Too Big to Jail? Key-Player Policies When the Network is Endogenous

This paper endogenizes the network for the seminal model presented in Ballester et al. (2006) by way of a simple simultaneous move game. Agents choose with whom to associate and how much effort to exert. Effort levels display local strategic complementarities and global strategic substitutes. I show that all pairwise Nash equilibrium networks are nested split graphs. As in Ballester et al. (2006), agents’ equilibrium effort levels are proportional to Bonacich centrality. However, their ranking now coincides with a simpler measure, which is also easier to identify: degree centrality. I then study key player policies, which aim at minimizing aggregate effort levels via the elimination of an agent. In the spirit of network formation, after an agent was eliminated from a pairwise Nash equilibrium network, the remaining agents may revise their effort decisions and adapt their linking behavior. It is shown that, if the parameter governing global strategic substitutes is sufficiently small, then eliminating a most central agent also decreases aggregate effort levels most. This mirrors results obtained by Ballester et al. (2006). However, when global strategic substitutes are large, then, different from Ballester et al. (2006), eliminating a most central agent may not be optimal. Eliminating a most central agent, who in equilibrium also exerts highest criminal effort, decreases competition/congestion effects and increases incentives of the remaining agents to create new links. The latter effect on the aggregate level of crime may outweigh the former. These results are relevant for a wide range of applications, such as juvenile delinquency and crime, R&D expenditure of firms, bank bailouts and trade.