Supermodular Network Games
V. Manshadi and R. Johari
Πi(xi, xj, xk) =
u(xi, xj+xk) – c(xi)
Supermodular games:
Games where nodes have strategic
complementarities
Network (or graphical) games:
Games where nodes interact through
network structure
A node’s actions can have significant
effects on distant nodes.
NEW INSIGHTS
j
Payoff of agent i:
i
1
k
We show:
• Membership in larger k-core implies higher actions in
equilibrium
• Higher centrality measure implies higher actions in
equilibrium
• If nodes don’t know network structure, largest
equilibrium depends on edge perspective degree
distribution
HOW IT WORKS:
ASSUMPTIONS AND LIMITATIONS:
We characterize equilibria in terms of
such global measures
We study equilibria of a static game between nodes.
The eventual goal is to understand dynamic network
games.
Local interaction does not imply
weak correlation between far
away nodes in cooperation
settings.
Centrality measures need to be
used to quantify the effect of
the network.
We assume utility exhibits strategic complementarities.
We exploit monotonicity of the best response to
prove our results:
The best action for node i is
increasing in its neighbor’s actions.
Centrality, coreness: Global measures
of power of a node
IMPACT
…
MAIN RESULT:
NEXT-PHASE GOALS
…
STATUS QUO
ACHIEVEMENT DESCRIPTION
This model assumed a static
interaction between the nodes.
Our end-of-phase goal is to
develop dynamic game models of
coordination on networks.
The power of a node in a networked coordination system depends on its centrality
(global properties) not just on its degree (a local property)
Motivation
 This work studies a benchmark model for cooperation in networked
systems.
 We consider large systems where each player only interacts with a
small number of other agents which are close to it. A network
structure governs the interactions.
 Graphical Games [Kearns et al. 02].
 Network Games [Galeotti et al. 08].
 The network structure has a significant effect on the equilibrium
 For what networks can epidemics arise?
 Graph theoretic conditions for a two action game [Morris 00].
 How about more general games? (continuous action space, more
general payoff functions, etc.)
 Does the equilibrium solely depend on local graph properties?
 What if the nodes do not know the entire network?
Model
 N-person game, each player’s action space is [0,1].
 graph G = (V,E) represents the interaction among nodes.
 Node i’s payoff depends on its own action xi , and the aggregate
actions of its neighbors ( j » i),
P
¦i(x) = u(xi; j»i xj ) ¡ c(xi)
 Node i’s payoff exhibits increasing differences in xi and x-i:
if xi ¸ xi’ and xi ¸ x-i, then
¦i(xi ; x0¡i ) ¡ ¦i (x0i ; x0¡i) ¸ ¦i (xi ; x¡i ) ¡ ¦i (x0i ; x¡i)
 k-core of G is the largest induced subgraph in which all nodes have
at least k neighbors.
 Coreness of node i, Cor(i), is the largest core that node i belongs to.
Preliminaries
 Define the largest best response (LBR) mapping as follows
n
o
BRi (x¡i) = max argmaxz2[0;1] [u(z; x¡i ) ¡ c(z)]
 Increasing differences property implies monotonicity in LBR
x¡i ¸ x0¡i ) BRi(x¡i) ¸ BRi(x0¡i)
 Game has a largest pure Nash equilibrium (LNE)
LNE is the fixed point of LBR initialized by all players playing 1
LNE is the Pareto preferred NE if i’s payoff is increasing in
1
LNE
0
LBR mapping
1
x¡i
Lower Bounding the LNE
Theorem: There exist thresholds 0 · °1 · °2 : : : · °K such that
if cor(i) = k, then LNEi ¸ °k.
 We compare LBR dynamics and k-LBR mapping defined as
n
o
BRk (x) = max argmaxz2[0;1] [u(z; kx) ¡ c(z)]
Time 0: Every player starts with playing 1,
A node i in k-core has at least k neighbors.
Time 1: x1i ¸ BRk (1),
1
BRk+1(x)
°k+1
At least k of i’s neighbors have at least k neighbors.
Time 2: x2i ¸ BRk (BRk (1)),
BRk (x)
°k
both sides are monotonically decreasing.
 xt # LNE; BR(t) (1) # °
k
i
0
1
Coreness and Bonacich Centrality
 A quadratic supermodular game
P
¦i (x) = xi j»i xj ¡
b 2
2 xi
+ axi
 Game has a unique NE which depends on Bonacich centrality,
LNE = ab B( 1b )
Given the adjacency matrix A, B(®) = (I ¡ ®A)¡11.
 Bi (®) is a weighted sum of all walks from any other node to i.
Weights are exponentially decreasing in path length.
Centrality of i heavily depends on centrality of i’s neighbors.
Lemma: if cor(i) = k, then Bi (®) ¸
1
1¡®k
Incomplete Information
 What if nodes do not know the entire network?
 The NE prediction can be misleading
 The LBR mapping takes too long to converge
 Model this scenario by a Bayesian supermodular game of
incomplete information
 Nature chooses the degree independently from degree distribution
(p0 , p1, … , pR)
 Each node knows its own degree and the degree distribution
 Node i forms beliefs about the degree of its neighbors based on the edge
perspective degree distribution (p’0 , p’1, … , p’R)
p0r = P(an edge is incident to a node of degree r) ¼ PRrpr
r=0
P’2
P’3
P’4
rpr
Monotonicity of LNE
 Largest symmetric BNE (LBNE) exists for the defined game.
Proposition I: LBNE is monotone in degree,
deg(i) ¸ deg(i0 ) ) LBNEi ¸ LBNEi0
 X2 with probability distribution p2 is first order stochastically dominated
(FOSD) by X1 with p1 ( p1 º p2) if
P (X1 ¸ x) ¸ P (X2 ¸ x)
Proposition II: LBNE is monotone is edge perspective degree distribution
p01 º p02 ) LBNE(p01) ¸ LBNE(p02)
 FOSD of edge perspective degree distributions is not equivalent to
FOSD of degree distributions
Summary and Future Work
 Supermodular games on graphs were proposed as a benchmark
model of cooperation in networked systems.
 Largest Nash equilibrium was studied in games of complete and
incomplete information about network.
 Local interaction does not imply weak correlation between far away nodes in
cooperation settings.
 Centrality measures need to be used to quantify the effect of the network.
Future Work:
 Model assumed a static interaction between nodes; develop dynamic
game models of coordination on networks.
 Centrality measures are not easy to compute; approximate the
centrality measures for real world networks such as sensor
networks.