20081116 auctions nikolenko_lecture09

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<ul><li> 1. : (2) FWorst-case , 2008 Worst-case </li></ul> <p> 2. : (2) F F(2)Outline 1 : F (2)2 F (2) DOP RSOP Worst-case 3. : (2) F F(2) N , . xi . N , . , . Worst-case 4. : (2) F F(2) , donations. :) . Worst-case 5. : (2) F F(2) , : N Revenue = T (x ) =i. xi =1 : . - , . ? Worst-case 6. : (2) F F(2) , , :Revenue = F(x ) = max p {- xi p}.p : x(i ) i - (, x(1) = maxi xi ). F(x ) = max ix(i ) .i Worst-case 7. : (2) F (2) FT (x ) F(x ) , , F(x ) T (x ). T (x ) , . ? Worst-case 8. : (2) F F(2)T (x ) F(x ) N 1T (x ) HN F(x ), HN = . ii =1 ( ). Worst-case 9. : (2) F (2) FT (x ) F(x ) . F(x ) = maxi ix(i ) . , NNT (x ) =xi=x( )i = i =1 i =1 NNix i F(x ) == HN F(x ). ii i =1i =1 Worst-case 10. : (2) F F(2)T (x ) F(x ) , , . xi = 1 . i , ix(i ) = 1 = F(x ) i . , T (x ) = HN . Worst-case 11. : (2) F F(2) . , , . , , x . , , ( ). Worst-case 12. : (2) F F(2) , .. x , . Hint: . Worst-case 13. : (2) F F(2) ? Worst-case 14. : (2) F F(2) ? , : - . , x - , y - z ... :) Worst-case 15. : (2) F F(2) (prot benchmark) G : RN R, x = (x1 , . . . , xN ) . , . T (x ) F(x ). Worst-case 16. : (2) F F(2) A G = maxx A(x ) . G( x) , A G . A G, G . Worst-case 17. : (2) F F(2) x , xi [1, h], N , o (log h ) T (x ). . . Worst-case 18. : (2) F (2) F . N = 2 T (x ) F(x ) : F(x ) T (x )/2. N =2 x = (x1 , x2 ), xi [1, h], , o (log h ) F(x ). Worst-case 19. : (2) F F(2) , xi . . , F(x ) . ? , ? , , . Worst-case 20. : (2) F F(2)F (2) F (2) :F (2) (x ) = max ix(i ) . i 2 , - F (2) A F (2) (x ) T (x )T (x ) F(x ) F (2) (x ) A(x ) . log n Worst-case 21. DOP : (2) F RSOP Outline 1 : F (2)2 F (2) DOP RSOP Worst-case 22. DOP : (2) F RSOP I , - F (2) . , x . Worst-case 23. DOP : (2) F RSOP I , 50 $10 50 $1. R1 R10 $1 $10 . R1 R10 ? Worst-case 24. DOP : (2) F RSOP I , 50 $10 50 $1. R1 R10 $1 $10 . R10 = 500, R1 = 100 ( $1). , F (2) (x ) = R10 = $500. Worst-case 25. DOP : (2) F RSOP I , 5 $10 95 $1. R1 R10 $1 $10 . R1 R10 ? Worst-case 26. DOP : (2) F RSOP I , 5 $10 95 $1. R1 R10 $1 $10 . R10 = 50, R1 = 100. , F (2) (x ) = R1 = $100. Worst-case 27. DOP : (2) F RSOP , x , . x opt(x ) = argmaxp {p - vi p }. Worst-case 28. DOP : (2) F RSOP , , i , , .. ti (b i ). : ti (b i ) = opt(b i ). (Deterministic Optimal Price auction, DOP). Worst-case 29. DOP : (2) F RSOP DOP-, DOP . ? - F (2) ? Worst-case 30. DOP : (2) F RSOP II , 10 $10 90 $1. DOP? Worst-case 31. DOP : (2) F RSOP II , 10 $10 90 $1. b1 89 $1 10 $10, opt(b 1 ) = $10. b10 90 $1 9 $10, opt(b 10 ) = $1. Worst-case 32. DOP : (2) F RSOP II , 10 $10 90 $1. , DOP $10, $1, $1 . $10 F (2) = 100. , . Worst-case 33. DOP : (2) F RSOP DOP , DOP . ? ... Worst-case 34. DOP : (2) F RSOP F (2) . Worst-case 35. DOP : (2) F RSOP , F (2) . b bi {1, h } ( ). nh (b ) b) h 1 n1 ( . Worst-case 36. DOP : (2) F RSOP nh (b ) b) h 1 n1 ( . , A ? , ti (b i ) i , nh (b i ) n1 (b i ). , t (nh , n1 ); , nh (b i ) n1 (b i )., , , t (nh , n1 ) {1, h}, ( bi {1, h}) . Worst-case 37. DOP : (2) F RSOP , . t (nh , n1 )? -, m t (m, 0) = h, h n, F (2) = hn. h-, ! Worst-case 38. DOP : (2) F RSOP , . t (nh , n1 )? , m t (0, m) = 1, 1 . F (2) = n, . :) Worst-case 39. DOP : (2) F RSOP m t (k , m k ).k = 0 1, k = m h. , - k = min{k : t (k , m k ) = h}. m +1 , nh (b ) = k , n1 (b ) = m k + 1. Worst-case 40. DOP : (2) F RSOP , 1, t (nh (b 1 ), n1 (b 1 )) = t (k , m k ) = h . , 1, . , h, t (nh (b h ), n1 (b h )) = t (k 1, m k + 1) = 1. k . , , h = n F (2) (x ) = nk , k . . Worst-case 41. DOP : (2) F RSOP ? , , . --. . , , ... . Worst-case 42. DOP : (2) F RSOP RSOP (Random SamplingOptimal Price, RSOP) . b , b b 1 2. t = opt(b ) t = opt(b ) . t b , t b. , RSOP , , . Worst-case 43. DOP : (2) F RSOP RSOP 4- F (2) . . b = ($1, $2). Worst-case 44. DOP : (2) F RSOP RSOP 4-F (2) . 1 , 2 RSOP 0. , b= {$1}, b = {$ 2}, t = $1, t = $2. Worst-case 45. DOP : (2) F RSOP RSOP 4- F (2) . RSOP $1 $2 $1. RSOP $.50. F (2) (b ) = $2. , RSOP 4-. Worst-case 46. DOP : (2) F RSOP RSOP 15- F (2) . b = (b1 , . . . , bN ), . b . , b1 , , ., b1 F (2) (b ) ( ). Worst-case 47. DOP : (2) F RSOP RSOP 15- F (2) . Xi {0, 1} , , i . ,X1 = 0. Si = i X j =1 i bi . Worst-case 48. DOP : (2) F RSOP RSOP 15- F (2) .Si , : Si , 0 i = 1; , 1 ;2 E , i Si ; i E .34 Worst-case 49. DOP : (2) F RSOP RSOP 15- F (2) . i , bi , .. j Si bi Sj bj . RSOP , Rev = (i Si )bi ( - bi - ). , Rev F15 (2). Worst-case 50. DOP : (2) F RSOP RSOP 15- F (2) .F (2) i , , , .. i bi jbj j 2. , i . B = {Si i2 }. i , , Pr[B] = 1 (B 2 i , , ). Worst-case 51. DOP : (2) F RSOP RSOP 15- F (2) ., F (2) i i Si S b b2 .3 E 3 S i 4i, 411 (i Si )bi ibi Si bi . 43 Worst-case 52. DOP : (2) F RSOP RSOP 15- F (2) . E 3 B 4 F (2)Rev = (i Si )bi 1 Si3i 6 , b F (2) F (2) E [Rev] Pr[E 3 B] ,4 6 15 .. Pr[B] = 1 , Pr[E 3 ] 0.9.2 4 Worst-case 53. DOP : (2) F RSOP , Pr[E 3 ] 0.9. 4 411/3 1/3 Pr[E 3 ] = 1 17 + 3 33 1 2 17 + 3 33 . 4 81 Worst-case 54. DOP : (2) F RSOP Pr[E ]. = k 1 k3 k ( ).4 Si i (k 1)(i Si ) Si 0. Worst-case 55. DOP : (2) F RSOP Zi = (k 1)(i Si ) Si .1 Zi 2 (k 1), 1; Z1 = k 1. Pr[E ] (ruin) ; p j = Pr [i : Zi = Z1 j ] . pk ( ). Worst-case 56. DOP : (2) F RSOP , 1, j pj = (p1 ) . p1 pk : 1 11kp1 =+ pk = (1 + p1 ). 2 22 Worst-case 57. DOP : (2) F RSOP 1 1k = 1 (1 + p1 ). p1 = 2 + 2 pk 2 p1 x k 2x + 1 [0, 1]. , (0, 1) ? Worst-case 58. DOP : (2) F RSOP , p11, 1, , . (Y1 , Y2 , . . .), Yi = Zi k + 2, Yi = 0, . , . Worst-case 59. DOP : (2) F RSOP W i = r Yi , r x k 2x + 1. (W1 , W2 , . . .) , .. E [Wi +1 |Wi ] = Wi . . Yi = 0 . Yi0 111E [r Zi + |Zi ] = r Zi 1 + r Zi +k 1 = (r 1 + r k 1 )r Zi 1= r Zi . 2 22 Worst-case 60. DOP : (2) F RSOP (W1 , W2 , . . .) , , E [Wt ] = W1 = r t 1. p1,t = Pr [i t : Zi = Z1 1] = Pr[Wt = 1]. At {Zt = Z1 1 ZsZt ,st }. Worst-case 61. DOP : (2) F RSOP At , p1,t = t Pr[A ], p = Pr[A ]. i =1 i 1 i =1i ,p1 = limt p1,t . p1,t = Pr[Wt = 1] E [Wt ] = r . ,p1 r1, , , p1 = r . Worst-case 62. DOP : (2) F RSOP !Lecture notes homepage: http://logic.pdmi.ras.ru/sergey/index.php?page=teaching , , : sergey@logic.pdmi.ras.ru, snikolenko@gmail.com smartnik. Worst-case </p>