Behaviour in Social Networks:Externalities, Altruism and Peer Effects

S. Currarini, E. Fumagalli and F. Panebianco

FEEM, April 2013

Motivation

Peer effects: behavioural complementarities due to• joint consumption• emulation• heard behaviour• ...

Incentive to act increases in neighbors’ actions.Common in risky or health related behavior:• adolescent groups (Powell et al., 2005)• smoking (Christakis and Fowler, 2007; Ali et al., 2011)• alcohol and drug consumption (Clark and Loheac,

2007, Mednick et al., 2010)• food disorders and obesity

”social multiplier” - Glaeser, Sacerdote andSchenkman, 2002, Cutler and Glaeser, 2007).Identification and measurement of peer effects(Manski 1993, Bramouille et al. 2009, Lee 2010)

Behaviors that generate peer effects often exertnegative externalities:• passive smoke• driving while drunk• sexual transmission of disease and unintended

pregnancies;• local pollution

Externalities build up and the stocks affects welfareand incentives:• own stock: e.g., passive smoke affects risk of illness

due to extra-cigarette;• neighbours’stocks: e.g., concern for friends’ health or

altruism).

Externalities and stocks create additional interactionchannels, as incentives to act may depend(negatively) on actions taken outside one’sneighborhood.

Smoking networks

Agents enjoy active smoke, and are affected bypassive smoke;The stock of passive smoke increases the risk ofillness due to an additional cigarette;Agents care for the effect of their smoke on theirfriends’ health (altruism);Incentive to smoke decrease with their friends offriends’ smoke.

d  

Figure : Indirect substitution effect

Stylized facts about smoking our research contributesto account for (Christakis and Fowler, 2007):• Smokers tend to move towards periphery of the

network.• Network affects behavior or behavior affects the

network?• C-F argue that marginalization of smokers is due to

link deletion - here we suggest that network affectsbehavior.

• People tend to quit smoking in clusters. Who? Why?How does position in the network affect smokingbehavior?

Theoretical issues:• Estimation of peer effects and causality;• Choice of instrumental variables;• Identification: which networks?

Local complementarities in production

Monopolists use neighbors’ products as inputs;Increases in i’s output increases demand for j’sproducts.Increases in j’s neighbor’s output increase demandfor j’s output, j’s price and i’s marginal cost.

International RelationsCountries take actions that affect neighbors;A country decides to react if the aggregate actionstaken by its neighbors exceeds a given threshold;If j and i are neighbors, and increase in j’s neighbor’sactions increases the probability that i’s additionalaction meets j’s threshold.

The Model

n agents in a fixed social network G with gij = {0, 1}.Each agent i takes action xi ∈ R.

Utility

Ui = αixi − γ0x2

i2 + φ ∑j∈N gijxixj

Figure : The network of social relations and peer effects

The Model

n agents in a fixed social network G with gij = {0, 1}.Each agent i takes action xi ∈ R.

Utility

Ui = αixi − γ0x2

i2 + φ ∑j∈N gijxixj − γ1

Q2i

2 − γ2 ∑j∈N gijQ2

j2

where for all h ∈ N we have defined

Qh ≡(

∑k∈N

ghkxk + xh

)

Setting φ̄ = φ− γ1 − γ2:

αxi − γ0+γ12 x2

i + φ̄ ∑j∈N

gijxixj − γ2 ∑k

g[2]ik xixk + h−i

Equilibrium

FOC for agent i

α− (γ0 + γ1)xi + φ̄ ∑j

gijxj − γ2 ∑k

g[2]ik xk = 0

Eq. FOC’s in matrix form:

α1̄ =[(γ1 + γ0)I− [φ̄G− γ2G2]

]x̄.

orα

γ1 + γ01̄ =

[I− φ̄

γ1 + γ0

(G− γ2

φ̄G2)]

x̄.

Equilibrium Behaviour:

α

γ1 + γ0

[I− φ̄

γ1 + γ0

(G− γ2

φ̄G2)]−1

1̄ = x̄.

Equilibrium

FOC for agent i

α− (γ0 + γ1)xi + φ̄ ∑j

gijxj − γ2 ∑k

g[2]ik xk = 0

Eq. FOC’s in matrix form:

α1̄ =[(γ1 + γ0)I− [φ̄G− γ2G2]

]x̄.

orα

γ1 + γ01̄ =

[I− φ̄

γ1 + γ0

(G− γ2

φ̄G2)]

x̄.

Equilibrium Behaviour:

α

γ1 + γ0

[I− φ̄

γ1 + γ0

(G− γ2

φ̄G2)]−1

1̄ = x̄.

Figure : Peer effects and indirect substitution

DefinitionThe Bonacich centrality matrix of g with parameter a isgiven by:

M(G, a) ≡ (I− aG)−1. (1)

mij = discounted sum of walks from i to j in G.

DefinitionThe vector of Bonacich centralities of g with parameter ais given by:

b(G, a) ≡M(G, a) · 1̄. (2)

Proposition 1 (Adaptation of Ballester et al. 2006)

Let γ1+γ0φ̄

> µ(G− γ2φ̄

G2). The unique interior NashEquilibrium of the game is given by:

x∗ =α

γ1 + γ0b(G− γ2

φ̄G2,

φ− γ1

γ1 + γ0). (3)

Issues and Research Questions

Social relations are given by the matrix G.

Social interaction is described by the matrixG− γ2

φ̄G2.

Centrality relates behavior to G− γ2φ̄

G2.

Need to relate behavior to the original social networkG.How does the ranking of behaviors within a networkdiffer?How does aggregate behavior in different networksG change?Who are the key players (those that if removed affectmost the aggregate behavior)?

Issues and Research Questions

Social relations are given by the matrix G.Social interaction is described by the matrixG− γ2

φ̄G2.

Centrality relates behavior to G− γ2φ̄

G2.

Need to relate behavior to the original social networkG.How does the ranking of behaviors within a networkdiffer?How does aggregate behavior in different networksG change?Who are the key players (those that if removed affectmost the aggregate behavior)?

Issues and Research Questions

Social relations are given by the matrix G.Social interaction is described by the matrixG− γ2

φ̄G2.

Centrality relates behavior to G− γ2φ̄

G2.

Need to relate behavior to the original social networkG.How does the ranking of behaviors within a networkdiffer?How does aggregate behavior in different networksG change?Who are the key players (those that if removed affectmost the aggregate behavior)?

Issues and Research Questions

Social relations are given by the matrix G.Social interaction is described by the matrixG− γ2

φ̄G2.

Centrality relates behavior to G− γ2φ̄

G2.

Need to relate behavior to the original social networkG.

How does the ranking of behaviors within a networkdiffer?How does aggregate behavior in different networksG change?Who are the key players (those that if removed affectmost the aggregate behavior)?

Issues and Research Questions

Social relations are given by the matrix G.Social interaction is described by the matrixG− γ2

φ̄G2.

Centrality relates behavior to G− γ2φ̄

G2.

Need to relate behavior to the original social networkG.How does the ranking of behaviors within a networkdiffer?

How does aggregate behavior in different networksG change?Who are the key players (those that if removed affectmost the aggregate behavior)?

Issues and Research Questions

Social relations are given by the matrix G.Social interaction is described by the matrixG− γ2

φ̄G2.

Centrality relates behavior to G− γ2φ̄

G2.

Need to relate behavior to the original social networkG.How does the ranking of behaviors within a networkdiffer?How does aggregate behavior in different networksG change?

Who are the key players (those that if removed affectmost the aggregate behavior)?

Issues and Research Questions

Social relations are given by the matrix G.Social interaction is described by the matrixG− γ2

φ̄G2.

Centrality relates behavior to G− γ2φ̄

G2.

Need to relate behavior to the original social networkG.How does the ranking of behaviors within a networkdiffer?How does aggregate behavior in different networksG change?Who are the key players (those that if removed affectmost the aggregate behavior)?

DefinitionThe weighted Bonacich centrality vector for G withparameter a and with weights vector w̄ is defined asfollows:

b(G, a, w̄) = M(G, a) · w̄ (4)

Proposition

The vector of equilibrium actions with altruism x̄ is given by:

α

γ1 + γ0b(G,

(φ− γ1)

γ1 + γ0, w̄) = x̄ (5)

where the vector of weights is given by:

w̄ = [I +γ2

γ1 + γ0M(G + G2)]−1 · 1̄. (6)

1

2

3

4

5

67

8

9

10

11

Three types of agents: 1, 2 and 3.

0.075 0.080 0.085 0.090 0.095 0.100Γ2

2

4

6

8

x

x3

x2

x1

Figure : Equilibrium actions with varying degrees of altruism.

Proposition

The marginal effect of γ2 on equilibrium behavior is given by:

∂x̄∗

∂γ2= − ∂

∂φb̄(G,

φ− γ1

γ1 + γ0, d) (7)

1

2 3 4 5

1

2

34

5

1

2

3 4

5

Network Players γ2 = 0 γ2 = 0.06x∗ x∗ x∗ x∗

Star 1 3.38 2.012-5 2.59 13.77 1.82 9.31

Papillon 1 3.53 1.632-5 2.95 15.29 1.69 8.42

Connected Star 1 3.73 1.342-5 3.39 17.29 1.45 7.17

Changing the Network: Density

Look at degree in regular networks.

Proposition

Consider a regular network, with γ1 + γ2 < φ.Equilibrium behaviour is a non monotonic function ofdegree, increasing for low degrees and decreasing for highdegrees. See here

0.1

0.2

0.8

d0.1*d0.2

*

0 1 2 3 4 5 6d

0.05

0.10

0.15

x*

Figure : Larger γ2 at lower curves (α = 2, γ1 = 0.3)

Adding and Severing Links

We know from Ballester et al. (2006) that if weincrease the entries of a network G, the resultingequilibrium is characterized by a larger aggregatebehavior (due to stronger complementarities).But how do we increase the terms of G̃ by adding andsevering links to G?

Proposition

Take r integer such that

(φ− γ1 − γ2) ≤ rγ2.

Consider G′ obtained by fully connecting an independent set ofnodes Z of size z in G, with z = r + 2.Then G̃′ < G̃ and x∗(G̃) > x∗(G̃′).

Key-Players

Theorem (Ballester et al. 2006)

If η(γ1+γ0)λγ2

> µ(C), the key player is the agent with thehighest intercentrality index, measured by ci = b2

i /mii.

1 2 3 4 5

Figure : Line network

Table : Key Player - Line network

γ2 bi ci0 3 > 2 > 1 3 > 2 > 1

0.05 1 > 2 > 3 1 > 2 > 3

Parametrization: φ = 1, γ0 = 0, γ1 = 0.9, α = 2

1

2

3

4

5

67

8

9

10

11

Table : Key Player - Ballester2006 network

γ2 bi ci0 2 > 1 > 3 2 > 1 > 30.002 2 > 1 > 3 2 > 3 > 10.01 2 > 3 > 1 2 > 3 > 10.02 3 > 2 > 1 3 > 2 > 1

Parametrization: φ = 1, γ0 = 0, γ1 = 0.95, α = 2

Policy-specific Key-Players

Definition

The α-key player is the agent i such that ∂x∂αi is maximal.

Proposition

The α-key player is the agent with the highest centralityin the network C.

Homophily, Polarization and Smoking

Two types: αH > αL.Pattern of connections matters for behaviour.Degree of Homophily q = share of same typeneighbors.Assume q is the same across agents and types.

(a)

1

2

3

4

5

6

7

8

(b)

1

2

3

4

5

6

7

8

(c)

12

34

56

78

Figure : Three networks with increasing degrees of homophily.

Equilbrium Behaviour is symmetric around commonmean:

xH = 12[ αH+αL

γ1+γ0−d(φ− γ1 − γ2

)+ γ2d2 + αH−αL

γ1+γ0+γ2d2(

1−2q)2

+d(

1−2q)(

φ−γ1−γ2) ]

xL = 12[ αH+αL

γ1+γ0−d(φ− γ1 − γ2

)+ γ2d2 + αL−αH

γ1+γ0+γ2d2(

1−2q)2

+d(

1−2q)(

φ−γ1−γ2) ] (8)

Proposition

Let d∗ be the degree for which equilibrium behavior ismaximal.

When d < d∗, polarization is monotonicallyincreasing in q.When d > d∗, polarization is non monotone in q,reaching its maximum at q̄ ∈ (1/2, 1], where q̄ isdecreasing in γ1, γ2 and d.The maximal polarization is independent of thedegree.

xh

xl

q

Average Action

0.0 0.2 0.4 0.6 0.8 1.0q

0.01

0.02

0.03

0.04

0.05

0.06

0.07xh xl

Figure : xH and xL

Empirical Analysis

From FOC we get

x = β1Gx + β2G2x + ρz + ζG∗z + ε

where

β1 =(φ− γ1 − γ2)

γ1 + γ0and β2 =

γ2

γ1 + γ0

Biases in the estimation of peer effect

Bias I

If γ2 = 0 then β1 = (φ−γ1)γ1+γ0

: downward bias

Bias IIIf γ2 is omitted then

x = β1Gx + ρz + ζG∗z + ε

andCov(x, Gx)

Var(Gx)= β1 + β2δG2x,Gx < β1

Identification

We analyze conditions of network so that theparameters are identified depending on G = G∗ andG 6= G∗, with and without altruismIf G = G∗, all regular networks are excludedIf G 6= G∗, a larger class of networks are excluded(e.g., star network).We derive an optimal set of instrument to solve theendogeny problem derived from the presence ofendogeneity of Gx and G2x.

Conclusions

Model with peer effects, negative externalities andaltruismAltruism generates new strategic interdependenciesof the substitute type with distance two neighbors.Central agents in a network need not be those withlargest behaviour.Adding links may reduce behavior. In regularnetworks: inverted bell pattern w.r.t. degreeHeterogeneous agents: behaviour is largest for large -but not complete - segregation of typesImplications for empirical work:• Peer effects may be underestimated;• Relation degree-behaviour non linear;• Two-distance neighbours may fail to be valid

instrument for estimating peer-effects.

Matrix G[k] identifies walks of order k

mij =∞

∑k=0

akG[k]

M =

mii mij mik · · · miw

mji mjj mjk · · · mjw

mki mkj mkk · · · mkw

· · · · · · · · · · · · · · ·mwi mwj mwk · · · mww

b̄ =

mii mij mik · · · miw

mji mjj mjk · · · mjw

mki mkj mkk · · · mkw

· · · · · · · · · · · · · · ·mwi mwj mwk · · · mww

·

11111

α− x∗[γ1 + γ0 − d(φ− γ1 − γ2) + γ2d2] = 0.

x∗ =α

γ1 + γ0 − d(φ− γ1 − γ2) + γ2d2 .

∂x∗

∂d=

α(φ− γ1 − γ2 − 2dγ2)

Den2 .

d < φ−γ1−γ2

2γ2⇒ ∂x∗

∂d > 0d = φ−γ1−γ2

2γ2⇒ ∂x∗

∂d = 0d > φ−γ1−γ2

2γ2⇒ ∂x∗

∂d < 0

Let:c = max{g̃ij} = 1c = min{g̃ij}θ = −min{0, c} > 0λ = c + θ

Define the matrix C as:

cij =g̃ij + θ

λ∈ [0, 1]. (9)

Proposition

Consider G and the matrix C defined as in (9). Letη(γ1+γ0)

λγ2> µ(C). The unique Nash equilibrium of the game is

given by:

x̄ =αηb(C, λγ2

η(γ1+γ0))

η(γ1 + γ0) + γ2θb( λγ1

, C). (10)

Rewrite FOC’s as:

x̄ =α

γ1 + γ0b(G,

φ− γ1

γ1 + γ0)− γ2

γ1 + γ0M(G + G2)x̄︸ ︷︷ ︸

z

The term:

zi = ∑j[(gij + g[2]ij )xj]

measures the aggregate equilibrium actions that agents ini’s neighborhood are exposed to.

The correction of equilibrium behavior is higher for thoseagents whose high centrality (in G) comes from paths thatlead to agents who are exposed to large amounts ofexternalities in equilibrium.