product adoption in social networks - hkbusecon.hkbu.edu.hk/eng/doc/netlearn170921m_slides.pdf ·...

Introduction Model Examples General Networks Tree Networks Examples The End

Product Adoption in Social Networks

Simon Board Moritz Meyer-ter-Vehn

UCLA

September 25, 2017


Motivation

How do societies learn about new innovations?

I New products, e.g. electric bikes

I New behaviors, e.g. smoking

I New principles, e.g. racial equality

We study a tractable model of learning on social networks

I Agents have opportunity to try product in random order.

I Before trying, agent sees which neighbors have adopted.

I After trying, agent adopts product if he likes it.

How does learning depend on network structure?


Results

Any network

I Characterize diffusion via system of ODEs.

I Show existence of a unique equilibrium.

Large random networks

I Product adoption improving over time.

I Product adoption improving in number of links.

I Impact of conversion rates on learning.

Network structure

I Directed vs undirected networks

I Direct information vs indirect information

I Cyclical vs acyclical


Literature

Social learning with sampling

I Smith and Sorensen (1996)

I Banerjee and Fudenberg (2004)

I Monzon and Rapp (2014)

Social learning on networks

I Celen and Kariv (2004)

I Acemoglu, Dahleh, Lobel and Ozdaglar (2011)

I Lobel and Sadler (2015, 2016)

Asymmetric social learning

I Guarino, Harmgart and Huck (2011)

I Herrera and Horner (2013)

I Hendricks, Sorensen, Wiseman (2012)


Yet another Paper on Social Learning?

“A significant gap in our knowledge concerns short-rundynamics and rates of learning in these models....Thecomplexity of Bayesian updating in a network makes thisdifficult, but even limited results would offer a valuablecontribution to the literature.”

The Oxford Handbook of the Economics of Networks 2016

Contribution

I Characterize such short-run dynamics via ODEs.

I Provide comparatives statics in time and linkage.

I Social learning on a given network.

I Modeling innovation: Action depends only on public info.


Model


Model

Players, Actions, Payoffs

I Product quality θ ∈ {L = 0, H = 1}, with Pr(z = H) = 12 .

I At time ti ∼ U [0, 1], agent i=1, . . . , N chooses to test or not.

I Adopts tested product with prob. zθ, where zL < zH .

I Benefit θ, and cost of trying ci ∼ F .

Information

I Commonly known directed network G = {(i, j)}.I Agent i observes which of his successors Si use product at ti.


The Inference Problem


The Inference Problem

Why has j not adopted the product?

I j tried product, but did not like it?

I j is not yet aware of product?

I j chose not to try product (because k did not adopt it)?


Examples


i→ j

Definition: Adoption Rate

xθi,t: Probability i has adopted product θ by time t

Leader, j:

xθj,t = Pr(j adopts|tests) Pr(j tests)

Follower, i:

xθi,t = Pr(i adopts|tests) Pr(i tests)


i→ j



Leader, j:

xθj,t = zθF (1/2)

Follower, i:

xθi,t = Pr(i adopts|tests) Pr(i tests)


i→ j



Leader, j:

xθj,t = zθF (1/2)

Follower, i:

xθi,t = zθ(Pr(j adopt) Pr(i tests|j adopt)+Pr(j not) Pr(i tests|j not)))


i→ j



Leader, j:

xθj,t = zθF (1/2)

Follower, i:

xθi,t = zθ

(xθj,tF

(xHj,t

xLj,t

)+ (1− xθj,t)F

(1− xHj,t1− xLj,t

))

where F(xH

xL

):= F

(xH

xH+xL

)


Adoption curves

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Time

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Pr(Try|A)

Pr(Try|N)Adoption Agent i

Adoption Agent jHigh Quality

Low Quality

Assumptions: c ∼ U [1/2, 2/3], zL = 1/2, zH = 1.


i� j

The ignorant neighbor

I Before ti, agent j never observes adoption by i.

I xθj,∅,t: Probability j has adopted product θ by t ≤ ti.

xθj,∅,t = zθ(Pr(i adopt) Pr(j tests|i adopt)+Pr(i not) Pr(j tests|i not)))


i� j




xθj,∅,t = zθ(Pr(j tests|i not)))


i� j




xθj,∅,t = zθF

(1− xHi,∅,t1− xLi,∅,t

)


i� j




xθ∅,t = zθF

(1− xH∅,t1− xL∅,t

)


i� j




xθ∅,t = zθF

(1− xH∅,t1− xL∅,t

)

Actual adoption rate

xθt = zθ

(xθ∅,tF

(xH∅,t

xL∅,t

)+ (1− xθ∅,t)F

(1− xH∅,t1− xL∅,t

))


Adoption curves

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Time

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Pr(Try|A)

Pr(Try|N)

Adoption Real Agent

Adoption Ignorant AgentHigh Quality

Low Quality



General Networks


Individual Adoption Rates . . . are not enough

A general formula for individual adoption rates

I xθA,t: Probability A ⊆ Si have adopted θ by t.

xθi = zθ∑A⊆Si

xθAF

(xHAxLA

)

But cannot recover joint xθA from marginals xθj


A Bigger State Space

State of network λ ∈ {∅, a, b}N

I λi = ∅: i hasn’t moved yet, t ≤ ti.I λi = a: i has moved, tried, and adopted the product.

I λi = b: i has moved, but not adopted the product.

Additional Notation

I Distribution x = (xλ), and xΛ :=∑

λ∈Λ xλ for sets Λ.

I If λi = a, b, write λ−i for “same state with λi = ∅”.

Inference

I If A ⊂ Si have adopted at ti, i knows λ ∈ Λ(i, A), namely:

λj = a for all j ∈ A λj 6= a for all j ∈ Si \A λi = ∅


ODE for General Networks

Theorem 1.

xθλ = − 1

1− t∑i:λi=∅

xθλ

+1

1− t∑i:λi=a

xθλ−izθF

(xHΛ(i,A)

xLΛ(i,A)

)

+1

1− t∑i:λi=b

xθλ−i

[1− zθF

(xHΛ(i,A)

xLΛ(i,A)

)]

Implications

I Existence, uniqueness, discrete-time approximation ...

I But: ODE cannot be computed, since it is 2 · 3N -dimensional.


Tree Networks


Trees

Large Random Networks

I Analysis is complicated by (a) Self-reference, (b) Correlation.

I Neither matters in large random networks.

(Homogeneous) Trees: Network G is

I . . . a tree if there is at most one path i→ . . .→ j.

I . . . homogeneous of degree k if every node has out-degree k.


Adoption in Trees

I Successors’ adoption (xθj)j∈Si conditionally independent.

I Probability A ⊆ Si adopt

xθA =∏j∈A

xθj∏

j∈Si\A

(1− xθj)

I Adoption rates

xθi = zθ∑A⊆Si

xθAF

(xHAxLA

)I 2N -dimensional ODE.


Adoption in Homogeneous Trees

I All nodes are symmetric.

I Probability ν of k adopt

xθ(ν,k) :=

(νk

)(xθ)ν(1− xθ)k−ν

I Adoption rate

xθ = zθk∑ν=0

xθ(ν,k)F

(xH(ν,k)

xL(ν,k)

)

I 2-dimensional ODE.


Adoption curves for the infinite line, k = 1.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Time

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Pr(Try|A)

Pr(Try|N)Adoption Rate

High Quality

Low Quality



Mean Field Approximation

Large (homogeneous) Random Networks

I Fix N nodes and randomly draw k out-links for every node i.

I xθ(i,G),t: Adoption rate of agent i in network G.

Theorem 2 (Mean Field Approximation).

Adoption in large random network converge to homog. tree, xθt :

limN→∞

Pr(|xθ(i,G),t − xθt | < ε) = 1 for all t, θ, ε > 0

Proof Sketch

I Large (hom) random network is “local” tree (with Pr→ 1).

I In discrete time t = ν/m (Euler) only “local” network matters.

I Discrete time solution converges to continuous time.


Speed of convergence - N -cyclesI Agents i ∈ N with links i→ i+ 1 mod n.I xθj,i,t: Probability j has adopted at t ≤ ti

xθj,i,t = zθ

(xθj+1,i,tF

(xHj+1,j,t

xLj+1,j,t

))

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Time

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

Adoption, n=1

Adoption, n=2

Adoption, n=3,4

High Quality

Low Quality


Comparative Statics(for finite trees)


Adoption curves

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Time

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Adoption, k=1

Adoption, k=5

Adoption, k=20

High Quality

Low Quality



Ranking Product Adoption

DefinitionInformation structure (µH , µL) has better product adoption than(µH , µL) if test chance is greater (smaller) for H (L).

Pr(test|H) ≥ Pr(test|H) and Pr(test|L) ≥ Pr(test|L)

Application

I Learning from successor adopt. (xHA,t, xLA,t)A⊆Si .

I Allows comparison across• Time t,• Agents i,• Networks G.

I E.g., i’s adoption improves over time iff for all t > s

xHi,t ≥ xHi,s and xLi,s ≥ xLi,t


Product Adoption improves in Blackwell-Order

Assumption F (c)c is convex, and F (c)(1− c) is concave.

I Satisfied if |F ′′(c)/F ′(c)| < 2




Lemma 1.If experiment (µH , µL) is Blackwell-sufficient for (µH , µL), then ithas better product adoption.

Proof

I Posterior ψ = ψ(A) = µH(A)µH(A)+µL(A)

sufficient for signal A.

I Try-out probability

Pr(test|H) = Pr(ψ ≤ c|H) =∑ψ

µH(ψ)F (ψ)





Proof




Pr(test|H) =∑ψ

µ(ψ)2ψF (ψ)





Proof




Pr(test|H) =∑ψ

µ(ψ)2ψF (ψ)

I Since 2ψF (ψ) convex, and µ(ψ) spread of µ(ψ):

Pr(test|H) =∑ψ

µ(ψ)2ψF (ψ) >∑ψ

µ(ψ)2ψF (ψ) = Pr(test|H)


Sufficiency for Dichotomies with Binary Signals

Lemma 2.A draw from (xH , xL) is sufficient for a draw from (xH , xL) iff

xH

xL≥ xH

xLand

1− xH

1− xL≤ 1− xH

1− xL

Thus: Adoption improves over time if trace (xHt , xLt )t concave.


Product Adoption as a Function of Time

Theorem 3.For any agent in any tree, product adoption improves over time.

For s < t: xHt ≥ xHs and xLs ≥ xLt

Induction over distance to leaves, m:

I Anchor, m = 0: No info., so xθ = zθ Pr(test) ≡ zθF (12).

I Step m→ m+ 1• Adoption improves in t for level-m agents j.• Trace (xHj,t, x

Lj,t)t concave.

• Lemma 2: (xHj,t, xLj,t) Blackw.-sufficient for (xHj,s, x

Lj,s) if t > s.

• Lemma 1: adoption improves in t for level-(m+ 1) agent i.


Product Adoption as a Function of Conversion Rates

I More informative conversion rates: zH ≥ zH and zL ≥ zL.

I Normalize product adoption xθt/zθ.

Theorem 4.Product adoption improves in informativeness of conversion rates.

For any t:˙xH

zH≥ xH

zHand

xL

zL≥

˙xL

zL(*)

Proof: Induction Step

I Assume (*) for all j ∈ Si. Then, a fortiori

xHj ≥ xHj and xLj ≥ xLj

I (xHj , xLj )j∈Si Blackwell-sufficient for (xHj , x

Lj )j∈Si .

I Lemma 1: (*) holds for i.


Product Adoption as a Function of Links

I Tree G with sub-tree G ⊆ G.

I Agent i ∈ G’s adoption rates: xθi in G, xθi in G.

Theorem 5.Product adoption improves in links.

˙xHi ≥ xHi and xLi ≥ ˙xLi (*)

Proof: Induction Step

I Assume (*) for all j ∈ Si. Then also

xHj ≥ xHj and xLj ≥ xLj

I (xHj , xLj )j∈Si Bl.-sufficient for (xHj , x

Lj )j∈Si (more and better).

I Lemma 1: (*) holds for i.


Back to the Examples


Reconsidering the Comparative Static in Links

Rationale for Theorem 5

I More links generate more information.

I More information improves product adoption.

But do more links generate more information?


The Self-Referential Link Harms Product Adoption

i has better adoption in i→ j than in i� j

I Idea: j’s choice is more informative in i→ j than in i� j.

I i→ j: i updates based on xθj,t, where xθj,t = zθF (1).

I i� j: i updates based on xθ∅,t, where xθ∅,t = zθF

(1−xH∅,t1−xH∅,t

).

I Thus, xθj,t = λtxθ∅,t with λt > 1 independent of θ.

I Conclude with Lemmas 1 and 2.

Contrast to Trees

I In tree, additional link generates additional information.

I But cannot learn from link back to oneself, j → i.

I Rather, j → i reduces probability that j generates information


The Correlating Link is Ambiguous

Information of i not Blackwell-ranked

I Correlated: Greater chance of best signal {j, k}.I Independent: Stronger inference from worst signal ∅.

Numerical Simulation: Independence slightly better


Conclusion

A Tractable Model of Social Learning on Networks

I Describe Adoption Rates via ODEs.

I Allows for comparative statics b/c of complementarity.

I Captures real network features

I Modeling innovation: Learn private information after move.

Quo vadis?

I Relax assumptions: Perfect info, homgeneity, undirected...

I Examples: Small networks with rich structure.


Relaxing Assumptions


Relaxing Three Modeling Assumptions

Homogeneous number of links

I Real networks heterogeneous.

I Heterogeneity covered, but ODE N -dimensional.

Common Knowledge

I May know popularity of friends, but friends of friends?

I Relaxing common knowledge simplifies analysis.

Directed Network

I Reasonable for blogs, Twitter, but not for friends.

I i� j allowed, but doesn’t arise in large random networks.


Poisson Network with Known Successors

Model

I Out-degrees drawn iid from P (·|k); no cycles.

I i observe only number of successors |Si| (and their adoption).

I Limit of large random network with link probability k/N .

Adoption rate

I Probability of ι successors: P (ι|k) := e−kkι/ι!

I Probability ν of ι have adopted: B(ν, ι;x) :=(ιν

)xν(1− x)ι−ν

xθ = zθ

( ∞∑ι=0

P (ι|k)

ι∑ν=0

B(ν, ι;xθ)F

(B(ν, ι;xH)

B(ν, ι;xL)

))


Poisson Network with Unknown Successors

Model

I Idea: Not so clear how many pertinent friends I have.

I i observe only adopting successors A ⊆ Si, but not Si.

Adoption rate

I Prob. ι adopting successors: P (ι|kxθ) := e−kxθ(kxθ)ι/ι!

xθ = zθ

( ∞∑ι=0

P (ι|kxθ)F(P (ι|kxH)

P (ι|kxL)

))

I Less separation than with known successors.


Undirected Poisson Networks

Complications

I i’s neighbors j have P (·|k) + 1 neighbors.

I Before ti, j conditions on λi = ∅.

But, assume neighbors are unknown

I Complications cancel, since j can’t see i before ti.

I j observes ν ∼ P (·|kxθ) adoptions from ι ∼ P (·|k) neighbors.

I Same adoption rates xθt as in directed Poisson network.


Examples


Complete Network

Definitions

I State: (α, β) agents have (adopted,passed).

I xθα,β,t: Prob. α others adopted, β others passed at t ≤ ti.I Aggregate xθα :=

∑β<n−α x

θα,β

Adoption Rates

xθα,β = − 1

1− t(n− 1− β − α)xθα,β

+1

1− t(n− 1− β − (α− 1))xθα−1,βz

θF

(xHα−1

xLα−1

)

+1

1− t(n− 1− (β − 1)− α))xθα,β−1

(1− zθF

(xHαxLα

))


3-Agent Networks

3 figures: (a) i→ j, k, (b) and j → k, (c) and j � k

...

I ...

product adoption in social networks - hkbusecon.hkbu.edu.hk/eng/doc/netlearn170921m_slides.pdf ·...

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