On the Difficulty of Characterizing Network Formation with Endogenous Behavior

Bolletta (2021, Math. Soc. Sci. 114:1-10) 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 one-dimensional types $\unicode{x2013}$ 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. We offer a counterexample showing that the claimed characterization is incorrect, highlight where the analysis errs, and discuss implications for network formation models.