@article {1502024,
	title = {Keeping Pace with Criminals: An Extended Study of Designing Patrol Allocation against Adaptive Opportunistic Criminals },
	journal = {Games Journal},
	year = {2016},
	abstract = {Game theoretic approaches have recently been used to model the deterrence effect of patrol officers{\textquoteright} assignments on opportunistic crimes in urban areas. One major challenge in this domain is modeling the behavior of opportunistic criminals. Compared to strategic attackers (such as terrorists) who execute a well-laid out plan, opportunistic criminals are less strategic in planning attacks and more flexible in executing well-laid plans based on their knowledge of patrol officers{\textquoteright} assignments. In this paper, we aim to design an optimal police patrolling strategy against opportunistic criminals in urban areas. Our approach is comprised by two major parts: learning a model of the opportunistic criminal (and how he or she responds to patrols) and then planning optimal patrols against this learned model. The planning part, by using information about how criminals responds to patrols, takes into account the strategic game interaction between the police and criminals. In more detail, first, we propose two categories of models for modeling opportunistic crimes. The first category of models learns the relationship between defender strategy and crime distribution as a Markov chain. The second category of models represents the interaction of criminals and patrol officers as a Dynamic Bayesian Network (DBN) with the number of criminals as the unobserved hidden states. To this end, we: (i) apply standard algorithms, such as Expectation Maximization (EM), to learn the parameters of the DBN; (ii) modify the DBN representation that allows for a compact representation of the model, resulting in better learning accuracy and the increased speed of learning of the EM algorithm when used for the modified DBN. These modifications exploit the structure of the problem and use independence assumptions to factorize the large joint probability distributions. Next, we propose an iterative learning and planning mechanism that periodically updates the adversary model. We demonstrate the efficiency of our learning algorithms by applying them to a real dataset of criminal activity obtained from the police department of the University of Southern California (USC) situated in Los Angeles, CA, USA. We project a significant reduction in crime rate using our planning strategy as compared to the actual strategy deployed by the police department. We also demonstrate the improvement in crime prevention in simulation when we use our iterative planning and learning mechanism when compared to just learning once and planning. Finally, we introduce a web-based software for recommending patrol strategies, which is currently deployed at USC. In the near future, our learning and planning algorithm is planned to be integrated with this software. This work was done in collaboration with the police department of USC.},
	author = {Zhang, Chao and Gholami, Shahrzad and Kar, Debarun and Sinha, Arunesh and Jain, Manish and Goyal, Ripple and Tambe, Milind}
}
