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Amazon.co.jp 2008.09.02 START

Amazon.co.jp Amazon.co.jp Amazon.co.jp Amazon Internet retailers are extremely hesitant about releasing specific sales data 1( )

ranking 500,000 100,000 Jan.1 Mar.1 Jun.1 Sep.1 Dec.1 2007 Amazon.co.jp 2( )

Stochastic ranking process a. b. 1 3( )

Stochastic ranking process a. b. 1 Amazon Amazon.co.jp Stochastic ranking process Pareto Amazon.co.jp 4( )

Amazon.co.jp Stochastic ranking process 5( )

Stochastic ranking process N ;1, 2,,N x (N) 1,0 w (N),,x(N) N,0 1,,w (N) N X (N) 1 (t),,x (N) ; 1 jump N (t) t [ 0] X (N) i (0) = x (N) i,0 ( i) τ (N) i,j, i =1, 2,,N, j =1, 2, ; i j 1 jump j τ (N) i,j+1 τ (N) i,j, j =0, 1, 2, (τ (N) i,0 =0) i, j j P[ τ (N) i,1 t ]=1 e w(n) i t 6( )

[ 1] X (N) i (τ (N) i,j )=1( i, j) [ 2] X (N) i (τ (N) i,j )=X (N) i jump 1 jump (τ (N) i,j 0) + 1 ( i, i,j ) 3 2 4 1 5 1 3 2 4 5 2 1 3 4 5 1 2 3 4 5 t=τ 1,1 t=τ 2,1 t=τ 1,2 t=τ 3,1 3 1 2 4 5 τ 1,1 <τ 2,1 <τ 1,2 <τ 3,1 < 7( )

x C (t) jump jump x C (t) jump jump 1 2 1 2 1 2 1 2 1 2 1 x C (t) 1 1 2 1 2 1 2 2 1 2 1 1 1 1 2 1 2 1 2 2 2 1 x (N) N C (t) =1+ χ (N) τ i=1 i t 1 t =0 X (N) C (t) =1 8( )

x C (t) N Jump λ (N) = 1 N y (N) C y C (t) =1 0 N i=1 δ w (N) i (t) = 1 N (x(n) C (t) 1) = 1 N e wt λ(dw) N λ N i=1 χ (N) τ i t y C (t) jump λ 9( )

y (N) i,0 = 1 N (x(n) i,0 1) N μ (N) y,0 (dw dy) = 1 N μ (N) y,t i δ w (N) i (dw) δ (N)(dy) μ y y,0 (dw) dy (N ) i,0 = 1 N (X(N) i Jump Y (N) i := 1 δ (N) N w i i μ y,t (dw) dy 1) δ (N) Y i (t) N μ y,t (dw) 10( )

U i t (y, t)+ j f j U j (y, t) U i y (y, t) = f iu i (y, t) (y, t) [0, 1) [0, ) 1 Burgers f i 0 i U i (y, t) t y i f j U j (0, 0) <, U i (y, 0) 0, smooth, j U j (y, 0) = 1 y Burgers j U i (0,t)=U i (0, 0), t 0 dy B dt (t) =v(y B(t),t); v(y, t) = j f j U j (y, t), y B (0) = y 0 jump μ y,t ({f i })= U i (y, t) y 11( )

jump U (dw; y, t)+ w U(dw ; y, t) U (dw; y, t) = wu(dw; y, t), t y (y, t) [0, 1) [0, ) U(dw; y, 0) 0 smooth, in y wu(dw;0, 0) <, U(R + ; y, 0) = 1 y Burgers U(dw;0,t)=U(dw;0, 0), t 0 μ y,t (dw) = U (dw; y, t) y 12( )

jump tail N Amazon.co.jp PDE 13( )

t =0 x C (t) Ny C (t) =N N 0 e wt λ(dw) Jump λ Pareto 14( )

Jump Pareto ( ) N 1/b Pareto w i = a, i =1,,N, a, b > 0 i w i i 80 20 a: = an 1/b b b b x C (t) Ny C (t) N(1 b(at) b Γ( b, at)) Γ N, a, b 15( )

2000 80 (n d 1 = 77) 500000 N =90 a =4 10 4 (1/a =3.5 ) b =0.6312 ( χ 2 /n d =1 ) 500 1000 1500 2000 16( )

50 1000 17( )

Chevalier Goolsbee b =1.2 Online bookstore brick-and-mortar bookstore CPI Brynjolfsson Hu Smith b =1.148 J. A. Hausman (1997) (consumer welfare) Long tail 0 input b Online retail Amazon.co.jp 18( )

y C (t) μ(dw; y, t) λ Pareto 0 <r<1 r S(r, 1) N (w,z) [0, ) [r,1) wμ z,t (dw) dz = NabΓ(1 b, q(r)) q(r) b 1 ; q(r) =at 1 (r), r =1 e q(r) + q(r) b Γ(1 b, q(r)) cf. w i 1 r S(r, 1) N ab b 1 (1 r(b 1)/b ) S(r, 1) (b =1.15, 1.2) S(r, 1) 1.4 1.5 1.4 1.3 1.2 1.1 0.2 0.4 0.6 0.8 19( )

b b b b>1: S tot = S(0, 1) Nab b 1 S(0, 0.2) b =2: 0.2 0.447 S tot 20 80 20 80 S(0.2, 1) b =1.2 (Chevalier, Goolsbee) 0.235, S tot S(0.2, 1) b =1.15 (Brynjolfsson, Hu, Smith) 0.189 S tot N b<1: S(r, 1) Nab b 1 (1 r(b 1)/b ): r =0 Amazon.co.jp b =0.6312 < 1 20( )

Amazon.co.jp C. Anderson, The Long tail Amazon.co.jp: b =0.6312 < 1 Amazon Amazon Amazon 21( )

2ch.net (2ch.net) web 1 stochastic ranking process 22( )

ranking 50 50 1 0 1 N = 795, (a,b ) = (3.3 10 4, 0.62) (1/a = 4 ) 12:00 24:00 time 1 1 10 23( )

stochastic ranking process Amazonl.co.jp λ (stochastic ranking process) online retail, long tail 24( )

web activity 25( )

8 λ stochastic ranking process 26( )

Stochastic ranking process (?) Stochastic ranking process long tail 27( )

NHK 2008 6 4 No.2592 Citation Statistics, (2008.6) International Mathematical Union (IMU), ICIAM, IMS citation data use and misuse 20 10 C. Anderson (long tail) Long tail 28( )

impact factor 29( )

Stochastic ranking process 365 30( )

End of slides. Click [END] to finish the presentation. K. Hattori, T. Hattori, Existence of an infinite particle limit of stochastic ranking process, Stochastic Processes and their Applications (2008), to appear. K. Hattori, T. Hattori, Equation of motion for incompressible mixed fluid driven by evaporation and its application to online rankings, preprint (2007). K. Hattori, T. Hattori, Mathematical analysis of long tail economy using stochastic ranking processes, preprint (2008). http://www.math.tohoku.ac.jp/~hattori/amazonj.htm Google END Bye