## How do your friends affect your school performance?

http://restud.oxfordjournals.org/content/76/4/1239.full

Paper Cited: Calvó-Armengol, Antoni, Eleonora Patacchini, and Yves Zenou. “Peer effects and social networks in education.” *The Review of Economic Studies* 76.4 (2009): 1239-1267.

In this academic paper, the authors develop a theoretical model that explains how each student in a network maximizes his utility by choosing his personal effort and peer effort simultaneously. The result reveals that, at the Nash equilibrium, a student’s choice is proportional to his location in a network (his Katz-Bonacich centrality measure). The paper also estimates peer effects and contextual effects using the famous Add Health data that maps a detailed network of adolescent friendship at school. The estimation result shows that, “a standard deviation increase in the Katz-Bonacich centrality increases the pupil school performance by more than 7% of one standard deviation” (Calvo-armengol, Patacchini, Zenou 2009).

This paper is very interesting in that it connects to two topics that we have studied in INFO 2040 — the topology of social networks and game theory. In terms of network topology, the Katz-Bonacich centrality measure, due to Katz (1953) and later extended by Bonacich (1987), provides a useful measure of a node’s importance by scoring its location in a network. It takes into account both direct and indirect friends of each individual by counting the total number of direct and indirect paths of any length that stem from that node and then discount the usefulness of far away indirect paths by using a geometrically decaying factor. In terms of game theory, the authors characterize a Nash Equilibrium where each student (player) chooses its individual effort level and peer effort level (strategies) simultaneously. Each individual maximizes its utility (payoff). The best response function for each individual then can be uniquely defined, thereby constituting a Nash equilibrium. The author then analytically relate equilibrium behavior to network location by showing that the Nash Equilibrium effort decision of each agent is proportional to his weighted centrality measure. This also allows for a structural estimation of the best response functions to identify peer effects and contextual effects.