Introduction Model Results Extensions

Viral Social Learning

David McAdams1 Yangbo Song2

1Fuqua School of Business, Duke University

2School of Management and Economics, CUHK (Shenzhen),

May 9, 2020

David McAdams, Yangbo Song Viral social learning May 9, 2020 1 / 20

Introduction Model Results Extensions

Motivation: idea/product that spreads like a virus

Example 1: job-market papers

Hiring committees learn about JMPs socially, as well as in applications

For a committee member: not knowing a paper prior to interview season isa bad signal

For a candidate: when to publish JMP matters

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Introduction Model Results Extensions

Motivation: idea/product that spreads like a virus

Example 2: influencer marketing

Customers find out new products from others’ adoption

If a better product is adopted faster, seeing it early is a good signal

Viral campaign vs. advertising campaign for producer

David McAdams, Yangbo Song Viral social learning May 9, 2020 3 / 20

Introduction Model Results Extensions

This paper

Questions

Demand side: lifecycle of viral diffusion?

Supply side: is a viral campaign optimal? For how long? Does it facilitatequality improvement?

Novelty of problem

Endogeneity of awareness and action sequence

Time-varying inference affected by producer’s strategy

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Introduction Model Results Extensions

Literature

Classical models of social learning

Foundation: Bikhchandani et al. (1992), Banerjee (1992), Smith andSorensen (2000)

Extensions: Banerjee (1993), Celen and Kariv (2004), Acemoglu et al.(2011), Song (2016)

SIR model of viral transmission

Theory: Kermack and McKendrick (1927, 1932, 1933)

Applications: Newman (2002), McAdams (2017a,b)

Other works on viral marketing and competitive contagion

Kempe et al. (2003, 2005), Mossel and Roch (2010)

Goyal et al. (2014)

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Introduction Model Results Extensions

Basic structure

One product with uncertain quality

Quality ω ∈ {g , b}, with probability α ∈ (0, 1) and 1− αSells at price p ≥ 0

Good product has value u, bad product u, assume p =u+

¯u

2

A unit mass of consumers

Each has a unit demand

At t = 0 of a continuous time line, a small ∆ > 0 fraction of consumersare exposed to the product

Independent private signals

Consumer i sees si ∈ {G ,B} upon exposure

Prob(si = G |g) = Prob(si = B|b) = ρ ∈ ( 12, 1)

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Introduction Model Results Extensions

Viral transmission

Three types of consumers

Susceptible: Sω(t). Sω(0) = 1−∆

Infected (adopting product): Iω(t)

Recovered (not adopting product): Rω(t)

Transmission rule

Each infected consumer meets other consumers randomly at rate β > 0

Sω(t), Iω(t) and Rω(t) are determined dynamically by

I ′ω(t) = βIω(t)Sω(t)pω(t)

R ′ω(t) = βIω(t)Sω(t)(1− pω(t))

S ′ω(t) = −βIω(t)Sω(t)

where pω(t) is the probability of adoption given ω.

David McAdams, Yangbo Song Viral social learning May 9, 2020 7 / 20

Introduction Model Results Extensions

Belief evolution

Upon awareness at t, a consumer’s interim belief q(t; ∅) is characterized by

q(t; ∅)1− q(t; ∅) =

αIg (t)Sg (t)

(1− α)Ib(t)Sb(t).

Hence, her posterior beliefs are

q(t;G) = q(t; ∅) ρ

ρq(t; ∅) + (1− ρ)(1− q(t; ∅))

q(t;B) = q(t; ∅) 1− ρ(1− ρ)q(t; ∅) + ρ(1− q(t; ∅))

.

David McAdams, Yangbo Song Viral social learning May 9, 2020 8 / 20

Introduction Model Results Extensions

Consumer’s decision

A consumer will

Herd on adoption if q(t; ∅) > ρ

Be sensitive to signals if q(t; ∅) ∈ (1− ρ, ρ)

Herd on non-adoption if q(t; ∅) < 1− ρq(t; ∅) = ρ or 1− ρ: to be discussed later.

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Introduction Model Results Extensions

Result 1: lifecycle

Unique equilibrium

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Introduction Model Results Extensions

Result 1: lifecycle

Two forces on consumer belief

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Introduction Model Results Extensions

Result 1: lifecycle

Belief evolution

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Introduction Model Results Extensions

Implications of consumer equilibrium

With viral social learning:

True quality remains unrevealed

Product obsolescence always occurs

Behavior of product reputation and adoption likelihood are sensitive toinitial beliefs

High α → both are monotone in timeLow α → neither is monotone in time

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Introduction Model Results Extensions

Result 2: robust occurrence of viral campaign

A model with endogenous campaign choice

Nature draws quality ω ∈ {g , b} with distribution (α, 1− α)

Producer’s choice (before knowing quality): t = T to stop viral campaignand switch to advertising campaign

Trade off in a viral campaign

Some adoption given bad signal

Long viral campaign → no adoption in advertising campaign

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Introduction Model Results Extensions

Result 2: robust occurrence of viral campaign

Theorem

The earliest optimal stopping time T ∗ is always positive.

Belief from advertising campaign is below α, and decreases in T

Always better to allow some viral spread

Two possibilities of exact optimumStop in the middle → consumers with good signal buy from advertisingcampaignNever stop → no one buys from advertising campaign

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Introduction Model Results Extensions

Result 2: robust occurrence of viral campaign

Numerically, stopping in the middle is almost always optimal.

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Introduction Model Results Extensions

Result 3: Goldilocks effect

A model with endogenous campaign choice + quality choice

Firm commits to marketing strategy: begin with viral campaign andswitch to advertising campaign at t = T

Firm observes cost c > 0 to improve quality

c ∼ F (c) on [0, c]

If there were no viral campaign, the market equilibrium induces a fractionα = F (2ρ− 1) of good quality firm.

David McAdams, Yangbo Song Viral social learning May 9, 2020 17 / 20

Introduction Model Results Extensions

Result 3: Goldilocks effect

Proposition

Suppose that α ∈ [ 12, ρ). A market equilibrium always exists, and in every

market equilibrium, α∗ ≤ α. The inequality is strict if α > 12.

David McAdams, Yangbo Song Viral social learning May 9, 2020 18 / 20

Introduction Model Results Extensions

Result 3: Goldilocks effect

Proposition

Suppose that α ∈ (1− ρ, 12). In every market equilibrium with profitable

advertising campaign, α∗ > α.

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Introduction Model Results Extensions

Extensions

Several topics for future research:

Pricing

Temporary infectiousness

Option to wait

Reversible adoption decisions

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