Strategic Interaction and Networks
By Yann Bramoullé, Rachel Kranton and Martin D’Amours
Presented by Daniel Chippin
February 12, 2015
Daniel Chippin
Bramoullé, Kranton and D’Amours
February 12, 2015
1 / 17
Summary
A characterization of Nash Equilibrium for games on a network
The best responses are decreasing and linear functions of the
neighbours actions.
Only thing that is necessary is the minimum eigenvalue of the
network.
Daniel Chippin
Bramoullé, Kranton and D’Amours
February 12, 2015
2 / 17
Setup
Basic Model
n agents linked on a network.
gij = gji = 1 if they are neighbours and gij = gji = 0 if they are not.
Each agent has an action xi ≥ 0
Daniel Chippin
Bramoullé, Kranton and D’Amours
February 12, 2015
3 / 17
Setup
Games with Linear Best Repies
The best response function is xi = max(0, 1 − δ
P
gij xj )
Define an active agent as one who has xi > 0, and A as the set of
active agents.
Daniel Chippin
Bramoullé, Kranton and D’Amours
February 12, 2015
4 / 17
Setup
Nash Equilibiria
PROPOSITION 1:
Actions x with active agents A is a Nash equilibrium if and only if
(i): (I + δGA )xA = 1
(ii): δGN−A,A xA ≥ 1
Daniel Chippin
Bramoullé, Kranton and D’Amours
February 12, 2015
5 / 17
Setup
Potential Function
The potential function is ϕ(x; δ, G ) = x T 1 − 12 x T (I + δG )x
NoticePthat the derivative of this function with respect to xi is:
1 − δ gij xj − xi
LEMMA 1:
The set of Nash equilibria for a given G and δ corresponds to the set of
maxima and saddle points of the potential function ϕ(x; δ, G ) on Rn+ .
Daniel Chippin
Bramoullé, Kranton and D’Amours
February 12, 2015
6 / 17
Results
Uniqueness
PROPOSITION 2:
If |λmin (G )| < 1/δ, there is a unique Nash equilibrium
Proof: If the condition is met then ϕ is concave.
PROPOSITION 3
For any δ and any graph G , if |λmin (G )| > 1/δ, there exists at least one
Nash equilibrium with inactive agents.
Proof: If the condition is met then ϕ is not concave so there must be a
direction which ϕ increases without bound.
Daniel Chippin
Bramoullé, Kranton and D’Amours
February 12, 2015
7 / 17
Results
Stability
An equilibrium stable if small changes in the agents’ actions lead back
to the original equilibrium.
LEMMA 2:
The set of stable equilibria for a given G and δ is equal to the strict
maxima of the potential function ϕ(x; δ, G ) on Rn+ . A stable equilibrium
exists for any G and almost any δ
Daniel Chippin
Bramoullé, Kranton and D’Amours
February 12, 2015
8 / 17
Results
More Stability
PROPOSITION 4:
Consider a graph G and a Nash equilibrium x with active agents A and
strictly inactive agents. x is stable if and only if |λmin (GA )| < 1/δ
Daniel Chippin
Bramoullé, Kranton and D’Amours
February 12, 2015
9 / 17
Results
Even More on Stability
PROPOSITION 5:
For |λmin (GA )| > 1/δ, all stable equilibria involve at least one inactive
agent.
PROPOSITION 6:
Consider a stable equilibrium x with active agents A. There is no other
equilibrium x 0 with active agents A0 such that A0 ⊂ A
Daniel Chippin
Bramoullé, Kranton and D’Amours
February 12, 2015
10 / 17
Results
Eigenvalues
|λmin (GA )| tends to be larger when networks are easier to subdivide
into two subgroups, with many intergroup links and few intragroup
links.
Let λmin (GA ) = εT G ε.
P
P
P
λmin (GA ) = εi εj gij + εi εj gij + εi εj gij
PROPOSITION 7:
For any graph G , let ε be an eigenvector for λmin (GA ) and let
R = i : ε ≥ 0 and S = i : ε < 0. Construct G 0 by removing links within R
and S, in any way. Then, |λmin (GA0 )| ≥ |λmin (GA )|.
Daniel Chippin
Bramoullé, Kranton and D’Amours
February 12, 2015
11 / 17
Results
Comparative Statics
PROPOSITION 8:
Consider a δ and G and a highest-aggregate-play equilibrium x ∗ (δ, G ).
Consider a δ 0 and G 0 where δ 0 ≥ δ P
and G is a subgraph
of G 0 and any
P
equilibrium vector x(δ 0 , G 0 ). Then
xi (δ 0 , G 0 ) ≤ xi∗ (δ, G ).
Daniel Chippin
Bramoullé, Kranton and D’Amours
February 12, 2015
12 / 17
Applications
Local Information Externalities
If 1/3 ≤ δ ≤ 1/2.24 then the left network has concentrated
innovation, and the right graph has diffuse innovation.
Daniel Chippin
Bramoullé, Kranton and D’Amours
February 12, 2015
13 / 17
Applications
Crime and Social Networks
P
xj ) − cxi (1 − φ gij xj )
P
Best response is then xi = max(0, 1−c
hij xj )) where
2α −
cφ
1
hij = 2 (1 − α gij )
Vi (xi , x−i ; α, φ, G ) = xi (1 − α
P
The model is calibrated with φ = 0.6 and α = 1
Daniel Chippin
Bramoullé, Kranton and D’Amours
February 12, 2015
14 / 17
Applications
Econometrics
Suppose you want to estimate a model xi = Zi β − δ
the condition that xi ≥ 0
P
gij xj + i with
There exist some techniques do analyse this, but require one to find
all the equilibrium.
Daniel Chippin
Bramoullé, Kranton and D’Amours
February 12, 2015
15 / 17
Applications
General Model
Heterogeneous targets: if |λmin (G )| < 1/δ then an equilibrium is unique
and exists. Otherwise it becomes necessary to implement an
upper bound as well as a lower bound.
Heterogeneous Payoff Impacts: Their results are robust toPa specific type
of weighting. fi (x−i , δi , G ) = max(0, 1 − δi
gij xj )
Daniel Chippin
Bramoullé, Kranton and D’Amours
February 12, 2015
16 / 17
Applications
Conclusion
PROPOSITION 2:
If |λmin (G )| < 1/δ, there is a unique Nash equilibrium
PROPOSITION 4:
Consider a graph G and a Nash equilibrium x with active agents A and
strictly inactive agents. x is stable if and only if |λmin (GA )| < 1/δ
Daniel Chippin
Bramoullé, Kranton and D’Amours
February 12, 2015
17 / 17