Effective Stackelberg Strategies on Epidemic Containment Games

This article studies Epidemic Containment (EC) games, where agents are the set of the vertices in a social network. Each agent has two strategies: being vaccinated or not. Each vaccinated agent is safe and the cost is 1. In EC games, spectral radii of connected components induced by all the non-vaccinated agents are used to measure the safety of the non-vaccinated agents. If the spectral radius of a connected component is strictly less than a given threshold, then all the agents in this component are safe and the cost is 0; otherwise, the agent is unsafe and the cost is $\infty$. Each agent wants to minimize her cost. Let $minNE$ and $maxNE$ denote the set of vaccinated agents in Nash Equilibria with the minimum and maximum total cost, respectively. This paper focuses on Stackelberg games, where a leader injects $k$ agents in advance and the rest agents perform $maxNE$ on the rest graph, where $k < |minNE|$. The leader aims to minimize the total cost of the leader and the rest agents. The theoretical results are on the scenario that the given threshold is less than 1. We first show that it is NP-hard to find the minimum total cost. Then we turn to find effective leader's strategies, with which the total cost is no more than $|maxNE|$ in the original EC game. The existence of effective leader's strategies on two kinds of networks is proved. One is a network including one-degree vertices; the other one includes a pair of vertices with inclusive social relationships. Besides, a complete characterization of effectiveness on bipartite graphs is established. For general cases, we propose a heuristic algorithm to find Stackelberg strategy. Numerical experiments are illustrated on small synthetic networks and four real networks, which demonstrates that the interventions are necessary.