A Variational Approach to Network Games

Abstract

Standard-Nutzungsbedingungen:Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden.Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen.Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in SummaryThis paper studies strategic interaction in networks. We focus on games of strategic substitutes and strategic complements, and departing from previous literature, we do not assume particular functional forms on players' payoffs. By exploiting variational methods, we show that the uniqueness, the comparative statics, and the approximation of a Nash equilibrium are determined by a precise relationship between the lowest eigenvalue of the network, a measure of players' payoff concavity, and a parameter capturing the strength of the strategic interaction among players. We apply our framework to the study of aggregative network games, games of mixed interactions, and Bayesian network games. Abstract. This paper studies strategic interaction in networks. We focus on games of strategic substitutes and strategic complements, and departing from previous literature, we do not assume particular functional forms on players' payoffs. By exploiting variational methods, we show that the uniqueness, the comparative statics, and the approximation of a Nash equilibrium are determined by a precise relationship between the lowest eigenvalue of the network, a measure of players' payoff concavity, and a parameter capturing the strength of the strategic interaction among players. We apply our framework to the study of aggregative network games, games of mixed interactions, and Bayesian network games.

A variational inequality framework for network games: Existence, uniqueness, convergence and sensitivity analysis

“…In fact our analysis suggests that for scalar games of strategic substitutes, the natural property is the uniform P-matrix condition. In addition to this substantial difference, our paper is distinct from Melo (2018) in the following ways: i) we do not consider only strong monotonicity but also uniform (block) P-functions, ii) we investigate how properties of the game Jacobian relate to network properties considering not only the minimum eigenvalue but also the spectral and infinity norm, iii) we consider networks that might be asymmetric and agents with possibly multidimensional strategy sets, iv) we consider games with mixed strategic effects for which the best response might not be monotone as a function of the neighbor aggregate, v) besides uniqueness and comparative statics, we also study convergence of best response dynamics. The recent paper Naghizadeh and Liu (2017a) focuses on a special case of scalar network games and derives conditions in terms of the absolute value of the elements of the adjacency matrix for uniqueness of Nash equilibrium.…”

“…The only papers that we are aware of that study properties of the Nash equilibrium in network games by using variational inequalities are Ui (2016), Melo (2018) and Naghizadeh and Liu (2017a). All these works consider network games with scalar non-negative strategies.…”

“…Specifically, Ui (2016) considers Bayesian network games with linear quadratic cost functions. Melo (2018) considers existence, uniqueness and comparative statics for scalar network games with symmetric unweighted networks, by focusing on strong monotonicity of the game Jacobian. For games with strategic substitutes, we show via a counterexample that strong monotonicity cannot be guaranteed under the minimum eigenvalue condition, not even for scalar network games with symmetric networks (see Example B.4).…”

“…A similar Lipschitz continuity result for the solution of general VIs was presented in (Dafermos, 1988, Theorem 2.1) under the assumption of strong monotonicity. Melo (2018) studies Lipschitz continuity of the Nash equilibrium in strongly monotone network games. With respect to both of these works, Theorem 4 proves Lipschitz continuity under the weaker uniform block P-function property allowing the analysis of games whose Jacobian is not strongly monotone, e.g.…”

Constrained public goods in networks

“…Other important related contributions have been made by Acemoglu, García‐Jimeno, and Robinson (), Acemoglu, Malekian, and Ozdaglar (), López‐Pintado (), Kinateder and Merlino (), and Sun () . For more recent contributions on the existence and uniqueness of a Nash equilibrium for network games, including public goods in networks, see Ramachandran and Chaintreau (), Naghizadeh and Liu (, ), Bodwin (), Melo (), Parise and Ozdaglar (, ), and Chen, Zenou, and Zhou ().…”