Supermodular Network Games V. Manshadi and R. Johari

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Supermodular Network GamesV. Manshadi and R.
Johari

• MAIN RESULT
• We assume utility exhibits strategic
  complementarities.
• We show
• Membership in larger k-core implies higher
  actions in equilibrium
• Higher centrality measure implies higher actions
  inequilibrium
• If nodes dont know network structure,
  largestequilibrium depends on edge perspective
  degree distribution
• HOW IT WORKS
• We exploit monotonicity of the best response
  toprove our results
• The best action for node i isincreasing in its
  neighbors actions.
• ASSUMPTIONS AND LIMITATIONS
• 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.
Payoff of agent i ?i(xi, xj, xk) u(xi, xjxk)
c(xi)
Supermodular gamesGames where nodes have
strategiccomplementarities Network (or
graphical) games Games where nodes interact
throughnetwork structure
A nodes actions can have significanteffects on
distant nodes. Centrality, coreness Global
measures of power of a node We characterize
equilibria in terms of such global measures
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)
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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?

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Model

• N-person game, each players action space is
  0,1.
• graph G (V,E) represents the interaction among
  nodes.
• Node is payoff depends on its own action xi ,
  and the aggregate actions of its neighbors (
  ),
• Node is payoff exhibits increasing differences
  in xi and x-iif xi xi and xi x-i, then
• 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.

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Preliminaries

• Define the largest best response (LBR) mapping
  as follows
• Increasing differences property implies
  monotonicity in LBR
• 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 is payoff is
  increasing in

LBR mapping
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Lower Bounding the LNE


Theorem There exist thresholds
such that if cor(i) k, then
.

• We compare LBR dynamics and k-LBR mapping
  defined as
• Time 0 Every player starts with playing 1,
• A node i in k-core has at least k neighbors.
• Time 1 ,
• At least k of is neighbors have at least k
  neighbors.
• Time 2 ,
• both sides are monotonically decreasing.

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Coreness and Bonacich Centrality

• A quadratic supermodular game
• Game has a unique NE which depends on Bonacich
  centrality,
• Given the adjacency matrix A,
  .
• 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
  is neighbors.

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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 (p0 , p1, , pR)

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Monotonicity of LNE

• Largest symmetric BNE (LBNE) exists for the
  defined game.
• with probability distribution is first
  order stochastically dominated (FOSD) by with
  ( ) if
• FOSD of edge perspective degree distributions is
  not equivalent to FOSD of degree distributions

Proposition I LBNE is monotone in degree,
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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.