Strategic network formation arises where agents receive benefit from connections to other agents, but also incur costs for forming links. We consider a new network formation game that incorporates an adversarial attack, as well as immunization against attack. An agents benefit is the expected size of her connected component post-attack, and agents may also choose to immunize themselves from attack at some additional cost. Our framework is a stylized model of settings where reachability rather than centrality is the primary concern and vertices vulnerable to attacks may reduce risk via costly measures. In the reachability benefit model without attack or immunization, the set of equilibria is the empty graph and any tree. The introduction of attack and immunization changes the game dramatically; new equilibrium topologies emerge, some more sparse and some more dense than trees. We show that, under a mild assumption on the adversary, every equilibrium network with $n$ agents contains at most $2n-4$ edges for $ngeq 4$. So despite permitting topologies denser than trees, the amount of overbuilding is limited. We also show that attack and immunization dont significantly erode social welfare: every non-trivial equilibrium with respect to several adversaries has welfare at least as that of any equilibrium in the attack-free model. We complement our theory with simulations demonstrating fast convergence of a new bounded rationality dynamic which generalizes linkstable best response but is considerably more powerful in our game. The simulations further elucidate the wide variety of asymmetric equilibria and demonstrate topological consequences of the dynamics e.g. heavy-tailed degree distributions. Finally, we report on a behavioral experiment on our game with over 100 participants, where despite the complexity of the game, the resulting network was surprisingly close to equilibrium.
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