Introduction
Related Literature
Homogenous preferences
Heterogenous Preferences
The Value of Network Information:
Assortative Mixing Makes the Difference
Mohamed Belhaj1
1 Aix-Marseille
2 Aix-Marseille
Frédéric Deroïan2
School of Economics, Centrale Marseille
School of Economics, GREQAM and CNRS
5th IOSE Conference, June 28 2016, Saint-Petersburg
Mohamed Belhaj, Frédéric Deroïan
Assortative Mixing
Summary
Introduction
Related Literature
Homogenous preferences
Heterogenous Preferences
Consumption decisions often interdependent:
Dropbox
Phone
Online game
Membership in social club (e.g., online dating services)
Video-conference technology
Purchase based on recommendation by friends
⇒ The network of interaction shapes individual decisions
Mohamed Belhaj, Frédéric Deroïan
Assortative Mixing
Summary
Introduction
Related Literature
Homogenous preferences
Heterogenous Preferences
Summary
Numerical technologies provide information on network
structure and individual characteristics
Consumers: Facebook, LinkedIN... observe the relation of
your relation, their purchases, ...
Firms: Big Data, Social Network Analysis Softwares
Mohamed Belhaj, Frédéric Deroïan
Assortative Mixing
Introduction
Related Literature
Homogenous preferences
Heterogenous Preferences
Summary
Information on network structure modifies consumer
decisions and firms’ strategies
Is information on network structure beneficial to
firms? to consumers?
Mohamed Belhaj, Frédéric Deroïan
Assortative Mixing
Introduction
Related Literature
Homogenous preferences
Heterogenous Preferences
Summary
Information on network structure modifies consumer
decisions and firms’ strategies
Is information on network structure beneficial to
firms? to consumers?
Mohamed Belhaj, Frédéric Deroïan
Assortative Mixing
Introduction
Related Literature
Homogenous preferences
Heterogenous Preferences
SETUP: Monopoly pricing, local complementarities,
heterogenous preferences for the good
Complete information (network and preferences known)
Incomplete information (only joint distribution of types
known)
We compare:
Monopoly profit
Consumer surplus (CS)
Mohamed Belhaj, Frédéric Deroïan
Assortative Mixing
Summary
Introduction
Related Literature
Homogenous preferences
Heterogenous Preferences
SETUP: Monopoly pricing, local complementarities,
heterogenous preferences for the good
Complete information (network and preferences known)
Incomplete information (only joint distribution of types
known)
We compare:
Monopoly profit
Consumer surplus (CS)
Mohamed Belhaj, Frédéric Deroïan
Assortative Mixing
Summary
Introduction
Related Literature
Homogenous preferences
Heterogenous Preferences
Main Results
The value of network information is positive when there is
Homogenous preferences:
Profit and CS: degree assortativity
Heterogenous preferences:
Profit:
Degree assortativity
Homophily
Preference/degree assortativity
CS: 3 above cdts + Preference/Bonacich centrality
assortativity
Almost all our conditions are independent of the intensity
of interaction
Mohamed Belhaj, Frédéric Deroïan
Assortative Mixing
Summary
Introduction
Related Literature
Homogenous preferences
Heterogenous Preferences
Main Results
The value of network information is positive when there is
Homogenous preferences:
Profit and CS: degree assortativity
Heterogenous preferences:
Profit:
Degree assortativity
Homophily
Preference/degree assortativity
CS: 3 above cdts + Preference/Bonacich centrality
assortativity
Almost all our conditions are independent of the intensity
of interaction
Mohamed Belhaj, Frédéric Deroïan
Assortative Mixing
Summary
Introduction
Related Literature
Homogenous preferences
Heterogenous Preferences
Main Results
The value of network information is positive when there is
Homogenous preferences:
Profit and CS: degree assortativity
Heterogenous preferences:
Profit:
Degree assortativity
Homophily
Preference/degree assortativity
CS: 3 above cdts + Preference/Bonacich centrality
assortativity
Almost all our conditions are independent of the intensity
of interaction
Mohamed Belhaj, Frédéric Deroïan
Assortative Mixing
Summary
Introduction
Related Literature
Homogenous preferences
Heterogenous Preferences
Summary
Empirical facts about social networks:
degree assortativity (Newman, 2002, Serrano et al, 2007)
Table 1: Degree Assortativity for different networks (Noldus
and Van Mieghem, (JCN, 2015))
Network assortativity
Physics coauthorship
0.363
Mathematics coauthorship
0.120
Company directors
0.276
World-Wide Web
-0.067
Neural network
-0.163
Criminal Network*
'0
Erdős-Rényi (ER) graph
'0
Barabási-Albert (BA) graph
'0
*Liu et al (2012)
homophily (McPherson et al, 2001)
Mohamed Belhaj, Frédéric Deroïan
Assortative Mixing
Introduction
Related Literature
Homogenous preferences
Outline
1
Introduction
2
Related Literature
3
Homogenous preferences
The Game
Nash Equilibrium Outcomes
Value of Network Information
4
Heterogenous Preferences
5
Summary
Mohamed Belhaj, Frédéric Deroïan
Assortative Mixing
Heterogenous Preferences
Summary
Introduction
Related Literature
Homogenous preferences
Heterogenous Preferences
Monopoly pricing under network externalities
Complete information: Candogan et al (2012), Bloch and
Querou (2013) (Bonacich centrality)
Incomplete information: Fainmesser and Galeotti (2015)
(degree)
Assortativity and Network
Homophily: Learning (Golub and Jackson, 2012), WOM
communication (Campbell, 2013)
Degree assortativity and network formation: Jackson and
Rogers (2007), Bramoullé et al (2012)
Assortative Matching and Complementarity: Becker
(1973), Durlauf and Seshadri (2003), Benabou (1996)
Mohamed Belhaj, Frédéric Deroïan
Assortative Mixing
Summary
Introduction
Related Literature
Homogenous preferences
The Game
Outline
1
Introduction
2
Related Literature
3
Homogenous preferences
The Game
Nash Equilibrium Outcomes
Value of Network Information
4
Heterogenous Preferences
5
Summary
Mohamed Belhaj, Frédéric Deroïan
Assortative Mixing
Heterogenous Preferences
Summary
Introduction
Related Literature
Homogenous preferences
Heterogenous Preferences
The Game
A two-stage game
Stage 1: Monopoly sets prices P = (p1 , p2 , ..., pn )
Stage 2: Consumers choose quantities (divisible good)
X = (x1 , x2 , .., xn )
P
Profit: i (pi − c)xi (c = 0 for the presentation)
Mohamed Belhaj, Frédéric Deroïan
Assortative Mixing
Summary
Introduction
Related Literature
Homogenous preferences
Heterogenous Preferences
The Game
The Game of Complete Information
Consumer utility (Candogan et al, 2012):
u(xi , x−i ) = xi −
1 2
x
2 i
|{z}
satiety effect
+δ
X
gij xi xj −pi xi
j∈N
|
{z
}
local peer effects
G = (gij ) the matrix of interaction
P
GT = G for the presentation, di = j gij : consumer i’s
degree
δ > 0 ⇒ positive externalities + complementarities
G: public information
Mohamed Belhaj, Frédéric Deroïan
Assortative Mixing
Summary
Introduction
Related Literature
Homogenous preferences
Heterogenous Preferences
The Game
The Game of Incomplete Information
G unknown
Public information: distribution of degrees
Each consumer knows his own degree
Monopoly knows each consumer’s degree (→ price
discrimination)
Mohamed Belhaj, Frédéric Deroïan
Assortative Mixing
Summary
Introduction
Related Literature
Homogenous preferences
Heterogenous Preferences
Summary
The Game
The Game of Incomplete Information
Bayesian Nash Equilibrium
T = {1, 2, ..., n − 1} the set of consumer types, i.e. degrees
st : nb of consumers of degree t ∈ T in network G
xt : consumption of degree t (assumed identical for
consumers of same degree)
g = 1T .G.1: sum of degrees
Mohamed Belhaj, Frédéric Deroïan
Assortative Mixing
Introduction
Related Literature
Homogenous preferences
Heterogenous Preferences
Summary
The Game
The Game of Incomplete Information
Expected utility of consumer i:
EUi (X ) = (1 − pi )xi −
X
xi2
+ δ xi .di .
2
t∈T
|
t st
xt
g
|{z}
Proba of degree-t neighbor
{z
}
av. neighbor consumption
X function of the statistics of the distribution of degrees
Mohamed Belhaj, Frédéric Deroïan
Assortative Mixing
Introduction
Related Literature
Homogenous preferences
Heterogenous Preferences
Summary
The Game
Modified Game of Complete Information
Equivalence with a modified game of complete information with
T
interaction matrix H = D gD
P
P
t st xt =
dj xj
t∈T
j∈N
EUi (X ) = (1 − pi )xi −
xi2
2
+ δ xi
P
j∈N
hij xj , where hij =
di dj
g
P
∀i, di = hij : for each consumer, same degree in both G
j
and H
Mohamed Belhaj, Frédéric Deroïan
Assortative Mixing
Introduction
Related Literature
Homogenous preferences
Nash Equilibrium Outcomes
Outline
1
Introduction
2
Related Literature
3
Homogenous preferences
The Game
Nash Equilibrium Outcomes
Value of Network Information
4
Heterogenous Preferences
5
Summary
Mohamed Belhaj, Frédéric Deroïan
Assortative Mixing
Heterogenous Preferences
Summary
Introduction
Related Literature
Homogenous preferences
Heterogenous Preferences
Summary
Nash Equilibrium Outcomes
Nash Equilibrium
W ∈ {G, H}
FOC, consumer i on network W :
X
xiBR = 1 − pi + δ
wij xj
j∈N
Assumption: δ × max(µ(G), µ(H)) < 1
(µ largest eigenvalue)
⇒ Existence and uniqueness
Mohamed Belhaj, Frédéric Deroïan
Assortative Mixing
(1)
Introduction
Related Literature
Homogenous preferences
Heterogenous Preferences
Nash Equilibrium Outcomes
Equilibrium Outcomes
Consider W ∈ {G, H}
B(W ) = (I − δW )−1 1 = 1 + δW 1 + δ 2 W 2 1 + δ 3 W 3 1 + ...
Proposition (Candogan et al)
Prices: pi (W ) =
1
2
Consumptions: xi (W ) = 21 Bi (W )
Mohamed Belhaj, Frédéric Deroïan
Assortative Mixing
Summary
Introduction
Related Literature
Homogenous preferences
Heterogenous Preferences
Nash Equilibrium Outcomes
Equilibrium Outcomes
Consider W ∈ {G, H}
B(W ) = 1 + δW 1 + δ 2 W 2 1 + δ 3 W 3 1 + ...
X
T
2
T
3
T
3
Bi (W )= n+δ 1
W 1} +δ2 1
| W
{z 1} +δ 1
| W
{z 1} +...
| {z
nb path of length 1
nb path of length 2
nb path of length 3
| i {z }
sum of centralities
Proposition (Candogan et al)
Profit: Π(W ) =
1
4
P
Bi (W )
i
Consumer surplus: CS(W ) =
Mohamed Belhaj, Frédéric Deroïan
Assortative Mixing
1
8
P
i
2
Bi (W )
Summary
Introduction
Related Literature
Homogenous preferences
Value of Network Information
Outline
1
Introduction
2
Related Literature
3
Homogenous preferences
The Game
Nash Equilibrium Outcomes
Value of Network Information
4
Heterogenous Preferences
5
Summary
Mohamed Belhaj, Frédéric Deroïan
Assortative Mixing
Heterogenous Preferences
Summary
Introduction
Related Literature
Homogenous preferences
Value of Network Information
White (2 links), Black (3 links)
Same degree distribution:
Mohamed Belhaj, Frédéric Deroïan
Assortative Mixing
Heterogenous Preferences
Summary
Introduction
Related Literature
Homogenous preferences
Heterogenous Preferences
Value of Network Information
⇒ Does degree assortativity matter?
Mohamed Belhaj, Frédéric Deroïan
Assortative Mixing
Summary
Introduction
Related Literature
Homogenous preferences
Heterogenous Preferences
Value of Network Information
Degree assortativity measures the tendency for nodes of
similar degrees to be connected by an edge
⇒ Pearson coefficient rD (G) (Newman, 2003)
Pearson
Proposition
Let H =
DD T
g
, Θ = G − H. We have
rD (G) proportional to D T ΘD
⇒ Degree Assortativity if D T ΘD > 0
Mohamed Belhaj, Frédéric Deroïan
Assortative Mixing
Summary
Introduction
Related Literature
Homogenous preferences
Heterogenous Preferences
Summary
Value of Network Information
Small Interaction
Recall
4 × Π(G)=n+ δ1T G1+δ 2 1T G2 1+δ 3 1T G3 1+δ 4 ..
4 × Π(H)=n+ δ1T H1+δ 2 1T H2 1+δ 3 1T H3 1+δ 4 ..
D = H.1 = G.1 ⇒:
1T (G − H)1 = 0
1T (G2 − H 2 )1 = 0
⇒
4 × (Π(G) − Π(H)) = δ 3 1T (G3 − H 3 )1 + δ 4 1T (G4 − H 4 )1 + ...
Mohamed Belhaj, Frédéric Deroïan
Assortative Mixing
Introduction
Related Literature
Homogenous preferences
Heterogenous Preferences
Value of Network Information
Small Interaction
δ'0:
ΠG − ΠH ' 1T (G3 − H 3 )1 = D T (G − H)D = D T ΘD
Degree assortativity discriminates
Mohamed Belhaj, Frédéric Deroïan
Assortative Mixing
Summary
Introduction
Related Literature
Homogenous preferences
Heterogenous Preferences
Summary
Value of Network Information
Theorem
If rD (G) > 0, we have both Π(G) > Π(H) and CS(G) > CS(H)
1
for all δ ∈]0, µ(G)
[.
Proof
The assortativity condition is independent of the intensity
of interaction
The assortativity condition is a necessary condition for
small interaction if ΘD 6= 0
Mohamed Belhaj, Frédéric Deroïan
Assortative Mixing
Introduction
Related Literature
Homogenous preferences
Heterogenous Preferences
Value of Network Information
Variance of consumption levels
Proposition
When rD (G) > 0, the network information increases the
variance of consumption levels.
Proof
Possible implications on inequalities?
Mohamed Belhaj, Frédéric Deroïan
Assortative Mixing
Summary
Introduction
Related Literature
Homogenous preferences
Heterogenous Preferences
Heterogenous Preferences
A = (a1 , a2 , · · · , an )
u(xi , x−i ) =
ai xi −
|{z}
preference
1 2
x
2 i
|{z}
satiety effect
+δ
X
gij xi xj −pi xi
j∈N
|
{z
}
local peer effects
Mohamed Belhaj, Frédéric Deroïan
Assortative Mixing
Summary
Introduction
Related Literature
Homogenous preferences
Heterogenous Preferences
Heterogenous Preferences
On network W ∈ {G, H}:
Prices: pi = a2i
P
P
Demand: xi = 12 BA,i
i
i
where BA = A + δWA + δ 2 W 2 A + ...
P
P
Profit: Π = 12 ai xi = 14 ai BA,i
i
i
Consumer surplus: CS =
Mohamed Belhaj, Frédéric Deroïan
Assortative Mixing
1P 2
xi
2
i
=
1
8
P
i
BA,i
2
Summary
Introduction
Related Literature
Homogenous preferences
Heterogenous Preferences
Summary
Assortative mixing between two characteristics
Degree Assortativity: rD (G) > 0 if D T ΘD > 0
Preference Assortativity: rA (G) > 0 if AT ΘA > 0
we generalize Pearson assortativity to 2 characteristics:
Preference / Degree Assortativity: rA,D (G) > 0 if AT ΘD > 0
Mohamed Belhaj, Frédéric Deroïan
Assortative Mixing
Introduction
Related Literature
Homogenous preferences
Heterogenous Preferences
Profit
Theorem
p
If rA (G) > 0, rD (G) > 0 and rA,D (G) > − rA (G) · rD (G), then
1
Π(G) > Π(H) for all δ ∈]0, µ(G)
[.
Proof:
Y = A + γD
∆Π > 0 if Y T ΘY ≥ 0
Mohamed Belhaj, Frédéric Deroïan
Assortative Mixing
Summary
Introduction
Related Literature
Homogenous preferences
Heterogenous Preferences
Summary
Do these three conditions guarantee an increase of consumer
surplus? NO! Aggregate demand can decrease
Mohamed Belhaj, Frédéric Deroïan
Assortative Mixing
Introduction
Related Literature
Homogenous preferences
Heterogenous Preferences
Consumer Surplus
Theorem
p
If rA (G) > 0, rD (G) > 0, rA,D (G) > − rA (G) · rD (G), and
rA,B(G) (G) > 0, we have CS(G) > CS(H).
Preference / Bonacich centrality assortativity guarantees an
increase of demand
rA,B(G) (G) depends on δ
Mohamed Belhaj, Frédéric Deroïan
Assortative Mixing
Summary
Introduction
Related Literature
Homogenous preferences
Heterogenous Preferences
Consumer Surplus
Conjecture: rA (G) ≥ 0, rD (G) ≥ 0 , rA,D (G) ≥ 0 altogether
guarantee an increase of CS?
Mohamed Belhaj, Frédéric Deroïan
Assortative Mixing
Summary
Introduction
Related Literature
Homogenous preferences
Heterogenous Preferences
Consumer Surplus
Networks satisfying ΘD = 0 (including regular networks):
rD (G) = 0, rD,B (G) = 0 (ΘD = 0)
∆Π > 0 and ∆CS > 0 if rA (G) > 0
Aggregate demand is constant
Mohamed Belhaj, Frédéric Deroïan
Assortative Mixing
Summary
Introduction
Related Literature
Homogenous preferences
Heterogenous Preferences
Summary
The value of information on the network structure is positive
under assortative mixing
Equivalence between game of incomplete information and
modified game of complete information
Relation between assortativity and the difference between
adjacency matrix of the two games
Mohamed Belhaj, Frédéric Deroïan
Assortative Mixing
Summary
Introduction
Related Literature
Homogenous preferences
Thank you!
Mohamed Belhaj, Frédéric Deroïan
Assortative Mixing
Heterogenous Preferences
Summary
Introduction
Related Literature
Homogenous preferences
Heterogenous Preferences
Knowing In-degree
Game of incomplete information
Consumer know the distribution of in-degree (they assume no
correlation between in-degrre and out-degree)
σk ,l : the covariance between in-degrees and out-degrees in
network G.
rK (G): In-degree Assortativity
Proposition
We have Π(G) ≥ Π(H̄) if both rK (G) > 0 and ρk ,l ≥ 0.
Mohamed Belhaj, Frédéric Deroïan
Assortative Mixing
Summary
Introduction
Related Literature
Homogenous preferences
Heterogenous Preferences
Summary
Partial Information
Game of complete information
Consumers know the number of links between each two types
(G = GT )
Q: the corresponding matrix of interaction (by agent) (Qij =
where si is the number of agents of type i) and ψij is the
number of links between type i and type j)
The Network Q has the same ‘assortativity’ as G.
All results (except CS/ heterogenous consumers) apply to
outcomes under Q versus outcomes under H
Mohamed Belhaj, Frédéric Deroïan
Assortative Mixing
ψij
si sj ,
Introduction
Related Literature
Homogenous preferences
Heterogenous Preferences
Z , Z 0 two vectors of scalar (possibly distinct) characteristics
The Pearson coefficient (Newman, 2003) generalized to two
characteristics:
rZ ,Z 0 (G) =
1 X 0
zz
σa σb 0
z,z
(e 0 − a b 0 )
| zz {z z z }
excess proportion of zz 0 -type links in G / random
ezz 0 : fraction of all edges in network G that join together
vertices of characteristics z and z 0
az the fraction of edges that start with a characteristic z
bz 0 the fraction of edges that end with a characteristic z 0
σa , σb standard deviations of the distributions az , bz 0
Proposition
Mohamed Belhaj, Frédéric Deroïan
Assortative Mixing
Summary
Introduction
Related Literature
Homogenous preferences
Heterogenous Preferences
Summary
Proof
X = 12 (I − δG)−1 1 = M1
Y = 12 (I − δH)−1 1
X = Y + δMΘY
We show that:
Π(G) − Π(H) = δ 2 k(G − H)Y k2M + δY T ΘY
Y = 1 + αD and Θ.1 = 0
Π(G) − Π(H) > 0 if D T ΘD ≥ 0
2
CS(G) − CS(H) = δ2 kM(G − H)Y k2 + 1 + νδ Π(G) − Π(H)
(not true under heterogeneous agents)
Theorem
Mohamed Belhaj, Frédéric Deroïan
Assortative Mixing
Introduction
Related Literature
Homogenous preferences
Heterogenous Preferences
Summary
Proof
Var (X ) = X T X −
(1T X )2
,
n
X = Y + δMΘY
#
1 T
T
2
2
2
⇒ Var (X ) − Var (Y ) = δ Y ΘM ΘY − (1 MΘY )
n
|
{z
}
=Var (MΘY )>0
1
+ 2 δ Y T MΘY − (1T Y )1T MΘY
n
|
{z
}
=φ
Y = 1 + αD ⇒ φ = α
1−δd
δ
X T ΘY (where d is average degree)
δd < 1 (as d ≤ µ(G)), thus φ > 0 when X T ΘY > 0, i.e.
rD (G) > 0
Proposition
Mohamed Belhaj, Frédéric Deroïan
Assortative Mixing