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... 32 IPV APV 33 (FPSB).... 34... 35 (SPSB).... 36... 37 47.... 48... 49.... 50 1
2/50 Auction: Theory and Practice 3 / 50 (WTO) 10 SDR 2007 1,600 Auction: Theory and Practice 4 / 50 2
Auction: Theory and Practice 5 / 50 (4) (5) = (12) (13) winner s curse Auction: Theory and Practice 6 / 50 3
F (x) {X 1,X 2,...,X n } X (1) X (2)... X (n) X (1) x F (1,n) (x) = Pr[X (1) x] = Pr[X 1 x,...,x n x] = = {F (x)} n n Pr[X i x] i=1 {X i } n i=1 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) [0,1] 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 (0, 1) E[X (k) ]=(n k +1)/(n +1) n 1 X n 1 f (n 1,n) (x) =n(n 1) F (x) {1 F (x)} n 2 f(x), x 0. 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) ]= 0 yf (n 1,n) (y)dy Auction: Theory and Practice 7 / 50 4
8/50 High Price Auction Low Price Auction Oral Format Sealed-Bid Format Auction: Theory and Practice 9 / 50 Independent Private Value Affiliated Private Value Common Value Auction: Theory and Practice 10 / 50 Auction: Theory and Practice 11 / 50 5
Auction: Theory and Practice 12 / 50 IPV 13/50 Auction: Theory and Practice 14 / 50 V Pr[V < v]=f V (v) V V [0, v] Auction: Theory and Practice 15 / 50 6
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 0 if b 1 < max 2 j n b j Auction: Theory and Practice 16 / 50 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 / 50 7
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 / 50 IPV: ExPay II (v) = v v = E[ Y (1,n 1) Y (1,n 1) <v] G n 1 (v) v = y g n 1(y) G n 1 (v) dy G n 1(v) = 0 v 0 yg n 1 (y)dy V i F V ( ), V i [0, v] v { v } n E[ExPay II (V )] = n yg n 1 (y)dy f V (v)dv 0 0 v { v } = n y f V (v)dv g n 1 (y)dy = v 0 0 = E[V (2,n) ] y 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 / 50 8
V Pr[V < v]=f V (v) V V [0, v] Auction: Theory and Practice 20 / 50 i v i b i 1 { v1 b Π 1 = 1 if b 1 > max 2 j n b j 0 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 / 50 9
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))=0 b = β(v) G n 1 (v) dβ(v) + g n 1 (v) β(v) =v g n 1 (v) dv d dv {G n 1(v) β(v)} = v g n 1 (v) β(0) = 0 β(v) = 1 G n 1 (v) v 0 y g n 1 (y)dy = E[Y (1,n 1) Y (1,n 1) <v]. (2) Auction: Theory and Practice 22 / 50 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 / 50 10
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) 0 v shading β I (v) n G n 1 (y) G n 1 (v) = ( ) n 1 FV (y) n 0, y < v F V (v) β I (v) n β II (v) 0 G n 1 (y)dy (4) n n n (5) Auction: Theory and Practice 24 / 50 11
α, β f V (v) = vα 1 (1 v) β 1, F V (v) = B(α, β) 1 β I (v) =v 1 G n 1 (v) v 0 G n 1 (y)dy v 0 f V (y)dy, G n (v) ={1 F V (v) } n 1, 0 v 1 1 1 β II (v) =v (6) n n E[ExPay I (V )] = n E[ExPay II (V )] = E[V (2,n) ] = 1 0 y n(n 1)f V (y) {1 F V (y)} {F V (y)} n 2 dy Bidding Function second price n = 5 n = 10 n = 50 n = 100 n = 1000 Bidding Function second price n = 5 n = 10 n = 50 n = 100 n = 1000 Bidding Function second price n = 5 n = 10 n = 50 n = 100 n = 1000 Valuation Valuation Valuation Density 0.0 0.5 1.0 1.5 2.0 2.5 (a,b)=(6,2) Density 0.0 0.5 1.0 1.5 2.0 2.5 (a,b)=(6,6) Density 0.0 0.5 1.0 1.5 2.0 2.5 (a,b)=(2,6) Cost Cost Cost Expected Revenue Expected Revenue Expected Revenue 0 200 400 600 800 1000 0 200 400 600 800 1000 0 200 400 600 800 1000 number of bidders number of bidders number of bidders Figure 1: High-Price Auction: Valuation = Beta Distribution Auction: Theory and Practice 25 / 50 12
High-Price Auction: (0, 1) { 0 v 0 1 0 <v<1 f V (v) =,F 0 otherwise V (v) = v 0 <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 0 G n 1 (y)dy = (7) β II (v) =v (8) n a n E[ExPay I (V )] = n E[ExPay II (V )] = E[V (2,n) ]= Low-Price Auction: λ { λe λ c c 0 f C (c) = 0 c<0,f C(c) =1 e λ c, G n 1 (v) =e λ(n 1) c,c 0. (11) β I (c) =c + 1 G n 1 (c) c G n 1 (y)dy = (9) n ; C (n 1,n) n n 1 b n E[ExRev I (C)] = E[C (n 1,n) ]= Auction: Theory and Practice 26 / 50 a V (2,n) (n 2+1, 2) b C (n 1,n) n n 1 13
IPV: Low Price c (0, 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))=0 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] (10) c E[ n ] c G n 1 (y)dy. (11) Auction: Theory and Practice 27 / 50 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 F C (c) 2 2 2 c Bidding Function second price n = 5 n = 10 n = 50 n = 100 n = 1000 Bidding Function second price n = 5 n = 10 n = 50 n = 100 n = 1000 Bidding Function second price n = 5 n = 10 n = 50 n = 100 n = 1000 Cost Cost Cost Density 0.0 0.5 1.0 1.5 2.0 2.5 (a,b)=(6,2) Density 0.0 0.5 1.0 1.5 2.0 2.5 (a,b)=(6,6) Density 0.0 0.5 1.0 1.5 2.0 2.5 (a,b)=(2,6) Cost Cost Cost Expected Payment Expected Payment Expected Payment 0 200 400 600 800 1000 0 200 400 600 800 1000 0 200 400 600 800 1000 number of bidders number of bidders number of bidders Figure 2: Low-Price Auction: Cost = Beta Distribution Auction: Theory and Practice 28 / 50 15
APV 29/50 Auction: Theory and Practice 30 / 50 : β I (x) = x 0 v(y, y; n)dl n 1 (y x), L n 1 (y x) =exp { x } g n 1 (t t) 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 Auction: Theory and Practice 31 / 50 a Pinkse and Tan (2005) The Affiliation Effect in First-Price Auctions, Econometrica 73, pp. 263-277. 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 / 50 17
IPV APV 33/50 IPV APV (FPSB) IPV (15) APV (14) 22 β (v) ={v β(v)} gn 1(v) G n 1 (v) b i = β(x i ) x i = β 1 (b i ) v = β(v)+ β (v) G n 1 (v) g n 1 (v) 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 / 50 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 / 50 18
IPV APV (SPSB) IPV (5) APV (13) APV Auction: Theory and Practice 36 / 50 Yahoo JAL ANA IPV GW APV Auction: Theory and Practice 37 / 50 Yahoo! 5000 100 250 Auction: Theory and Practice 38 / 50 19
l i p i,l n l β 1 E[p i,l n l ] = β 0 + β 1 n l ρ τ (p i,l n l ) = β 0,τ + β 1,τ n l ρ τ (p n) n p 100τ IPV Auction: Theory and Practice 39 / 50 Bidding Prices and Number of Bidders Figure 4: Auction: Theory and Practice 40 / 50 20
JAL Bidding Price Empirical CDF: JAL Figure 5: JAL Auction: Theory and Practice 41 / 50 ANA Bidding Price Empirical CDF: ANA Figure 6: ANA Auction: Theory and Practice 42 / 50 21
JAL Effect of nbidders Figure 7: JAL Auction: Theory and Practice 43 / 50 JAL Slopes of Percentile Curves: JAL Figure 8: JAL Auction: Theory and Practice 44 / 50 22
ANA Effect of nbidders Figure 9: ANA Auction: Theory and Practice 45 / 50 ANA Slopes of Percentile Curves: ANA Figure 10: ANA Auction: Theory and Practice 46 / 50 23
47/50 PPS 2004 (PPS) PPS PPS PPS PPS PPS PPS Auction: Theory and Practice 48 / 50 24
n PPS( ) PPS δ δ δ PPS 14 Ultra High Voltage Consumer s Cost, Social Cost High Voltage 12 Cost (Yen/kWh) 10 8 6 14 12 10 8 6 14 12 10 n = 4 n = 6 n = 10 Type Consumer Social 8 6 0.0 0.1 0.2 0.3 0.4 0.5 0.0 0.1 0.2 0.3 0.4 0.5 Preference Rate (δ) Figure 11: Auction: Theory and Practice 49 / 50 25
p B PPS p B+ PPS p A Ultra High Voltage Inefficient Allocations High Voltage 0.6 0.5 Probability 0.4 0.3 0.2 0.1 0.0 0.6 0.5 0.4 0.3 0.2 0.1 0.0 0.6 0.5 0.4 0.3 n=4 n=6 n=10 Probability Total p B p B+ p A 0.2 0.1 0.0 0.0 0.1 0.2 0.3 0.4 0.5 0.0 0.1 0.2 0.3 0.4 0.5 Preference Rate (δ) Figure 12: Auction: Theory and Practice 50 / 50 26