Bayesian Security Games for Controlling Contagion Jason Tsai 1 , Yundi Qian 1 , Yevgeniy Vorobeychik 2 , Christopher Kiekintveld 3 , Milind Tambe 1 1 University of Southern California, Los Angeles, CA {jasontts,yundi.qian,tambe} @usc.edu 2 Sandia National Laboratories, Livermore, CA yvorobe@sandia.gov 3 University of Texas at El Paso, El Paso, TX cdkiekintveld@utep.edu Abstract—Inﬂuence blocking games have been used to model adversarial domains with a social component, such as counterin- surgency. In these games, a mitigator attempts to minimize the efforts of an inﬂuencer to spread his agenda across a social network. Previous work has assumed that the inﬂuence graph structure is known with certainty by both players. However, in reality, there is often signiﬁcant information asymmetry between the mitigator and the inﬂuencer. We introduce a model of this information asymmetry as a two-player zero-sum Bayesian game. Nearly all past work in inﬂuence maximization and social network analysis suggests that graph structure is fundamental in strategy generation, leading to an expectation that solving the Bayesian game exactly is crucial. Surprisingly, we show through extensive experimentation on synthetic and real-world social networks that many common forms of uncertainty can be addressed near- optimally by ignoring the vast majority of it and simply solving an abstracted game with a few randomly chosen types. This suggests that optimal strategies of games that do not model the full range of uncertainty in inﬂuence blocking games are typically robust to uncertainty about the inﬂuence graph structure. Index Terms—Game theory, Social contagion, Inﬂuence max- imization I. I NTRODUCTION Social contagion has long been of great interest in the literature on marketing, the spread of rumors, and, recently, in the context of Arab Spring [11], [13], [19]. Our speciﬁc focus is on counterinsurgency, which we view as a competition for the support of local leaders. Counterinsurgency can be modeled as a game with two strategic players, the insurgents and the peacekeepers, in which the insurgents aim to spread their views, unrest, etc. among the local population, while the peacekeepers wish to minimize the resulting contagion by engaging in their own inﬂuence campaign [8], [7], [20]. The key computational question we address is: given limited resources, how to select which of the local leaders to inﬂuence to minimize the global impact of the insurgency. These ‘inﬂuence blocking’ games have received recent attention in the security games literature [20], where they have been modeled using graphs with nodes representing the tribal leaders and edges representing possible transmission of inﬂuence. However, this line of work has assumed that full information about network structure is available to both players. In practice, informational challenges abound in coun- terinsurgency, where the insurgents are typically an indigenous group that has an informational advantage, and the mitigators are often uncertain about the the social network [8]. We model counterinsurgency as an inﬂuence blocking game with asymmetric information. Speciﬁcally, we assume that the inﬂuencer (an insurgent group) has perfect knowledge of the inﬂuence graph structure, while the mitigator is uncertain about it. In the resulting Bayesian game, an inﬂuencer type is a particular instantiation of the inﬂuence graph, and the mitigator must reason over the distribution over these graphs (i.e., inﬂuencer types) in order to compute an optimal strategy. Past work in inﬂuence maximization and social network analysis highlight the importance of graph structure in strat- egy generation [11], [3], [6]. In addition, previous work on Bayesian security games has shown that not accounting for even small degrees of payoff uncertainty can lead to large drops in solution quality [12]. Thus, we expect strategies generated without modeling most of the uncertainty about graph structure to do far worse than the optimal solution to the Bayesian game. Supporting this intuition, we show that there are cases where a mitigator who has incorrect information about a single edge can suffer unbounded loss and that quantifying the impact of changing a single edge in a given graph is #P-Hard. We also show empirically that, indeed, under our models of uncertainty, optimal mitigator strategies for different inﬂuencer types are vastly different. However, while past work has focused on sophisticated algorithms for Bayesian security games [9], [12], [22], we showcase the opposite approach that runs directly counter to what intuition and our initial experiments suggest: ignoring the vast majority of uncertainty has minimal impact. Speciﬁcally, we show through extensive experiments that computing a miti- gation strategy based on a game with only a few randomly sam- pled inﬂuencer types yields near-optimal rewards for widely varied models of uncertainty. We experiment on 3 different synthetic graph models with and without resource imbalances on both sides, 5 models of uncertainty, weighted/unweighted counting of nodes, varied edge weight distributions, varied graph sizes, varied degrees of uncertainty, and varied degrees of sampling. We also conduct experiments on two real-world social networks using two different models graph construction. In all, we studied over 200 experimental settings and con- sistently observe the same result: simple sampling techniques perform near-optimally. This suggests that even in domains as challenging as ours, models which ignore uncertainty may nevertheless be robust to it.