Multi-robot patrolling is the problem of repeatedly visiting a group of regions of interest in an environment with a group of robots to prevent intrusion. Early works have proposed deterministic patrolling algorithms which could be learned by an adversary observing them over time. More recent works provide non-deterministic patrolling schemes which only work for perimeter patrolling and require coordination and synchronization. In this paper, we investigate the problem of finding robust and scalable strategies for multi-robot patrolling under an adversarial environment. So, we present algorithms to find different decentralized strategies for a patroller in the form of Markov chains which use convex optimization to minimize the average commute time for an environment, a subset of the environment, or a specific region of an environment when we use uniform distribution over all regions for both patroller and adversary. Additionally, we use these strategies in a game theoretical setup to form a payoff matrix to obtain an optimal mixed strategy for patroller. We also propose an algorithm to find a decentralized strategy for patroller in the form of Markov chain which converges very fast as it is the optimal response from patroller against the adversary when we use non-uniform distribution
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