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Strategic network formation arises in settings where agents receive some benefit from their connectedness to other agents, but also incur costs for forming these links. We consider a new network formation game that incorporates an adversarial attack, as well as immunization or protection against the attack. An agent's network 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 can be viewed as a stylized model of settings where reachability rather than centrality is the primary interest as in many technological networks such as the Internet, and vertices may be vulnerable to attacks such as viruses, but may also reduce risk via potentially costly measures such as an anti-virus software. Our main theoretical contributions include a strong bound on the edge density at equilibrium. In particular, we show that under a very mild assumption on the adversary's attack model, every equilibrium network contains at most only $$2n-4$$2n-4 edges for $$n \\ge 4$$nï¾¿4, where n denotes the number of agents and this upper bound is tight. We also show that social welfare does not significantly erode: every non-trivial equilibrium with respect to several adversarial attack models asymptotically has social welfare at least as that of any equilibrium in the original attack-free model. We complement our sharp theoretical results by 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.

Strategic Network Formation with Attack and Immunization