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Daniella Stevenson
6 years ago
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7 d t = 1,, T d i j J i,d,t J i,d,t+ = J i,d,t j j a i,d,t j i t d γ i > 0 i j t d U i,j,d,t = μ i,j,d + β i,j,d,t γ i + ε i,j,d,t ε i,j,d,t t ε i,,d,t μ β μ μ i,j,d d t d, t

8 j μ i,j,d i j d z j ν j,d v i μ i,j,d = z j + ν j,d v i sign z j = sign v i v i = 1 z j = 1 z j = 1 j z j ν j,d j d ν j,d N 0, τ ν τ ν l ν j,d l j j s j,l,d ν j,d N z j + ν j,d, τ s s j,l,d s j l d τ s ω l,j l j ω l,j ω j,l ω

9 τ s n i,j,d,t E μ i,j,d I i,d,t = z j v i + s τ s n i,j,d,t + τ i,j,d,t z j v i ν t d n i,j,d,t j t s i,j,d,t I i,d,t t n s I i,d,t z j v i t s i,j,d,t v i τ s τ ν n i,j,d,t n i,j,d,t = 0 j z j v i. β β β R + β i,j,d,t F d I i,d,t, λ i, α j

10 α j (0, 1) j α j λ i i F d I i,d,t t d j i j t d E β i,j,d,t I i,d,t = α j N K i,d,t 1 + A i,d,t + α j K i,d,t λ i + u κ + K i,d,t λ i i,d,t j t d A i,d,t α j K i,d,t u i,d,t N

11 κ j j α j /(1 + A i,d,t + α j ) j α j A i,d,t j α j N K i,d,t j λ i u i,d,t μ β i d μ β V(I t, ε t ) = max ε,t, j J t E β j,t I t + E μ j I t γ + ε j,t +V(I, ε ) f I I t, j g (ε ) di dε t I t n t, s t, K t, h t, u t

12 f I I t, j I t j h t A t j h j,t α j g (ε) ε n t s t K t h t u t h t+ h t j h j s t l ω l,j l l j n t,j = n t,j + 1 s t,j n t,j = n t,j s t,j = s t,j f j j j α j

13 ω j α j β = 0

15 60% 35% 5%

16 t Sites per session sites 3 sites 4 5 sites Sessions Male Female x y 2.7 {.5,.5} D i i t = 1,, T d a i,d,t

17 x y

18 ,,,,,,,,,,,,,,,,,,, i d n i,j,d,t ω w j,d log 1 + words j,d K i,d, w j,d i j

19 i j j Pr i a = j n i,j > 0 = d t 1 a i,d,t = j n i,j,d,t > 0 d t 1 n i,j,d,t > 0 i j Pr i a = j n i,j = 0 = d t 1 a i,d,t = j n i,j,d,t = 0 d t 1 n i,j,d,t = 0 i Pr i (a > 0 n a > 0) Pr i (a > 0 n a = 0) i a n a > 0 Δ i = Pr i (a > 0 n a > 0) Pr i (a > 0 n a = 0) i Δ i > 0 Δ i < 0 Δ i 0 Δ i 39% Δ i < 0 31% Δ i = 0 30% Δ i > 0 40% Pr a = j n j > 0 Pr a = j n j = 0 2.0% l l = 1,, 50 Δ i < 0 Δ i = 0 Δ i > 0

20 Cumulative share of consumers 100% 75% 50% 25% 0% Difference in probability of visiting linked site after seeing one or more links ( i ) x i 5.6% v i λ i γ i D i v i N η v + D i φ v, ζ v, log λ i N η λ + D i φ λ, ζ λ, log γ i N η γ + D i φ γ, ζ γ γ i γ i exp γ w γ i,d = γ i d γ w > 0

21 α j w j,d i K i,d, d i K i,d, w j,d j K i,d,t α j K i,d,t K i,d, K i,d, t = 1 K i,d,t t > 1 α j α j ε i,d,j,t EV (0, 1) j I i,d,t V j I i,d,t + ε i,d,j,t V j I i,d,t V j I i,d,t = E β i,d,j,t I i,d,t + E μ i,d,j I i,d,t γ i,d + log exp V ȷ (I ) f I I i,d,t, ȷ di ȷ J i,d,t j j t j ε i,d,j,t θ

22 z j, α j φ v, φ λ, φ γ v i λ i γ i η λ, η γ ζ λ, ζ γ γ w τ s a = a i,d,t I = I i,d,t T i,d exp V j I i,d,t θ L (θ a, I) 1 + ȷ exp V J ȷ I i,d,t θ i,d,t i d t j J i,d,t 1a i,d,t =j I u K s L (θ a, n, w, h) n w h u K s u K s (θ, I) p (θ D) θ p (θ a, n, h, w, D) L (θ a, I) f (I n, h, w) p (θ D) ds du dk

23 z j φ v τ s α j α j α j η λ φ λ ζ λ α j α j α j η λ α j φ λ ζ λ η γ φ γ ζ γ γ w N = 30 κ = 4 τ ν = 1 j z j = 0 η v = 0

24 ζ v = 1 α j

25 f (x) f (x) f (x) i j z j v i τ s

26 α j z j Information quantity ( α ) egotastic dlisted thesuperficial celebuzz perezhilton Match location ( z ) Posterior density v λ γ... (.) (.) (.)... (.) (.) (.)... (.) (.) (.)... (.) (.) (.)... (.) (.) (.)... (.) (.) (.)... (.) (.) (.) η... (.) (.) ζ... (.) (.) %.%.%

27 Match uncertainty remaining 100% 90% 80% 70% 60% Number of links observed y n =,, x μ j,d n j,d =,, μ j,d n j,d = (τ s n + ) z j x z j z j < 0 z j > 0 v i v i z j < 0 z j > 0

28 τ s τ s 217 α j y λ i α j λ i α j λ i λ i α j λ i α j α j

29 γ i log γ i γ w z j α j ω j x y z j z j z j 0 z j x

30 Information quantity ( α ) E D Match location ( z ) Link frequency: 20% 40% 60% S C P

31 S 3 5 = 15 S S ω n ω S = 200

32 a , a , a..... % a a., a, a. X Y Y X X X t = % a

33 0.25% 2% 2% 2% 2% 0.5% 0.35%

34 t t % 5.5% 0.2% 0.3% 0.67% 2.5% 79% 71% 0.47% 0.07% 2.11% 0.34%

39 d N N t j k t {0, 1} N k b, = 0 b j ι j {0, 1} N u R N + d i j t N β j,t = u b ι j,b 1 k b,t b= u ι j β j,t = K u K t u t K t = N k b= b,t t u t = N u K t b= bk b,t K u j K t u t K u j K u

40 ι j u ι j j u b j j π b [0, 1] α j (0, 1) j b j Pr[ι j,b = 1 π b ] ρ j,b ρ j,b Pr ι j,b = 1 π b = 1 (1 π b ) α j j α j 1 ρ j,b π b j α j 0 π b > ρ j,b 0 α j j u b σ σ u b σ Expo σ d σ u b ι j u b σ π b ι j u b i.i.d. π b, Beta (α, 1), α > 0

41 k b,t = 0 π b,t k b,t = 0, h t Beta (α, 1 + A t ) h t t A t J j= h t,jα j α j j N K t EK K t I t, j = (N K t )1 Bα,+A t +α j B(α,+A t ) B (, ) I t t K t h t A t N K t j α α j α = 1 EK K t I t, j = (N K t ) α j +A t +α j i.i.d. σ InvGa (κ + 1, κ λ), κ > 0, λ > 0 λ i κ K t u t σ t I t InvGa (κ + K t + 1, κ λ + K t u t ) I t K t u t j

42 j α j E β j,t I t = +A t +α j (N K t ) λ + K t κ +K t (u t λ) K u I t u K, I t K I t u K, I t K K u K K t K t u t κ λ+k u κ λ+k t u κ+kt+ t κ λ+k u p u K, K, I t = > K K u t K t u t B(κ +K t +,K K t ) δ ut u K = K t K > K t u u = u t β j,t F β j,t I t, λ, α j = N Gaβ j,t K K t, λ + K t κ +K t (u t λ)binomk K t N K t, K =K t α j +A t +α j I t n t, s t, K t, u t, h t A t j h t,j α j I I t j f I I t, j = f n, s, K, u, h n t, s t, K t, u t, h t, j f I I t, j = p s n, n t, s t p n n t, j p u K, K t, u t p K K t, h t, j p h h t, j p u K, K t, u t p K K t, h t, j h j p h h t, j = δ h t,j, h h t h j 1 n s n ȷ j n ȷ n t,ȷ + 1 ω j,ȷ n t,ȷ 1 ω j,ȷ p n ȷ n t, j = ω j,ȷ δ nt,ȷ + n ȷ + 1 ω j,ȷ δ n t,ȷ n ȷ

43 ȷ s ȷ s ȷ = s t,ȷ N z p s j + τ sn t,j s t,j z j j n τ j, n t,j, s t,j = s n t,j +τ ν, τ s + n t,j τ s + τ ν, n j = n t,j + 1 δ st,j s j, n j = n t,j w j,d K i,d,t K i,d, E K i,d, = Nα j /(1 + α j ) i K i,d, K i,d, j d i K i,d, E K i,d, w j,d = N q w j,d q w j,d j d K i,d, K i,d,t t > 1 q w j,d α j (0, 1) q w j,d w j,d 0, qw j,d = 1 c w + w j,d c w w j,d d i K i,d, = 0 max w j,d i K i,d, = N/2 N κ λ i N κ λ i

44 κ N = 30 κ = 4 α j α α = 1 ν j,d s j,k,d τ ν = 1 τ s z j v i z j v i v i η v = 0 ζ v = 1 K u s u s σ σ σλ u u = uλ u λ s j,l,d s j,l,d z j ν j,d τ s s j,l,d z j ν j,d E s j,l,d = 0 V s j,l,d = 1 s ν j,d logit α j N (0, 1) z j N (0, 1) τ s Ga (.4, 5) E τ s = 2 η, φ ζ N 0, 10 ζ, ζ χ λ γ γ w N (0, 1) φ v N (0, 1)

45 z j α j

46 τ s =.2 τ s = 2 λ = 0 β = 0 γ = 2 v = 2 z = 0 ω L,R = 1 ω R,L = 0 Probability of initiating a session (No links) Noisy Informative Links Behavior Myopic Forward looking

47 Share of sessions starting at linking site (No links) Noisy Informative Links Behavior Myopic Forward looking Number of sites visited during session (No links) Noisy Informative Links Behavior Myopic Forward looking

48 0.62 Share of sessions visiting linked site (No links) Noisy Informative Links Behavior Myopic Forward looking Conditional probability of visiting the linked site Myopic Forward looking (No links) Noisy Informative (No links) Noisy Informative Links Signal Lower than expected match (No links) Higher than expected match

49 p (θ W ) D θ D θ D θ D θ f (, ) θ θ D θ θ c θ c D θ c t θ c θ c θ θ π b,t k b,t = 0, h t Beta (α, 1 + A t ) h t t A t J j= h tα j α j α t α j t b t 1 (1 π b ) α (1 π b ) α t = (1 π b ) A t

50 Θ W t θ θ (t ) b θ p(θ W ) D θ μ, Σ f(θ, D θ ) θ c N(μ, Σ) p(θ c W ) D θ c μ, Σ f(θ c, D θ c) α p(θc )N(θ μ, ) p(θ)n(θ c μ,) u U(0, 1) u < α θ (t) b θ c θ (t) b θ I p(i θ (t) ) W f(i, θ (t) ) W { W, I, θ (t) } Θ θ (t)

51 π b b (1 π b ) A t π α b (1 π b ) [B (α, 1)] = (1 π b) (A t+) (1 π b) A t π α b (1 π b ) [B (α, 1)] dπ B (α, A t + 1) b Beta (α, A t + 1) π α b N K t j E K K t I t, j = (N K t ) 1 B α, 1 + A t + α j B (α, 1 + A t ) B (, ) I t t K t h t A t b j t ρ j,b,t π b,t = 1 1 π b,t α j π b,t E ρ j,b,t I t = 1 1 π b,t α j Beta π b,t α, 1 + A t dπ b,t = 1 B α, 1 + α j + A t B (α, 1 + A t ) N K t K t u t I t K t u t σ t I t InvGa (κ + K t + 1, κ λ + K t u t ) u b σ t K t K t u t σ t Ga (K t, σ t ) σ σ κ + K t + 1 κ λ + K t u t u b σ u u b u b

52 j j α j E β j,t I t = (N K t ) λ + (u 1 + A t + α j κ + K t λ) t K t E β j,t I t = E K u K t u t I t, j = E K E u K, I t I t, j K t u t E u K, I t = 1 K E [ σ t I t ] (K K t ) + u t K t = κ λ + K t u t 1 K t κ + K t K K t + u t K E β j,t I t = E K κ λ + K t u t κ + K t 1 K t K + u t K t K I t, j K t u t = E K K t I t, j κ λ + K t u t κ + K t E K K t I t, j u K, I t K K u K K t K t u t κ λ+k u κ λ+k t u κ+kt+ t κ λ+k u p u K, K, I t = > K K u t K t u t B(κ +K t +,K K t ) δ ut u K = K t β j,t K, σ t, I t Ga (K K t, σ t ) σ t I t InvGa (κ + K t + 1, κ λ + K t u t ) p β j,t, σ t K, I t σ t p β j,t K, I t = K K t κ +K t + β j,t κ β j,t +κ λ+k t u λ+k t u t t β t,j +κ λ+k t u t β t,j B (κ + K t + 1, K K t ) β j,t = u K u t K t u K K > K t

53 z 1 z 2 z 3 z 4 z density value α 1 α 2 α 3 α 4 α density value ζ λ 2 ζ γ 2 τ s 15 6 density 10 density value value η λ η γ γ w 2 30 density 1 density value value

54 5 φ γ 1 φ γ 2 φ γ 3 φ γ 4 φ γ 5 φ γ 6 φ γ 7 4 density value φ λ 1 φ λ 2 φ λ 3 φ λ 4 φ λ 5 φ λ 6 φ λ density value φ v 1 φ v 2 φ v 3 φ v 4 φ v 5 φ v 6 φ v density value

.14% duke.edu idc.ac.il

.14% duke.edu idc.ac.il .14% roos@rsm.nl mela@ duke.edu ronshachar@ idc.ac.il .54%.01%.18%.14% 2.3%.05% d t = 1,, T d i j j = 0 J i,d,t J i,d,t+1 = J i,d,t j j a i,d,t j i t d j t d μ i,j,d i j γ i > 0 ε i,j,d,t t i j t d