2 / 5 Auction: Theory and Practice 3 / 5 (WTO) 1 SDR 27 1,6 Auction: Theory and Practice 4 / 5 2

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1 June 22, IPV 13 SPSB SPSB SPSB FPSB FPSB FPSB Low Price Auction APV IPV APV 33 (FPSB) (SPSB)

2 2 / 5 Auction: Theory and Practice 3 / 5 (WTO) 1 SDR 27 1,6 Auction: Theory and Practice 4 / 5 2

3 Auction: Theory and Practice 5 / 5 (4) (5) = (12) (13) winner s curse Auction: Theory and Practice 6 / 5 3

4 F(x) {X 1,X 2,...,X n } X (1) X (2)... X (n) X (1) x n F (1,n) (x) = Pr[X (1) x] = Pr[X 1 x,...,x n x] = Pr[X i x] {X i } n i=1 = {F(x)} n X (1) f (1,n) (x) = (d/dx){f(x)} n = n {F(x)} n 1 f(x) X (2) X (2) x x x F (2,n) (x) = {F(x)} n + n C 1 {F(x)} n 1 {1 F(x)} X (2) f (2,n) (x) = n(n 1){F(x)} n 2 {1 F(x)} f(x) = n{1 F(x)} f (1,n 1) (x) k X k f (k,n) (x) = n! (k 1)!(n k)! {F(x)}n k {1 F(x)} k 1 f(x) i=1 [,1] f(x) = 1 F(x) = x f (k,n) (x) = n! (k 1)!(n k)! xn k (1 x) k 1 = xk 1 (1 x) n k B(n k +1,k) x (,1) E[X (k) ] = (n k+1)/(n+1) n 1 X f (n 1,n) (x) = n(n 1) F(x) {1 F(x)} n 2 f(x), x. λ f(x) = λe λ x F(x) = 1 e λ x f (n 1,n) (x) = λn(n 1) e λ(n 1) x {1 e λ x } n 1 E[X (n 1) ] = yf (n 1,n) (y)dy Auction: Theory and Practice 7 / 5 4

5 8 / 5 High Price Auction Low Price Auction Oral Format Sealed-Bid Format Auction: Theory and Practice 9 / 5 Independent Private Value Affiliated Private Value Common Value Auction: Theory and Practice 1 / 5 Auction: Theory and Practice 11 / 5 5

6 Auction: Theory and Practice 12 / 5 IPV 13 / 5 Auction: Theory and Practice 14 / 5 V Pr[V < v] = F V (v) V V [, v] Auction: Theory and Practice 15 / 5 n 1 1 v 1 b 1 1 { v1 max Π 1 = 2 j n b j if b 1 > max 2 j n b j if b 1 < max 2 j n b j Auction: Theory and Practice 16 / 5 6

7 IPV: β II (v) = v (1) β i max j i b j 1. v i > β i v i i v i β i v i > z i > β i z i i v i β i v i > β i > z i z i i 2. β i > v i v i i β i > v i > z i z i i β II (v) = v β II (v) = v Auction: Theory and Practice 17 / 5 IPV: 1 1 v G n 1 (v) = Pr[V 2 v,v 3 v,...,v n v] n = Pr[V j v] = {F V (v)} n 1 j=2 n 1 n 1 Y (1,n 1) max 2 j n V j Y (1,n 1) G n 1 (v) = Pr[Y (1,n 1) v] Auction: Theory and Practice 18 / 5 7

8 IPV: ExPay II (v) = v v = E[Y (1,n 1) Y (1,n 1) < v] G n 1 (v) = = v v y g n 1(y) G n 1 (v) dy G n 1(v) yg n 1 (y)dy V i F V ( ), V i [, v] v { v } n E[ExPay II (V)] = n yg n 1 (y)dy f V (v)dv = n = v v y = E[V (2,n) ] { v y } f V (v)dv g n 1 (y)dy y {n {1 F V (y)}g n 1 (y)}dy n {V i } n i=1 V (2,n) n V (2,n) n n n Auction: Theory and Practice 19 / 5 V Pr[V < v] = F V (v) V V [, v] Auction: Theory and Practice 2 / 5 8

9 i v i b i 1 { v1 b Π 1 = 1 if b 1 > max 2 j n b j if b 1 < max 2 j n b j v β(v) β(v) β(v) max β(v j) = β( max V j) = β(y (1,n 1) ) 2 j n 2 j n Auction: Theory and Practice 21 / 5 IPV: v 1 b E[Π 1 (v)] = (v b) Pr[β(Y (1,n 1) ) < b] = (v b) Pr[Y (1,n 1) < β 1 (b)] = (v b) G n 1 (β 1 (b)) β 1 (b) b = β(v) g n 1 (v) = G n 1(v) g n 1 (β 1 (b)) β (β 1 (b)) (v b) G n 1(β 1 (b)) = b = β(v) G n 1 (v) dβ(v) dv +g n 1 (v) β(v) = v g n 1 (v) d dv {G n 1(v) β(v)} = v g n 1 (v) β() = β(v) = 1 G n 1 (v) v y g n 1 (y)dy = E[Y (1,n 1) Y (1,n 1) < v]. (2) Auction: Theory and Practice 22 / 5 9

10 IPV: i v i v ExPay I (v) = v v = E[Y (1,n 1) Y (1,n 1) < v] G n 1 (v), (Y (1,n 1) max 2 j n V j) ExPay I (v) = ExPay II (v) n E[ExPay I (V)] = E[V (2,n) ] (3) E[ n ] Auction: Theory and Practice 23 / 5 β I (v) = v β II (v) = v 1 G n 1 (v) v v β I (v) v { } Gn 1 (y) dy G n 1 (v) v shading G n 1 (y)dy (4) β I (v) n G n 1 (y) G n 1 (v) = ( ) n 1 FV (y) n, y < v F V (v) β I (v) n β II (v) n n n (5) Auction: Theory and Practice 24 / 5 1

11 α,β f V (v) = vα 1 (1 v) β 1, F V (v) = B(α, β) v f V (y)dy, G n (v) = {1 F V (v)} n 1, v 1 1 β I (v) = v 1 G n 1 (v) v G n 1 (y)dy 1 1 β II (v) = v (6) n n E[ExPay I (V)] = n E[ExPay II (V)] = E[V (2,n) ] = 1 y n(n 1)f V (y) {1 F V (y)} {F V (y)} n 2 dy Ba\\af_ Fmf[lagf k][gf\ hja[] f = 5 f = 1 f = 5 f = 1 f = 1 Ba\\af_ Fmf[lagf k][gf\ hja[] f = 5 f = 1 f = 5 f = 1 f = 1 Ba\\af_ Fmf[lagf k][gf\ hja[] f = 5 f = 1 f = 5 f = 1 f = 1 VYdmYlagf VYdmYlagf VYdmYlagf D]fkalq (Y,Z)=(6,2) D]fkalq (Y,Z)=(6,6) D]fkalq (Y,Z)=(2,6) Cgkl Cgkl Cgkl Eph][l]\ R]n]fm] Eph][l]\ R]n]fm] Eph][l]\ R]n]fm] fmez]j g^ Za\\]jk fmez]j g^ Za\\]jk fmez]j g^ Za\\]jk Figure 1: High-Price Auction: Valuation = Beta Distribution Auction: Theory and Practice 25 / 5 11

12 High-Price Auction: (,1) { v 1 < v < 1 f V (v) =, F otherwise V (v) = v < v < 1, G n 1 (v) = {F V (v)} n 1 1 v 1 (4) β I (v) = v 1 G n 1 (v) v G n 1 (y)dy = (7) β II (v) = v (8) a n n E[ExPay I (V)] = n E[ExPay II (V)] = E[V (2,n) ] = Low-Price Auction: λ { λe λ c c f C (c) = c <, F C(c) = 1 e λ c, G n 1 (v) = e λ(n 1) c,c. (11) β I (c) = c+ 1 G n 1 (c) c G n 1 (y)dy = (9) b n ; C (n 1,n) n n 1 n E[ExRev I (C)] = E[C (n 1,n) ] = a V (2,n) (n 2+1,2) b C (n 1,n) n n 1 Auction: Theory and Practice 26 / 5 12

13 IPV: Low Price c (, c) 1 b Y (n 1,n 1) n 1 E[Π 1 (c)] = (b c) Pr[β(Y (n 1,n 1) ) > b] = (b c) Pr[Y (n 1,n 1) > β 1 (b)] = (b c) G n 1 (β 1 (b)) β 1 (b) b = β(c) G n 1 (x) n 1 x G n 1 (x) = {1 F C (x)} n 1 g n 1 (c) = G n 1(c) g n 1 (β 1 (b)) β (β 1 (b)) (b c)+g n 1(β 1 (b)) = b = β(c) d dv {G n 1(c) β(c)} = c g n 1 (c) c c β( c) = c β(c) = = c+ 1 G n 1 (c) c c 1 G (n 1,n 1) (c) y g n 1 (y)dy = E[Y (n 1,n 1) Y (n 1,n 1) > c] (1) c c G n 1 (y)dy. (11) E[ n ] Auction: Theory and Practice 27 / 5 13

14 F C (c) c 1 β(c) = c+ G n 1 (y)dy G n 1 (c) c 1 = c+ {1 F C (c)} n 1 {1 F C (y)} n 1 dy c F C (c) Bidding Function second price n = 5 n = 1 n = 5 n = 1 n = 1 Bidding Function second price n = 5 n = 1 n = 5 n = 1 n = 1 Bidding Function second price n = 5 n = 1 n = 5 n = 1 n = 1 Cost Cost Cost Density (a,b)=(6,2) Density (a,b)=(6,6) Density (a,b)=(2,6) Cost Cost Cost Expected Payment Expected Payment Expected Payment number of bidders number of bidders number of bidders Figure 2: Low-Price Auction: Cost = Beta Distribution Auction: Theory and Practice 28 / 5 14

15 APV 29 / 5 Auction: Theory and Practice 3 / 5 : x { x } g n 1 (t t) β I (x) = v(y,y;n)dl n 1 (y x), L n 1 (y x) = exp y G n 1 (t t) dt (12) G n 1 (y x) = Pr[Y (1,n 1) y X 1 = x] x : β II (x) = v(x,x;n) (13) v(x,y;n) = E[V 1 X 1 = x,y (1,n 1) = y] a a Pinkse and Tan (25) The Affiliation Effect in First-Price Auctions, Econometrica 73, pp Auction: Theory and Practice 31 / 5 15

16 Affiliation i X i n : {X 1,X 2,...,X n } (Affiliation) multivariate total positivity n {X 1,X 2,...,X n } κ( ) x > x E[κ(Y (1,n 1) ) X 1 = x ] E[κ(Y (1,n 1) ) X 1 = x], Y (1,n 1) = max 2 j n X j Y (1,n 1) = max 2 j n X j G n 1 (y x) = Pr[Y (1,n 1) y X 1 = x] x V 1 v 1 (X) = v(x 1,X ( 1) ) X ( 1) V 1 v 1 (X) = v(x 1,X ( 1) ) X ( 1) x y v(x,y;n) = E[V 1 X 1 = x,y (1,n 1) = y] (14) IPV V 1 v 1 (X) = v(x 1 ) v(x,y;n) = E[V 1 X 1 = x,y (1,n 1) = y] v(x) (15) n Auction: Theory and Practice 32 / 5 16

17 IPV APV 33 / 5 IPV APV (FPSB) IPV (15) APV (14) 22 β (v) = {v β(v)} gn 1(v) G n 1 (v) v = β(v)+ β (v) G n 1 (v) g n 1 (v) b i = β(x i ) x i = β 1 (b i ) v i = b i + G n 1( β 1 (b i ) ) g n 1 ( β 1 (b i ) ) = b i + G( b i ) g( b i ) APV Auction: Theory and Practice 34 / 5 IPV paradigm APV paradigm v i = b i + G( b i ) g( b i ) {v i } n i=1 APV IPV Kolmogorov-Smirnov Figure 3: / Auction: Theory and Practice 35 / 5 17

18 IPV APV (SPSB) IPV (5) APV (13) APV Auction: Theory and Practice 36 / 5 Yahoo JAL ANA IPV GW APV Auction: Theory and Practice 37 / 5 Yahoo! Auction: Theory and Practice 38 / 5 18

19 l i p i,l n l β 1 E[p i,l n l ] = β +β 1 n l ρ τ (p i,l n l ) = β,τ +β 1,τ n l ρ τ (p n) n p 1τ IPV Auction: Theory and Practice 39 / 5 Figure 4: Auction: Theory and Practice 4 / 5 19

20 JAL Figure 5: JAL Auction: Theory and Practice 41 / 5 ANA Figure 6: ANA Auction: Theory and Practice 42 / 5 2

21 JAL Figure 7: JAL Auction: Theory and Practice 43 / 5 JAL Figure 8: JAL Auction: Theory and Practice 44 / 5 21

22 ANA Figure 9: ANA Auction: Theory and Practice 45 / 5 ANA Figure 1: ANA Auction: Theory and Practice 46 / 5 22

23 47 / 5 PPS 24 (PPS) PPS PPS PPS PPS PPS PPS (ex. ) Auction: Theory and Practice 48 / 5 23

24 n PPS( ) PPS δ δ δ PPS 14 Ultra High Voltage Consumer s Cost, Social Cost High Voltage 12 Cost (Yen/kWh) n = 4 n = 6 n = 1 Type Consumer Social Preference Rate (δ) Figure 11: Auction: Theory and Practice 49 / 5 24

25 p B PPS p B+ PPS p A Ultra High Voltage Inefficient Allocations High Voltage.6.5 Probability n=4 n=6 n=1 Probability Total p B p B+ p A Preference Rate (δ) Figure 12: Auction: Theory and Practice 5 / 5 25

2/50 Auction: Theory and Practice 3 / 50 (WTO) 10 SDR ,600 Auction: Theory and Practice 4 / 50 2

2/50 Auction: Theory and Practice 3 / 50 (WTO) 10 SDR ,600 Auction: Theory and Practice 4 / 50 2 stakagi@econ.hokudai.ac.jp June 24, 2011 2.... 3... 4... 7 8... 9.... 10... 11... 12 IPV 13 SPSB... 15 SPSB.... 17 SPSB.... 19 FPSB... 20 FPSB.... 22 FPSB.... 23... 24 Low Price Auction.... 27 APV 29...