Z 1 • Can view xτ (k + 1) as random number picked ∼ (a, a + ε ] • Drop out if it is picked ∼ (a, a + 1] () ( + 1) 1 1/ • 1 Pr[lτ (a) = ⊥] = =ε 1/ε
  84. 84. Introduction Tools The Mechanism Summary Summary 1. Pr[lτ (a) = ⊥] = ε 2. lτ (a) ≤ a 3. lτ (a1 ) lτ (a2 ) ⇒ a1 lτ (a2 ) − 1
  85. 85. Introduction Tools The Mechanism Summary Summary 1. Pr[lτ (a) = ⊥] = ε 2. lτ (a) ≤ a 3. lτ (a1 ) lτ (a2 ) ⇒ a1 lτ (a2 ) − 1
  86. 86. Introduction Tools The Mechanism Summary Summary 1. Pr[lτ (a) = ⊥] = ε 2. lτ (a) ≤ a 3. lτ (a1 ) lτ (a2 ) ⇒ a1 lτ (a2 ) − 1
  87. 87. Introduction Tools The Mechanism Summary Multiplicative Consensus Function • Mechanism uses multiplicative variant. • Lτ (a) = lτ (logN a) ; N is some constant 1. Pr[Lτ (a) = ⊥] = ε 2. Lτ (a) ≤ a 3. Lτ (a1 ) Lτ (a2 ) ⇒ a1 Lτ (a2 )/N
  88. 88. Introduction Tools The Mechanism Summary Multiplicative Consensus Function • Mechanism uses multiplicative variant. • Lτ (a) = lτ (logN a) ; N is some constant 1. Pr[Lτ (a) = ⊥] = ε 2. Lτ (a) ≤ a 3. Lτ (a1 ) Lτ (a2 ) ⇒ a1 Lτ (a2 )/N
  89. 89. Introduction Tools The Mechanism Summary Multiplicative Consensus Function • Mechanism uses multiplicative variant. • Lτ (a) = lτ (logN a) ; N is some constant 1. Pr[Lτ (a) = ⊥] = ε 2. Lτ (a) ≤ a 3. Lτ (a1 ) Lτ (a2 ) ⇒ a1 Lτ (a2 )/N
  90. 90. Introduction Tools The Mechanism Summary Multiplicative Consensus Function • Mechanism uses multiplicative variant. • Lτ (a) = lτ (logN a) ; N is some constant 1. Pr[Lτ (a) = ⊥] = ε 2. Lτ (a) ≤ a 3. Lτ (a1 ) Lτ (a2 ) ⇒ a1 Lτ (a2 )/N
  91. 91. Introduction Tools The Mechanism Summary Multiplicative Consensus Function • Mechanism uses multiplicative variant. • Lτ (a) = lτ (logN a) ; N is some constant 1. Pr[Lτ (a) = ⊥] = ε 2. Lτ (a) ≤ a 3. Lτ (a1 ) Lτ (a2 ) ⇒ a1 Lτ (a2 )/N
  92. 92. Introduction Tools The Mechanism Summary Outline Introduction Problem 3 Kinds of Truthfulness Related Work Tools ∆-Perturbed Maximizer Consensus Function with Drop-outs The Mechanism Social Choice Function Feasibility Approximation Ratio Universal Truthfulness Summary
  93. 93. Introduction Tools The Mechanism Summary Combination of Allocation • τ ∼ [0, 1] and then x • N = 2(log m + 3)n/ε • for each bidder i • Li = Lτ (v (−i) ) max • v (−i) : maxj=i vj (m) largest max bid excluded is max • if Li = ⊥ → si = 0 • if Li = ⊥ • call ∆i -perturbed maximizer ; ∆i = Li /N • get allocation s(i) • si = s(i) i • Summary : bidder i get ith element of allocation from ∆i -perturbed maximizer
  94. 94. Introduction Tools The Mechanism Summary Combination of Allocation • τ ∼ [0, 1] and then x • N = 2(log m + 3)n/ε • for each bidder i • Li = Lτ (v (−i) ) max • v (−i) : maxj=i vj (m) largest max bid excluded is max • if Li = ⊥ → si = 0 • if Li = ⊥ • call ∆i -perturbed maximizer ; ∆i = Li /N • get allocation s(i) • si = s(i) i • Summary : bidder i get ith element of allocation from ∆i -perturbed maximizer
  95. 95. Introduction Tools The Mechanism Summary Combination of Allocation • τ ∼ [0, 1] and then x • N = 2(log m + 3)n/ε • for each bidder i • Li = Lτ (v (−i) ) max • v (−i) : maxj=i vj (m) largest max bid excluded is max • if Li = ⊥ → si = 0 • if Li = ⊥ • call ∆i -perturbed maximizer ; ∆i = Li /N • get allocation s(i) • si = s(i) i • Summary : bidder i get ith element of allocation from ∆i -perturbed maximizer
  96. 96. Introduction Tools The Mechanism Summary Combination of Allocation • τ ∼ [0, 1] and then x • N = 2(log m + 3)n/ε • for each bidder i • Li = Lτ (v (−i) ) max • v (−i) : maxj=i vj (m) largest max bid excluded is max • if Li = ⊥ → si = 0 • if Li = ⊥ • call ∆i -perturbed maximizer ; ∆i = Li /N • get allocation s(i) • si = s(i) i • Summary : bidder i get ith element of allocation from ∆i -perturbed maximizer
  97. 97. Introduction Tools The Mechanism Summary Combination of Allocation • τ ∼ [0, 1] and then x • N = 2(log m + 3)n/ε • for each bidder i • Li = Lτ (v (−i) ) max • v (−i) : maxj=i vj (m) largest max bid excluded is max • if Li = ⊥ → si = 0 • if Li = ⊥ • call ∆i -perturbed maximizer ; ∆i = Li /N • get allocation s(i) • si = s(i) i • Summary : bidder i get ith element of allocation from ∆i -perturbed maximizer
  98. 98. Introduction Tools The Mechanism Summary Combination of Allocation • τ ∼ [0, 1] and then x • N = 2(log m + 3)n/ε • for each bidder i • Li = Lτ (v (−i) ) max • v (−i) : maxj=i vj (m) largest max bid excluded is max • if Li = ⊥ → si = 0 • if Li = ⊥ • call ∆i -perturbed maximizer ; ∆i = Li /N • get allocation s(i) • si = s(i) i • Summary : bidder i get ith element of allocation from ∆i -perturbed maximizer
  99. 99. Introduction Tools The Mechanism Summary Combination of Allocation • τ ∼ [0, 1] and then x • N = 2(log m + 3)n/ε • for each bidder i • Li = Lτ (v (−i) ) max • v (−i) : maxj=i vj (m) largest max bid excluded is max • if Li = ⊥ → si = 0 • if Li = ⊥ • call ∆i -perturbed maximizer ; ∆i = Li /N • get allocation s(i) • si = s(i) i • Summary : bidder i get ith element of allocation from ∆i -perturbed maximizer
  100. 100. Introduction Tools The Mechanism Summary Only 2 Dierent Allocations • i∗ is bidder who bid the highest vi (m) • ∀i = i∗ , v (−i) = vmax (recall: v(−i) largest max bid excluded is) max max • same Li = Lτ (vmax ) ⇒ same ∆i ⇒ same allocation s(i) ∗) • Only s(i might be dierent − ∗ −         −     ∗     −   (i)   −  (i∗ )  , s  ∗   −  s   , s   −     ∗     −    −   ∗  (i∗ th)  ∗  − ∗ − ∗) • s(i) , s(i are feasible allocations (not allocate more than m) • results from Knapsack
  101. 101. Introduction Tools The Mechanism Summary Only 2 Dierent Allocations • i∗ is bidder who bid the highest vi (m) • ∀i = i∗ , v (−i) = vmax (recall: v(−i) largest max bid excluded is) max max • same Li = Lτ (vmax ) ⇒ same ∆i ⇒ same allocation s(i) ∗) • Only s(i might be dierent − ∗ −         −     ∗     −   (i)   −  (i∗ )  , s  ∗   −  s   , s   −     ∗     −    −   ∗  (i∗ th)  ∗  − ∗ − ∗) • s(i) , s(i are feasible allocations (not allocate more than m) • results from Knapsack
  102. 102. Introduction Tools The Mechanism Summary Only 2 Dierent Allocations • i∗ is bidder who bid the highest vi (m) • ∀i = i∗ , v (−i) = vmax (recall: v(−i) largest max bid excluded is) max max • same Li = Lτ (vmax ) ⇒ same ∆i ⇒ same allocation s(i) ∗) • Only s(i might be dierent − ∗ −         −     ∗     −   (i)   −  (i∗ )  , s  ∗   −  s   , s   −     ∗     −    −   ∗  (i∗ th)  ∗  − ∗ − ∗) • s(i) , s(i are feasible allocations (not allocate more than m) • results from Knapsack
  103. 103. Introduction Tools The Mechanism Summary Only 2 Dierent Allocations • i∗ is bidder who bid the highest vi (m) • ∀i = i∗ , v (−i) = vmax (recall: v(−i) largest max bid excluded is) max max • same Li = Lτ (vmax ) ⇒ same ∆i ⇒ same allocation s(i) ∗) • Only s(i might be dierent − ∗ −         −     ∗     −   (i)   −  (i∗ )  , s  ∗   −  s   , s   −     ∗     −    −   ∗  (i∗ th)  ∗  − ∗ − ∗) • s(i) , s(i are feasible allocations (not allocate more than m) • results from Knapsack
  104. 104. Introduction Tools The Mechanism Summary Only 2 Dierent Allocations • i∗ is bidder who bid the highest vi (m) • ∀i = i∗ , v (−i) = vmax (recall: v(−i) largest max bid excluded is) max max • same Li = Lτ (vmax ) ⇒ same ∆i ⇒ same allocation s(i) ∗) • Only s(i might be dierent − ∗ −         −     ∗     −   (i)   −  (i∗ )  , s  ∗   −  s   , s   −     ∗     −    −   ∗  (i∗ th)  ∗  − ∗ − ∗) • s(i) , s(i are feasible allocations (not allocate more than m) • results from Knapsack
  105. 105. Introduction Tools The Mechanism Summary Outline Introduction Problem 3 Kinds of Truthfulness Related Work Tools ∆-Perturbed Maximizer Consensus Function with Drop-outs The Mechanism Social Choice Function Feasibility Approximation Ratio Universal Truthfulness Summary
  106. 106. Introduction Tools The Mechanism Summary Feasibility ∑n si ≤ m i=0 1. Li = Li∗ = ⊥ • s(i) = s(i∗) = s • so s is feasible. 2. Li = ⊥or Li∗ = ⊥ • s is [feasible allocation+empty allocation] • #items is not increased ⇒ feasible
  107. 107. Introduction Tools The Mechanism Summary Feasibility ∑n si ≤ m i=0 1. Li = Li∗ = ⊥ • s(i) = s(i∗) = s • so s is feasible. 2. Li = ⊥or Li∗ = ⊥ • s is [feasible allocation+empty allocation] • #items is not increased ⇒ feasible
  108. 108. Introduction Tools The Mechanism Summary Feasibility ∑n si ≤ m i=0 1. Li = Li∗ = ⊥ • s(i) = s(i∗) = s • so s is feasible. 2. Li = ⊥or Li∗ = ⊥ • s is [feasible allocation+empty allocation] • #items is not increased ⇒ feasible
  109. 109. Introduction Tools The Mechanism Summary Feasibility ∑n si ≤ m i=0 1. Li = Li∗ = ⊥ • s(i) = s(i∗) = s • so s is feasible. 2. Li = ⊥or Li∗ = ⊥ • s is [feasible allocation+empty allocation] • #items is not increased ⇒ feasible
  110. 110. Introduction Tools The Mechanism Summary Feasibility ∑n s i ≤ m i=0 (2) 3 Li = Li∗ = ⊥ • Li∗ = Lτ (v(−i∗ ) ) Lτ (vmax ) = Li max • v(−i∗ ) Li /N = ∆i max • because if Lτ (a1 ) Lτ (a2 ), then a1 Lτ (a2 )/N • Max bid of all bidder i except i∗ are ∆i ⇒ get nothing (will use again) • ∑i si = si∗ ≤ m
  111. 111. Introduction Tools The Mechanism Summary Feasibility ∑n s i ≤ m i=0 (2) 3 Li = Li∗ = ⊥ • Li∗ = Lτ (v(−i∗ ) ) Lτ (vmax ) = Li max • v(−i∗ ) Li /N = ∆i max • because if Lτ (a1 ) Lτ (a2 ), then a1 Lτ (a2 )/N • Max bid of all bidder i except i∗ are ∆i ⇒ get nothing (will use again) • ∑i si = si∗ ≤ m
  112. 112. Introduction Tools The Mechanism Summary Feasibility ∑n s i ≤ m i=0 (2) 3 Li = Li∗ = ⊥ • Li∗ = Lτ (v(−i∗ ) ) Lτ (vmax ) = Li max • v(−i∗ ) Li /N = ∆i max • because if Lτ (a1 ) Lτ (a2 ), then a1 Lτ (a2 )/N • Max bid of all bidder i except i∗ are ∆i ⇒ get nothing (will use again) • ∑i si = si∗ ≤ m
  113. 113. Introduction Tools The Mechanism Summary Feasibility ∑n s i ≤ m i=0 (2) 3 Li = Li∗ = ⊥ • Li∗ = Lτ (v(−i∗ ) ) Lτ (vmax ) = Li max • v(−i∗ ) Li /N = ∆i max • because if Lτ (a1 ) Lτ (a2 ), then a1 Lτ (a2 )/N • Max bid of all bidder i except i∗ are ∆i ⇒ get nothing (will use again) • ∑i si = si∗ ≤ m
  114. 114. Introduction Tools The Mechanism Summary Feasibility ∑n s i ≤ m i=0 (2) 3 Li = Li∗ = ⊥ • Li∗ = Lτ (v(−i∗ ) ) Lτ (vmax ) = Li max • v(−i∗ ) Li /N = ∆i max • because if Lτ (a1 ) Lτ (a2 ), then a1 Lτ (a2 )/N • Max bid of all bidder i except i∗ are ∆i ⇒ get nothing (will use again) • ∑i si = si∗ ≤ m
  115. 115. Introduction Tools The Mechanism Summary Outline Introduction Problem 3 Kinds of Truthfulness Related Work Tools ∆-Perturbed Maximizer Consensus Function with Drop-outs The Mechanism Social Choice Function Feasibility Approximation Ratio Universal Truthfulness Summary
  116. 116. Introduction Tools The Mechanism Summary (1 − 4ε )−Approximation Ratio • Prob no one drop out ≥ 1 − 2ε • Prob bidder i∗ drop out = ε • Prob bidder i = i∗ drop out = ε • They are completely correlated. • Will prove that if none drop out ⇒ approx ratio ≥ (1 − 2ε ) • Overall approx ratio ≥ (1 − 2ε )2 ≥ (1 − 4ε )
  117. 117. Introduction Tools The Mechanism Summary (1 − 4ε )−Approximation Ratio • Prob no one drop out ≥ 1 − 2ε • Prob bidder i∗ drop out = ε • Prob bidder i = i∗ drop out = ε • They are completely correlated. • Will prove that if none drop out ⇒ approx ratio ≥ (1 − 2ε ) • Overall approx ratio ≥ (1 − 2ε )2 ≥ (1 − 4ε )
  118. 118. Introduction Tools The Mechanism Summary (1 − 4ε )−Approximation Ratio • Prob no one drop out ≥ 1 − 2ε • Prob bidder i∗ drop out = ε • Prob bidder i = i∗ drop out = ε • They are completely correlated. • Will prove that if none drop out ⇒ approx ratio ≥ (1 − 2ε ) • Overall approx ratio ≥ (1 − 2ε )2 ≥ (1 − 4ε )
  119. 119. Introduction Tools The Mechanism Summary (1 − 4ε )−Approximation Ratio • Prob no one drop out ≥ 1 − 2ε • Prob bidder i∗ drop out = ε • Prob bidder i = i∗ drop out = ε • They are completely correlated. • Will prove that if none drop out ⇒ approx ratio ≥ (1 − 2ε ) • Overall approx ratio ≥ (1 − 2ε )2 ≥ (1 − 4ε )
  120. 120. Introduction Tools The Mechanism Summary ∗) Analyze Allocation s(i • Additive error Li∗ ≤ 2(log m + 3)n∆i∗ = 2(log m + 3)n N = ε Li∗ 2(log m+3)n/ε • Li∗ = Lτ (v(−i∗ ) ) ≤ v(−i∗ ) ≤ vmax ≤ opt max max ∗ • v(s(i ) ) ≥ opt − ε Li∗ ≥ (1 − ε )opt
  121. 121. Introduction Tools The Mechanism Summary ∗) Analyze Allocation s(i • Additive error Li∗ ≤ 2(log m + 3)n∆i∗ = 2(log m + 3)n N = ε Li∗ 2(log m+3)n/ε • Li∗ = Lτ (v(−i∗ ) ) ≤ v(−i∗ ) ≤ vmax ≤ opt max max ∗ • v(s(i ) ) ≥ opt − ε Li∗ ≥ (1 − ε )opt
  122. 122. Introduction Tools The Mechanism Summary ∗) Analyze Allocation s(i • Additive error Li∗ ≤ 2(log m + 3)n∆i∗ = 2(log m + 3)n N = ε Li∗ 2(log m+3)n/ε • Li∗ = Lτ (v(−i∗ ) ) ≤ v(−i∗ ) ≤ vmax ≤ opt max max ∗ • v(s(i ) ) ≥ opt − ε Li∗ ≥ (1 − ε )opt
  123. 123. Introduction Tools The Mechanism Summary 2 Cases When None Drop out 1. Li = Li∗ ∗ • Mechanism yields s = s(i ) • Social welfare ≥ (1 − ε )opt 2. Li = Li∗ • Max bid of all bidder i except i∗ are ∆i ∗ • si∗ = s(i i∗ ) and si = 0 for all i = i∗ (i∗ ) • Social welfare is vi∗ (si∗ ) (i∗ ) ∗ ∗ vi∗ (si∗ ) = v(s(i ) ) − ∑ vi (s(i ) ) ≥ (1 − ε )opt − ε opt i=i∗ = (1 − 2ε )opt
  124. 124. Introduction Tools The Mechanism Summary 2 Cases When None Drop out 1. Li = Li∗ ∗ • Mechanism yields s = s(i ) • Social welfare ≥ (1 − ε )opt 2. Li = Li∗ • Max bid of all bidder i except i∗ are ∆i ∗ • si∗ = s(i i∗ ) and si = 0 for all i = i∗ (i∗ ) • Social welfare is vi∗ (si∗ ) (i∗ ) ∗ ∗ vi∗ (si∗ ) = v(s(i ) ) − ∑ vi (s(i ) ) ≥ (1 − ε )opt − ε opt i=i∗ = (1 − 2ε )opt
  125. 125. Introduction Tools The Mechanism Summary 2 Cases When None Drop out 1. Li = Li∗ ∗ • Mechanism yields s = s(i ) • Social welfare ≥ (1 − ε )opt 2. Li = Li∗ • Max bid of all bidder i except i∗ are ∆i ∗ • si∗ = s(i i∗ ) and si = 0 for all i = i∗ (i∗ ) • Social welfare is vi∗ (si∗ ) (i∗ ) ∗ ∗ vi∗ (si∗ ) = v(s(i ) ) − ∑ vi (s(i ) ) ≥ (1 − ε )opt − ε opt i=i∗ = (1 − 2ε )opt
  126. 126. Introduction Tools The Mechanism Summary 2 Cases When None Drop out 1. Li = Li∗ ∗ • Mechanism yields s = s(i ) • Social welfare ≥ (1 − ε )opt 2. Li = Li∗ • Max bid of all bidder i except i∗ are ∆i ∗ • si∗ = s(i i∗ ) and si = 0 for all i = i∗ (i∗ ) • Social welfare is vi∗ (si∗ ) (i∗ ) ∗ ∗ vi∗ (si∗ ) = v(s(i ) ) − ∑ vi (s(i ) ) ≥ (1 − ε )opt − ε opt i=i∗ = (1 − 2ε )opt
  127. 127. Introduction Tools The Mechanism Summary Outline Introduction Problem 3 Kinds of Truthfulness Related Work Tools ∆-Perturbed Maximizer Consensus Function with Drop-outs The Mechanism Social Choice Function Feasibility Approximation Ratio Universal Truthfulness Summary
  128. 128. Introduction Tools The Mechanism Summary Mechanism is Truthful i (Direct Characterization) 1. Payment pi : not depend on vi (can depend on s = f (vi , v−i ) and v−i ) (i) pi = qs (v−i ) 2. Social choice function maximizes each bidders utility : for all i (i) f (v) = argmax(vi (s) − qs (v−i )) • Informal explanation • is utility is vi (s) − qs , if he lies vi he may get vi (s ) − qs • f already chooses s which maximizes vi (s) − qs • ⇒ tell the true vi to f for optimizing his utility
  129. 129. Introduction Tools The Mechanism Summary Mechanism is Truthful i (Direct Characterization) 1. Payment pi : not depend on vi (can depend on s = f (vi , v−i ) and v−i ) (i) pi = qs (v−i ) 2. Social choice function maximizes each bidders utility : for all i (i) f (v) = argmax(vi (s) − qs (v−i )) • Informal explanation • is utility is vi (s) − qs , if he lies vi he may get vi (s ) − qs • f already chooses s which maximizes vi (s) − qs • ⇒ tell the true vi to f for optimizing his utility

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