stakagi@econ.hokudai.ac.jp June 19, 29 2............................................................................ 3.................................................................. 4 5...................................................................... 6........................................................................ 7............................................................................ 8.................................................................. 9 IPV 1......................................................... 12 IPV:............................................. 14 IPV:................. 16......................................................... 17 IPV:............................................. 19 IPV:................. 2..................................................................... 21 IPV:........................................................... 23........................................................................... 26 APV 27 Affiliation...................................................................... 29 APV:............................................................. 3 APV:............................................................. 31 APV:................................................................... 32 IPV APV 35 IPV APV..................................................... 36..................................... 37............................................................................... 38 1
2 / 38 (WTO) 1 SDR 27 1,6 Auction: Theory and Practice 3 / 38 Auction: Theory and Practice 4 / 38 2
5 / 38 High Price Auction Low Price Auction Oral Format Sealed-Bid Format Auction: Theory and Practice 6 / 38 Independent Private Value Common Value Affiliciated Private Value Auction: Theory and Practice 7 / 38 3
Auction: Theory and Practice 8 / 38 Auction: Theory and Practice 9 / 38 IPV 1 / 38 Auction: Theory and Practice 11 / 38 Pr[V < v] = F V (v) V [, v] Auction: Theory and Practice 12 / 38 4
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 13 / 38 IPV: β II (v) = v (1) Auction: Theory and Practice 14 / 38 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 Y 1 max 2 j n V j a Y 1 G n 1 (v) Pr[Y 1 < v] = G n 1 (v) a G n 1 (v) n 1 v Auction: Theory and Practice 15 / 38 IPV: EP II (v) = v v 1 v v = E[ Y 1 Y 1 < v ] G n 1 (v) = y g n 1 (y)dy G n 1 (v) = y g n 1 (y)dy G n 1 (v) V i F V ( )on [, v] v n E[EP II (V )] = n = n = v v { v y } y g n 1 (y)dy } { v y f V (v)dv f V (v)dv (2) g n 1 (y)dy (3) y {n {1 F V (y)}g n 1 (y)}dy = E[V (2,n) ] (4) n {V i } n i=1 V (2,n) n V (2,n) Auction: Theory and Practice 16 / 38 6
Pr[V < v] = F V (v) V [, v] Auction: Theory and Practice 17 / 38 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 ) 2 j n 2 j n Auction: Theory and Practice 18 / 38 7
IPV: v 1 b E[Π 1 (v)] = (v b) Pr[β(Y 1 ) < b] = (v b) Pr[Y 1 < β 1 (b)] β 1 (b) b = β(v) = (v b) G n 1 ( β 1 (b) ) 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 Y 1 < v]. (5) Auction: Theory and Practice 19 / 38 IPV: i v i v m I (v) = v v = E[ Y 1 Y 1 < v ] G n 1 (v), (Y 1 max 2 j n V j) m I (v) = m II (v) n E[m I (V )] = E[V (2,n) ] (6) Auction: Theory and Practice 2 / 38 8
β I (v) = β II (v) = v 1 G n 1 (v) v y g n 1 (y)dy = v 1 G n 1 (v) v G n 1 (y)dy (7) v β I (v) v {G n 1(y)/G n 1 (v)}dy v β II (v) shading β I (v) n G n 1 (y) G n 1 (v) = ( ) FV (y) n 1 n, y < v F V (v) β I (v) n β II (v) n n (8) Auction: Theory and Practice 21 / 38 (, 1) { 1 < v < 1 f V (v) = otherwise v F V (v) = v < v < 1 1 v 1, G n 1 (v) = {F V (v)} n 1 β I (v) = v 1 G n 1 (v) v G n 1 (y)dy = v v n = n 1 n v (9) β II (v) = v (1) Auction: Theory and Practice 22 / 38 9
IPV: r β I (v; r) = r Gn 1(r) G n 1 (v) + 1 G n 1 (v) v r y g n 1 (y)dy, v r β II (v; r) = v m(v; r) = r G n 1 (r) + v r y g n 1 (y)dy, v r S [, v] v n r {1 F V (r)} G n 1 (r) + n y {1 F V (y)} g n 1 (y)dy r v S v S = n r {1 F V (r)} G n 1 (r) + n y {1 F V (y)} g n 1 (y)dy + {F V (r)} n v r r r n {1 (r v ) λ(r )} {1 F V (r )} G n 1 (r ) =, λ(r ) = f V (r ) 1 F V (r ) r = v + 1 λ(r ) r Auction: Theory and Practice 23 / 38 1
IPV: r Auction: Theory and Practice 24 / 38 λ(v) = f V (v) 1 F V (v) = 1 1 v = r = v + 1 r = r = 1 + v 2 n = 2, v = S = n r {1 F V (r )} G n 1 (r ) + v r y {1 F V (y)} g n 1 (y)dy + {F V (r )} n v = 5 12 r = S = v y {1 F V (y)} g n 1 (y)dy = 1 3 = 4 12 Auction: Theory and Practice 25 / 38 11
N k p k β I (v) = = N 1 k=1 N 1 k=1 p k G k (v) N 1 j= p j G j (v) E[Y 1,k Y 1,k < v] (Y 1,k max V j) 2 j k+1 p k G k (v) N 1 j= p j G j (v) β(k) I β II (v) = v By-It-Now Option (v) β (k) (v) k I BIN BIN Auction: Theory and Practice 26 / 38 APV 27 / 38 Auction: Theory and Practice 28 / 38 12
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 ) X 1 = x ] E[ κ(y 1 ) X 1 = x], Y 1 = max 2 j n X j Y 1 = max 2 j n X j G n 1 (y x) = Pr[Y 1 y X 1 = x] x 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 = y ] (11) IPV V 1 v 1 (X) = v(x 1 ) v(x, y; n) = E[ V 1 X 1 = x, Y 1 = y ] = E[ V 1 X 1 = x] v(x) (12) n a Auction: Theory and Practice 29 / 38 a APV v(x, y; n) n IPV v(x, y; n) n 13
APV: x b b β 1 (b) Y 1 β 1 (b) β(y 1 ) Π = β 1 (b) {v(x, y; n) β(y)} g n 1 (y x)dy = β 1 (b) {v(x, y; n) v(y, y; n)} g n 1 (y x)dy Π b = 1 { ( v x, β 1 β (b); n ) v ( β 1 (b), β 1 (b); n )} g n 1 (β 1 (b) x) = (b) y < x v(x, y; n) v(y, y; n) > y > x v(x, y; n) v(y, y; n) < y = x β II (x) = v(x, x; n) (13) Auction: Theory and Practice 3 / 38 APV: x b z b = β(z) β(z) Π = z {v(x, y; n) β(z)} g n 1 (y x)dy = z v(x, y; n) g n 1 (y x)dy β(z) G n 1 (z x) z Π z = {v(x, z; n) β(z)} g n 1 (z x) β (z) G n 1 (z x) = z > x (z < x) Π z > (Π z < ) z = x β (x) = {v(x, x; n) β(x)} gn 1(x x) G n 1 (x x) β() = 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 (14) Auction: Theory and Practice 31 / 38 14
APV: IPV = APV a x EP II (x) = E[β II (Y 1 ) X 1 = x, Y 1 < x] EP II (x) = E[β II (Y 1 ) X 1 = x, Y 1 < x] = E[v(Y 1, Y 1 ; n) X 1 = x, Y 1 < x] x = v(y, y; n) dg n 1(y x) G n 1 (x x) x b EP I (x) = β I (x) EP I (x) = β I (x) = x v(y, y; n)dl n 1 (y x) EP II (x) EP I (x) c Auction: Theory and Practice 32 / 38 a b x c Krishna (22) Auction Theory, Academic Press. (14) (13) a b Auction: Theory and Practice 33 / 38 a Pinkse and Tan (25) The Affiliation Effect in First-Price Auctions, Econometrica 73, pp. 263-277. b winner s curse 15
v(x) = V V ( x ) β F x v (x v) =, x v v V F v (v) = 1 v α, v 1, α > 2 β I (x; n) = (n 1) β + (max{1, x}) (n 1) β 1 v(x, x; n) 1 + (n 1) β v(x, x; n) = α + β n max{1, x} α + β n 1 n α = 2.5, β =.5 { } n β α S n = 1 (α + n β β)(α + n β 1) α 1 Auction: Theory and Practice 34 / 38 16
er sred sred Bid dingpricesv.s.numberofbid Bidding Function 1.2 1.4 1.6 1.8 2. 2.2 n=2 n=4 n=6..5 1. 1.5 2. 2.5 Signal x Figure 1: venueofsellerv.s.numberofbid Revenue 1.44 1.46 1.48 1.5 1.52 1.54 5 1 15 2 Number of Bidders Figure 2: 17
IPV APV 35 / 38 IPV APV IPV (12) APV (11) β (x) = {v(x, x) β(x)} g(x x) G(x x) v(x, x) = β(x) + β (x) G(x x) g(x x) b i = β(x i ) x i = β 1 (b i ) v(x i, x i ) = b i + G(β 1 (b i ) β 1 (b i )) g(β 1 (b i ) β 1 (b i )) = b i + G(b i b i ) g(b i b i ) APV IPV (8) APV (13) APV Auction: Theory and Practice 36 / 38 Yahoo JAL ANA IPV GW APV Auction: Theory and Practice 37 / 38 18
l i p i,l β 1 n l E[p i,l n l ] = β + β 1 n l ρ τ (p i,l n l ) = β + β 1 n l ρ τ (p n) n p 1τ IPV Auction: Theory and Practice 38 / 38 Figure 3: 19
Figure 4: JAL Figure 5: ANA 2
QuantileRegresionLines:JAL Bidding Prices 1 2 3 4 5 6 7 1% 3% 5% 7% 9% 2 4 6 8 1 12 Number of Bidders Figure 6: JAL Estimates of Coefficient on nbidders 4 3 2 1 1 SlopeofRegresionCurves:JAL Coefficients (quantile) Lower & Upper C.I. Coefficients (mean) Lower & Upper C.I..2.4.6.8 Percent Figure 7: JAL 21
QuantileRegresionLines:ANA Bidding Prices 2 4 6 8 1% 3% 5% 7% 9% 2 4 6 8 1 Number of Bidders Figure 8: ANA SlopeofRegresionCurves:ANA Estimates of Coefficient on nbidders 6 4 2 Coefficients (quantile) Lower & Upper C.I. Coefficients (mean) Lower & Upper C.I..2.4.6.8 Percent Figure 9: ANA 22