Strategic Interaction and Networks

Abstract

We dedicate this paper to the memory of Toni Calvó-Armengol. As will be clear in the subsequent pages, he has made a lasting contribution to our thinking about networks and economics. We deeply miss his insight and his company. We are grateful to Alexander Groves for research assistance and to Noga Alon, Roland Bénabou, Francis Bloch, Andrea Galeotti, Sanjeev Goyal, Matthew Jackson, Antoine Loeper, Brian Rogers, and Yves Zenou and participants in seminars and conferences at Barcelona, Berkeley, Caltech, Cambridge, Duke, École Polytechnique Abstract:This paper brings a general network analysis to a wide class of economic games. A network, or interaction matrix, tells who directly interacts with whom. A major challenge is determining how network structure shapes overall outcomes. We have a striking result. Equilibrium conditions depend on a single number: the lowest eigenvalue of a network matrix. Combining tools from potential games, optimization, and spectral graph theory, we study games with linear best replies and characterize the Nash and stable equilibria for any graph and for any impact of players' actions. When the graph is sufficiently absorptive (as measured by this eigenvalue), there is a unique equilibrium. When it is less absorptive, stable equilibria always involve extreme play where some agents take no actions at all. This paper is the first to show the importance of this measure to social and economic outcomes, and we relate it to different network link patterns.

Networks of Many Public Goods with Non-Linear Best Replies

“…This is because individual preferences are generally specified such that best response functions are piece-wise linear, regardless of whether the agents' strategy space is uni- (Bramoullé and Kranton, 2007;Ballester and Calvó-Armengol, 2010;Bramoullé et al, 2014) or multidimensional (Ilkiliç, 2011). In this section, the existence of a PSNE in a network game of strategic substitutes (i.e., the voluntary contribution game) is established when the strategy spaces of the agents are multidimensional and heterogeneous, and the set of best response functions define non-linear mappings.…”

“…This question has been studied in detail for the case of a single public good. Several conditions have been established, whether the best replies are linear (see, e.g., Bloch and Zenginobuz, 2007;Ballester and Calvó-Armengol, 2010;Bramoullé et al, 2014) or non-linear (see, e.g., Rébillé and Richefort, 2014;Allouch, 2015). However, the more realistic case of several public goods has received much less attention.…”

“…5 Under complete information 6 , a uniqueness condition that depends on network structure only is established for three cases: linear best responses and unipartite network (Corbo et al, 2007;Ballester and Calvó-Armengol, 2010;Bramoullé et al, 2014), linear best responses and bipartite network (Ilkiliç, 2011), and non-linear best responses and unipartite network (Rébillé and Richefort, 2014;Allouch, 2015). Here we also study a fourth case, of non-linear best responses and bipartite network, which generalizes the three other cases.…”

“…Network games of Cournot competition and common property resources can be defined as games of strategic substitutes and negative externalities (see, e.g., Ilkiliç, 2011;Bramoullé et al, 2014).…”

Influence networks and public goods

“…Galeotti et al (2010) also shrink considerably the potentially large set of equilibria that arise under complete information by assuming incomplete knowledge by part of the consumers with respect to the network. Bramoullé et al (2014) focus on games of strategic substitutes on networks with linear best-reply functions which has recently been extended to nonlinear settings by Allouch (2015). We contribute to this vast literature by analyzing local public goods in a random network context.…”