Inoculation strategies for bounded degree graphs

Abstract

We study the inoculation game, a game-theoretic abstraction of epidemic containment played on an undirected graph G: each player is associated with a node in G and can either acquire protection from a contagious process or risk infection. After decisions are made, an infection starts at a random node v and propagates through all unprotected nodes reachable from v. It is known that the price of anarchy (PoA) in n-node graphs can be as large as $\Theta \left(n\right)$. Our main result is a tight upper bound of $O\left(\sqrt{n\Delta }\right)$ on the PoA, where Δ is the maximum degree of the graph. Indeed, we provide constructions of graphs with maximum degree Δ for which the PoA is $\Omega \left(\sqrt{n\Delta }\right)$. We also study additional factors that can reduce the PoA, such as higher thresholds for contagion and varying the costs of becoming infected vs. acquiring protection.