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Marketing via Friends: Strategic Diffusion of Information in Social Networks with Homophily Roman Chuhay Higher School of Economics, ICEF, CAS 2011 Roman Chuhay (ICEF, CAS, HSE) Marketing via Friends 2011 1 / 20 Homophily Homophily is a

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Marketing via Friends: Strategic Diffusion of Information in Social Networks with Homophily

Roman Chuhay

Higher School of Economics, ICEF, CAS 2011

Roman Chuhay (ICEF, CAS, HSE) Marketing via Friends 2011 1 / 20

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SLIDE 2

Homophily

Homophily is a tendency of people to interact more with those who are similar to them.

Roman Chuhay (ICEF, CAS, HSE) Marketing via Friends 2011 2 / 20

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Homophily

Homophily is a tendency of people to interact more with those who are similar to them.

◮ In our context - people with similar preferences tend to interact more. Roman Chuhay (ICEF, CAS, HSE) Marketing via Friends 2011 2 / 20

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Homophily

Homophily is a tendency of people to interact more with those who are similar to them.

◮ In our context - people with similar preferences tend to interact more.

ρ = 0 ρ = 0.5 ρ = 0.75 ρ = 1.0

Roman Chuhay (ICEF, CAS, HSE) Marketing via Friends 2011 2 / 20

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SLIDE 5

Literature review

WOM literature:

◮ Theoretical: Campbell (2010), Goyal and Galeotti (2009), Lopez-Pintado &

Watts (2009)

◮ Empirical: Leskovec et al. (2008), Iribarren and Moro (2007) Roman Chuhay (ICEF, CAS, HSE) Marketing via Friends 2011 3 / 20

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Literature review

WOM literature:

◮ Theoretical: Campbell (2010), Goyal and Galeotti (2009), Lopez-Pintado &

Watts (2009)

◮ Empirical: Leskovec et al. (2008), Iribarren and Moro (2007)

Homophily:

◮ Information diffusion: Currarini, Jackson & Pin (2009), Jackson & Golub

(2010)

◮ Social norms and preferences: Christakis & Fowler (2007), Fiore and Donath

(2005)

Roman Chuhay (ICEF, CAS, HSE) Marketing via Friends 2011 3 / 20

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SLIDE 7

Model

Network structure:

Roman Chuhay (ICEF, CAS, HSE) Marketing via Friends 2011 4 / 20

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SLIDE 8

Model

Network structure: There is measure γ of consumers of type A and 1 − γ of type B.

Roman Chuhay (ICEF, CAS, HSE) Marketing via Friends 2011 4 / 20

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SLIDE 9

Model

Network structure: There is measure γ of consumers of type A and 1 − γ of type B. Consumers are embedded into undirected network of social contacts:

Roman Chuhay (ICEF, CAS, HSE) Marketing via Friends 2011 4 / 20

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Model

Network structure: There is measure γ of consumers of type A and 1 − γ of type B. Consumers are embedded into undirected network of social contacts:

◮ Degree distribution p(k). Roman Chuhay (ICEF, CAS, HSE) Marketing via Friends 2011 4 / 20

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SLIDE 11

Model

Network structure: There is measure γ of consumers of type A and 1 − γ of type B. Consumers are embedded into undirected network of social contacts:

◮ Degree distribution p(k). ◮ Vector (ρA, ρB) identifies proportion of consumers of the same type in the

neighborhood of a randomly chosen consumer of type A and B.

Roman Chuhay (ICEF, CAS, HSE) Marketing via Friends 2011 4 / 20

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Model cont’d

Consumers:

Roman Chuhay (ICEF, CAS, HSE) Marketing via Friends 2011 5 / 20

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Model cont’d

Consumers: Heterogenous preferences towards the product

Roman Chuhay (ICEF, CAS, HSE) Marketing via Friends 2011 5 / 20

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Model cont’d

Consumers: Heterogenous preferences towards the product

◮ Across types: ⋆ Type A prefers high values of characteristic w ⋆ Type B prefers low values of characteristic w Roman Chuhay (ICEF, CAS, HSE) Marketing via Friends 2011 5 / 20

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Model cont’d

Consumers: Heterogenous preferences towards the product

◮ Across types: ⋆ Type A prefers high values of characteristic w ⋆ Type B prefers low values of characteristic w ◮ Within types: ⋆ Reservation price ¯

Pj

⋆ Threshold level of the product’s characteristic ¯

wj, s.t. induces a consumer to buy the product

Roman Chuhay (ICEF, CAS, HSE) Marketing via Friends 2011 5 / 20

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Model cont’d

Consumers: Heterogenous preferences towards the product

◮ Across types: ⋆ Type A prefers high values of characteristic w ⋆ Type B prefers low values of characteristic w ◮ Within types: ⋆ Reservation price ¯

Pj

⋆ Threshold level of the product’s characteristic ¯

wj, s.t. induces a consumer to buy the product

Consumers can buy the product only if they learn about it from:

◮ Direct advertisement. ◮ Observing a neighbor who has already acquired the product. Roman Chuhay (ICEF, CAS, HSE) Marketing via Friends 2011 5 / 20

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Model cont’d

Monopolist:

Roman Chuhay (ICEF, CAS, HSE) Marketing via Friends 2011 6 / 20

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Model cont’d

Monopolist: Knows degree distribution p(k) and homophily levels (ρA, ρB).

Roman Chuhay (ICEF, CAS, HSE) Marketing via Friends 2011 6 / 20

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Model cont’d

Monopolist: Knows degree distribution p(k) and homophily levels (ρA, ρB). Maximizes profits by choosing:

Roman Chuhay (ICEF, CAS, HSE) Marketing via Friends 2011 6 / 20

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Model cont’d

Monopolist: Knows degree distribution p(k) and homophily levels (ρA, ρB). Maximizes profits by choosing:

◮ Price P ∈ [0, 1] Roman Chuhay (ICEF, CAS, HSE) Marketing via Friends 2011 6 / 20

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Model cont’d

Monopolist: Knows degree distribution p(k) and homophily levels (ρA, ρB). Maximizes profits by choosing:

◮ Price P ∈ [0, 1] ◮ Characteristic of the product w ∈ [0, 1] Roman Chuhay (ICEF, CAS, HSE) Marketing via Friends 2011 6 / 20

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Model cont’d

Monopolist: Knows degree distribution p(k) and homophily levels (ρA, ρB). Maximizes profits by choosing:

◮ Price P ∈ [0, 1] ◮ Characteristic of the product w ∈ [0, 1]

Cost of production is 0.

Roman Chuhay (ICEF, CAS, HSE) Marketing via Friends 2011 6 / 20

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Model cont’d

Monopolist: Knows degree distribution p(k) and homophily levels (ρA, ρB). Maximizes profits by choosing:

◮ Price P ∈ [0, 1] ◮ Characteristic of the product w ∈ [0, 1]

Cost of production is 0. To induce sales the monopolist advertises product to infinitesimal part of the population.

Roman Chuhay (ICEF, CAS, HSE) Marketing via Friends 2011 6 / 20

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Baseline model assumptions

Population:

Roman Chuhay (ICEF, CAS, HSE) Marketing via Friends 2011 7 / 20

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Baseline model assumptions

Population: Consumers of type A and B constitute half of the population γ = 0.5 and consequently ρ = ρA = ρB

Roman Chuhay (ICEF, CAS, HSE) Marketing via Friends 2011 7 / 20

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Baseline model assumptions

Population: Consumers of type A and B constitute half of the population γ = 0.5 and consequently ρ = ρA = ρB Preferences:

Roman Chuhay (ICEF, CAS, HSE) Marketing via Friends 2011 7 / 20

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Baseline model assumptions

Population: Consumers of type A and B constitute half of the population γ = 0.5 and consequently ρ = ρA = ρB Preferences: Reservation price ¯ Pj and threshold value of characteristic ¯ wj are i.i.d. U[0, 1]

Roman Chuhay (ICEF, CAS, HSE) Marketing via Friends 2011 7 / 20

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Baseline model assumptions

Population: Consumers of type A and B constitute half of the population γ = 0.5 and consequently ρ = ρA = ρB Preferences: Reservation price ¯ Pj and threshold value of characteristic ¯ wj are i.i.d. U[0, 1] Consumers buy the product with probability:

Roman Chuhay (ICEF, CAS, HSE) Marketing via Friends 2011 7 / 20

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Baseline model assumptions

Population: Consumers of type A and B constitute half of the population γ = 0.5 and consequently ρ = ρA = ρB Preferences: Reservation price ¯ Pj and threshold value of characteristic ¯ wj are i.i.d. U[0, 1] Consumers buy the product with probability:

◮ Type A: qA = Pr(w ≥ ¯

wj ∩ P ≤ ¯ Pj) = (1 − P)w

Roman Chuhay (ICEF, CAS, HSE) Marketing via Friends 2011 7 / 20

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Baseline model assumptions

Population: Consumers of type A and B constitute half of the population γ = 0.5 and consequently ρ = ρA = ρB Preferences: Reservation price ¯ Pj and threshold value of characteristic ¯ wj are i.i.d. U[0, 1] Consumers buy the product with probability:

◮ Type A: qA = Pr(w ≥ ¯

wj ∩ P ≤ ¯ Pj) = (1 − P)w

◮ Type B: qB = Pr(w ≤ ¯

wj ∩ P ≤ ¯ Pj) = (1 − P)(1 − w)

Roman Chuhay (ICEF, CAS, HSE) Marketing via Friends 2011 7 / 20

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Illustration of the monopolist problem

Monopolist chooses optimally characteristic w and price P to maximize profits:

w 0.

1 P qA 1 P qB

A B B A A A A B B B A B B A A

Preferences frontier Induced network of potential buyers

Roman Chuhay (ICEF, CAS, HSE) Marketing via Friends 2011 8 / 20

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Illustration of the monopolist problem

Monopolist chooses optimally characteristic w and price P to maximize profits:

w 0.25

1 P qA 1 P qB

A B B A A A A B B B A B B A A

Preferences frontier Induced network of potential buyers

Roman Chuhay (ICEF, CAS, HSE) Marketing via Friends 2011 8 / 20

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Illustration of the monopolist problem

Monopolist chooses optimally characteristic w and price P to maximize profits:

w 0.5

1 P qA 1 P qB

A B B A A A A B B B A B B A A

Preferences frontier Induced network of potential buyers

Roman Chuhay (ICEF, CAS, HSE) Marketing via Friends 2011 8 / 20

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Illustration of the monopolist problem

Monopolist chooses optimally characteristic w and price P to maximize profits:

w 0.75

1 P qA 1 P qB

A B B A A A A B B B A B B A A

Preferences frontier Induced network of potential buyers

Roman Chuhay (ICEF, CAS, HSE) Marketing via Friends 2011 8 / 20

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Illustration of the monopolist problem

Monopolist chooses optimally characteristic w and price P to maximize profits:

w 1.

1 P qA 1 P qB

A B B A A A A B B B A B B A A

Preferences frontier Induced network of potential buyers

Roman Chuhay (ICEF, CAS, HSE) Marketing via Friends 2011 8 / 20

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Illustration of the monopolist problem

Monopolist chooses optimally characteristic w and price P to maximize profits:

P P

1 P qA 1 P qB

A B B A A A A B B B A B B A A

Preferences frontier Induced network of potential buyers

Roman Chuhay (ICEF, CAS, HSE) Marketing via Friends 2011 8 / 20

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Cascade of sales per advertisement

We modify Newman’s mixing patterns model by incorporating consumers decision to buy the product (probability that a node is operational).

Roman Chuhay (ICEF, CAS, HSE) Marketing via Friends 2011 9 / 20

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Cascade of sales per advertisement

We modify Newman’s mixing patterns model by incorporating consumers decision to buy the product (probability that a node is operational). Expected size of cascade of sales per advertisement: s(qA, qB, ρ, z1, z2, γ) = (γ 1 − γ)

  • I + z2

1

z2

  • I − z2

z1

  • qAρ

qA(1 − ρ) qB(1 − ρ) qBρ −1 − I qA qB

  • where z1 and z2 are expected numbers of first and second neighbors.

Roman Chuhay (ICEF, CAS, HSE) Marketing via Friends 2011 9 / 20

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The global cascade phase

The global cascade: advertisement to infinitesimal part of the population leads to acquisition of the product by non-zero proportion of the population.

Roman Chuhay (ICEF, CAS, HSE) Marketing via Friends 2011 10 / 20

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The global cascade phase

The global cascade: advertisement to infinitesimal part of the population leads to acquisition of the product by non-zero proportion of the population.

Proposition

If z2

z1 > min{2, ρ−1} there exist combinations of product characteristic and price

(w, P) such that global cascade of sales arises.

Roman Chuhay (ICEF, CAS, HSE) Marketing via Friends 2011 10 / 20

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The global cascade phase

The global cascade: advertisement to infinitesimal part of the population leads to acquisition of the product by non-zero proportion of the population.

Proposition

If z2

z1 > min{2, ρ−1} there exist combinations of product characteristic and price

(w, P) such that global cascade of sales arises. Necessary condition for existence of the giant component of connected consumers, z2

z1 > 1.

Roman Chuhay (ICEF, CAS, HSE) Marketing via Friends 2011 10 / 20

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The global cascade phase

The global cascade: advertisement to infinitesimal part of the population leads to acquisition of the product by non-zero proportion of the population.

Proposition

If z2

z1 > min{2, ρ−1} there exist combinations of product characteristic and price

(w, P) such that global cascade of sales arises. Necessary condition for existence of the giant component of connected consumers, z2

z1 > 1.

The existence of the global cascade in the case when z2

z1 < 2 hinges on the

homophily level.

Roman Chuhay (ICEF, CAS, HSE) Marketing via Friends 2011 10 / 20

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Optimal design strategy:

1 2

1 Ρ

1 2

1 w

Proposition

The optimal characteristic of the product is the following correspondence: w ∗ =    [0, 1], ρ = 1

2

1/2, ρ < 1

2

{0, 1}, ρ > 1

2

Roman Chuhay (ICEF, CAS, HSE) Marketing via Friends 2011 11 / 20

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Optimal design strategy: Intuition

ρ = 0 ρ = 1

Roman Chuhay (ICEF, CAS, HSE) Marketing via Friends 2011 12 / 20

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Optimal design strategy: Intuition

ρ = 0 ρ = 1

Case ρ = 0:

◮ All neighbors are of different type. ◮ Spreading depends on the attractiveness of the product to both types. Roman Chuhay (ICEF, CAS, HSE) Marketing via Friends 2011 12 / 20

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Optimal design strategy: Intuition

ρ = 0 ρ = 1

Case ρ = 0:

◮ All neighbors are of different type. ◮ Spreading depends on the attractiveness of the product to both types.

Case ρ = 1:

◮ Two clusters of consumers of the same type. ◮ Specialized design is optimal. Roman Chuhay (ICEF, CAS, HSE) Marketing via Friends 2011 12 / 20

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Optimal pricing strategy:

P PFI 0.5 Ρ 0.5 0.32 P

Proposition

The optimal price P∗ is lower than in the case of full information and for ρ > 1

2 is strictly decreasing function in the level of homophily.

For two degree distributions p(k) and p’(k) and corresponding optimal prices P∗ and P′∗ if p(k) is a mean preserving spread of p’(k) then P∗ < P′∗.

Roman Chuhay (ICEF, CAS, HSE) Marketing via Friends 2011 13 / 20

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Demand function

Q(P, ρ, z1, z2) =   

1−P 2

  • 1 +

z1(1−P) 2−z2/z1(1−P)

  • ,

ρ ≤ 1

2 1−P 2

  • 1 +

z1(1−P)

1 ρ −z2/z1(1−P)

  • ,

ρ > 1

2

Roman Chuhay (ICEF, CAS, HSE) Marketing via Friends 2011 14 / 20

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Demand function

Q(P, ρ, z1, z2) =   

1−P 2

  • 1 +

z1(1−P) 2−z2/z1(1−P)

  • ,

ρ ≤ 1

2 1−P 2

  • 1 +

z1(1−P)

1 ρ −z2/z1(1−P)

  • ,

ρ > 1

2

Proposition

The demand function Q(P, ρ, z1, z2) has following properties:

1

Decreasing and convex in price P.

2

Increasing and convex in homophily level ρ, for ρ > 1

2.

3

The absolute value of the price elasticity of demand is:

P 1 − P

  • 1 + z1
  • 1

z1 − (1 − P)z2ρ − 1 z1 + (1 − P)(z12 − z2)ρ

  • ,

which is higher than price elasticity in the case of full information

P 1−P and is

increasing in homophily level ρ, for ρ > 1

2.

Roman Chuhay (ICEF, CAS, HSE) Marketing via Friends 2011 14 / 20

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Demand: Intuition

Demand is decreasing and convex in P.

A 1 B 2 B 3 A 4 A 5 A 6 B 7 B 8 B 9 B10 A11 B12 B13 A14 A15

Induced network of buyers.

Roman Chuhay (ICEF, CAS, HSE) Marketing via Friends 2011 15 / 20

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Demand: Intuition

Demand is decreasing and convex in P.

A 1 B 2 B 3 A 4 A 5 A 6 B 7 B 8 B 9 B10 A11 B12 B13 A14 A15

Induced network of buyers.

Roman Chuhay (ICEF, CAS, HSE) Marketing via Friends 2011 15 / 20

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Demand: Intuition

Demand is decreasing and convex in P.

A 1 B 2 B 3 A 4 A 5 A 6 B 7 B 8 B 9 B10 A11 B12 B13 A14 A15

Induced network of buyers.

Roman Chuhay (ICEF, CAS, HSE) Marketing via Friends 2011 15 / 20

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Demand: Intuition

Demand is decreasing and convex in P.

A 1 B 2 B 3 A 4 A 5 A 6 B 7 B 8 B 9 B10 A11 B12 B13 A14 A15

Induced network of buyers.

Roman Chuhay (ICEF, CAS, HSE) Marketing via Friends 2011 15 / 20

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Demand: Intuition

Demand is decreasing and convex in P.

A 1 B 2 B 3 A 4 A 5 A 6 B 7 B 8 B 9 B10 A11 B12 B13 A14 A15

Induced network of buyers.

Roman Chuhay (ICEF, CAS, HSE) Marketing via Friends 2011 15 / 20

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SLIDE 55

Demand: Intuition

Demand is decreasing and convex in P.

A 1 B 2 B 3 A 4 A 5 A 6 B 7 B 8 B 9 B10 A11 B12 B13 A14 A15

Induced network of buyers.

Roman Chuhay (ICEF, CAS, HSE) Marketing via Friends 2011 15 / 20

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SLIDE 56

Demand: Intuition

Demand is decreasing and convex in P.

A 1 B 2 B 3 A 4 A 5 A 6 B 7 B 8 B 9 B10 A11 B12 B13 A14 A15

Induced network of buyers.

Roman Chuhay (ICEF, CAS, HSE) Marketing via Friends 2011 15 / 20

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Welfare

Proposition

Consumers and producers surplus are increasing functions in the homophily level

  • f the society.

Roman Chuhay (ICEF, CAS, HSE) Marketing via Friends 2011 16 / 20

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Welfare

Proposition

Consumers and producers surplus are increasing functions in the homophily level

  • f the society.

Monopolist surplus is increasing in the level of homophily

Roman Chuhay (ICEF, CAS, HSE) Marketing via Friends 2011 16 / 20

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Welfare

Proposition

Consumers and producers surplus are increasing functions in the homophily level

  • f the society.

Monopolist surplus is increasing in the level of homophily

◮ Demand is increasing in the level of homophily.

PS(P∗(ρ), ρ, z1, z2) = P∗(ρ) × Q(P∗(ρ), ρ, z1, z2)

Roman Chuhay (ICEF, CAS, HSE) Marketing via Friends 2011 16 / 20

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Welfare

Proposition

Consumers and producers surplus are increasing functions in the homophily level

  • f the society.

Monopolist surplus is increasing in the level of homophily

◮ Demand is increasing in the level of homophily.

PS(P∗(ρ), ρ, z1, z2) = P∗(ρ) × Q(P∗(ρ), ρ, z1, z2)

Consumer surplus is increasing in the level of homophily

Roman Chuhay (ICEF, CAS, HSE) Marketing via Friends 2011 16 / 20

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Welfare

Proposition

Consumers and producers surplus are increasing functions in the homophily level

  • f the society.

Monopolist surplus is increasing in the level of homophily

◮ Demand is increasing in the level of homophily.

PS(P∗(ρ), ρ, z1, z2) = P∗(ρ) × Q(P∗(ρ), ρ, z1, z2)

Consumer surplus is increasing in the level of homophily

◮ Demand is increasing - more consumers buy the product. ◮ The optimal price is decreasing in the level of homophily.

CS(P∗(ρ), ρ, z1, z2) = 1

P∗(ρ)

Q(P, ρ, z1, z2)dP

Roman Chuhay (ICEF, CAS, HSE) Marketing via Friends 2011 16 / 20

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Model Extensions

Targeted advertisement.

◮ Targeting advertisement is always optimal. ◮ For high enough connectivity the optimal design is the same as without

targeting.

Roman Chuhay (ICEF, CAS, HSE) Marketing via Friends 2011 17 / 20

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Model Extensions

Targeted advertisement.

◮ Targeting advertisement is always optimal. ◮ For high enough connectivity the optimal design is the same as without

targeting.

Non-linear probability frontier.

◮ Bend inward frontier - results are the same. ◮ Bend outward frontier - similar shape, but design is continuous. Roman Chuhay (ICEF, CAS, HSE) Marketing via Friends 2011 17 / 20

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Model Extensions

Targeted advertisement.

◮ Targeting advertisement is always optimal. ◮ For high enough connectivity the optimal design is the same as without

targeting.

Non-linear probability frontier.

◮ Bend inward frontier - results are the same. ◮ Bend outward frontier - similar shape, but design is continuous.

Monopolist benefits from one group.

◮ Low levels of homophily - compromise design is still optimal. ◮ High levels of homophily - compromise design is optimal when audience is

small.

Roman Chuhay (ICEF, CAS, HSE) Marketing via Friends 2011 17 / 20

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Selling to one type.

Price P is fixed and the monopolist maximizes sales to consumers of type B.

Roman Chuhay (ICEF, CAS, HSE) Marketing via Friends 2011 18 / 20

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SLIDE 66

Selling to one type.

Price P is fixed and the monopolist maximizes sales to consumers of type B.

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Ρ w

Proposition

There is threshold value ˆ ρ1(z1, z2) such that if ρ < ˆ ρ1(z1, z2) the optimal characteristic w ∗

1 belongs to the interval

  • 0, 1

2

  • , while if ρ > ˆ

ρ1(z1, z2) then w ∗

1 = 0

Roman Chuhay (ICEF, CAS, HSE) Marketing via Friends 2011 18 / 20

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SLIDE 67

Selling to one type.

Consumers of type A constitute 80% of the population (γ = 0.8). The monopolist maximizes sales to consumers of type B. The solution:

0.75 0.80 0.85 0.90 0.95 1.00 0.0 0.2 0.4 0.6 0.8 1.0 ΡA w

Roman Chuhay (ICEF, CAS, HSE) Marketing via Friends 2011 19 / 20

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SLIDE 68

Selling to one type.

Consumers of type A constitute 80% of the population (γ = 0.8). The monopolist maximizes sales to consumers of type B. The solution:

0.75 0.80 0.85 0.90 0.95 1.00 0.0 0.2 0.4 0.6 0.8 1.0 ΡA w

Roman Chuhay (ICEF, CAS, HSE) Marketing via Friends 2011 19 / 20

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Conclusions

Roman Chuhay (ICEF, CAS, HSE) Marketing via Friends 2011 20 / 20

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Conclusions

For low levels of homophily the compromise design of product is preferred to specialized products even if there is no cost of producing more than one type

  • f product.

Roman Chuhay (ICEF, CAS, HSE) Marketing via Friends 2011 20 / 20

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SLIDE 71

Conclusions

For low levels of homophily the compromise design of product is preferred to specialized products even if there is no cost of producing more than one type

  • f product.

Price elasticity of demand is increasing in the homophily level.

Roman Chuhay (ICEF, CAS, HSE) Marketing via Friends 2011 20 / 20

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SLIDE 72

Conclusions

For low levels of homophily the compromise design of product is preferred to specialized products even if there is no cost of producing more than one type

  • f product.

Price elasticity of demand is increasing in the homophily level. Monopolist and consumers benefit from increase in the level of homophily.

Roman Chuhay (ICEF, CAS, HSE) Marketing via Friends 2011 20 / 20

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SLIDE 73

Conclusions

For low levels of homophily the compromise design of product is preferred to specialized products even if there is no cost of producing more than one type

  • f product.

Price elasticity of demand is increasing in the homophily level. Monopolist and consumers benefit from increase in the level of homophily. A product designed to attract both types of consumers may be optimal even if a monopolist benefits only from one group of consumers.

Roman Chuhay (ICEF, CAS, HSE) Marketing via Friends 2011 20 / 20