On the Difficulty of Characterizing Network Formation with Endogenous Behavior

Abstract

Bolletta (2021) studies a model in which a network is strategically formed and then agents play a linear best-response investment game in it. The model is motivated by an application in which people choose both their study partners and their levels of educational effort. Agents have different onedimensional types-private returns to effort. A main result claims that (pairwise Nash) stable networks have a locally complete structure consisting of possibly overlapping cliques: if two agents are linked, they are part of a clique composed of all agents with types between theirs. A counterexample shows that the claimed characterization is incorrect. We specify where the analysis errs and discuss implications for network formation models. Bolletta (2021) studies the joint strategic choice of network links and productive effort, motivated by applications in the economics of education. The theoretical findings are used in interpreting empirical studies on peer effects, such as Carrell et al. (2013). This fits into a literature on network games 1 and games on endogenous networks-see, e.g., König et al. (2014), Hiller (2017), and Badev (2021.The paper examines a stylized network formation process. At each time, starting from a status quo network, two (random) agents i and j can form a new link between them by mutual consent; alternatively, either can unilaterally sever any of their own existing links. After each such stage, agents play the static Nash equilibrium of a coordination game in the current network: each wants to match an average of a private ideal effort (its "type") and average behavior in its neighborhood. Higher types are inclined to make higher efforts in the effort game, holding spillovers fixed. In assessing the consequences of their decisions in the formation stage, players anticipate equilibrium behavior in the action-choice game on the resulting network. The process is said to reach an equilibrium if no pair of agents wish to change their linking decisions at the formation stage. In the leading application, links are study-partner relationships and the action is educational efforts.A basic theoretical question is: which networks can be equilibrium outcomes? A main claim of Bolletta (2021), stated in its Proposition 1, is that equilibrium networks have a specific structure, called locally complete. This means that, at any equilibrium, if two agents are connected, they and all agents with types between theirs form a clique-a completely connected subnetwork.The purpose of this note is to point out that this proposition is false. We do this by providing an explicit counterexample: an equilibrium outcome that is not locally complete. We also show that this outcome can be reached by the dynamic process studied in Bolletta (2021) starting from the empty network. It is thus a natural outcome reachable from a "neutral" status quo. Our note also examines arguments that underlie the proposition in order to highlight where the analysis errs.We give a brief roadmap. In Section 1, we establish key notation and review ...