1. 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 in equilibrium If nodes don’t know network structure, largest equilibrium depends on edge perspective degree distribution HOW IT WORKS: We exploit monotonicity of the best response to prove our results: The best action for node i is increasing in its neighbor’s actions. ASSUMPTIONS AND LIMITATIONS: We study equilibria of a static game between nodes. The eventual goal is to understand dynamic network games. Payoff of agent i: Π i (x i, x j, x k ) = u(x i, x j +x k ) – c(x i ) Supermodular Network Games V. Manshadi and R. Johari 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) A node’s actions can have significant effects on distant nodes. Centrality, coreness: Global measures of power of a node We characterize equilibria in terms of such global measures 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. IMPACT NEXT-PHASE GOALS ACHIEVEMENT DESCRIPTION STATUS QUO NEW INSIGHTS TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAA A A A A A AA Supermodular games: Games where nodes have strategic complementarities Network (or graphical) games: Games where nodes interact through network structure … … i j k … … 1 … …

2. 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?

3. 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 x i, and the aggregate actions of its neighbors ( ),  Node i’s payoff exhibits increasing differences in x i and x - i : if x i ¸ x i ’ and x i ¸ 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.

4. 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 i’s payoff is increasing in LNE 0 1 1 LBR mapping

5.  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 i’s neighbors have at least k neighbors.  Time 2:,  both sides are monotonically decreasing.  0 1 1 Lower Bounding the LNE Theorem: There exist thresholds such that if cor(i) = k, then.

6.  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 i’s neighbors. Coreness and Bonacich Centrality Lemma: if cor(i) = k, then

7. 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 (p 0, p 1, …, p R )  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 ) P’ 3 P’ 2 P’ 4

8.  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 Monotonicity of LNE Proposition I: LBNE is monotone in degree, Proposition II: LBNE is monotone is edge perspective degree distribution

9. 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.