Cops and Invisible Robbers: the Cost of Drunkenness

We examine a version of the Cops and Robber (CR) game in which the robber is invisible, i.e., the cops do not know his location until they capture him. Apparently this game (CiR) has received little attention in the CR literature. We examine two variants: in the first the robber is adversarial (he actively tries to avoid capture); in the second he is drunk (he performs a random walk). Our goal in this paper is to study the invisible Cost of Drunkenness (iCOD), which is defined as the ratio ct_i(G)/dct_i(G), with ct_i(G) and dct_i(G) being the expected capture times in the adversarial and drunk CiR variants, respectively. We show that these capture times are well defined, using game theory for the adversarial case and partially observable Markov decision processes (POMDP) for the drunk case. We give exact asymptotic values of iCOD for several special graph families such as $d$-regular trees, give some bounds for grids, and provide general upper and lower bounds for general classes of graphs. We also give an infinite family of graphs showing that iCOD can be arbitrarily close to any value in [2,infinty). Finally, we briefly examine one more CiR variant, in which the robber is invisible and "infinitely fast"; we argue that this variant is significantly different from the Graph Search game, despite several similarities between the two games

Statistical Model Checking for Cops and Robbers Game on Random Graph Models

Cops and robbers problem has been studied over the decades with many variants and applications in graph searching problem. In this work, we study a variant of cops and robbers problem on graphs. In this variant, there are di�erent types of cops and a minimum number of each type of cops are required to catch a robber. We studied this model over various random graph models and analyzed the properties using statistical model checking. To the best of our knowledge this variant of the cops and robber problem has not been studied yet. We have used statistical techniques to estimate the probability of robber getting caught in di�erent random graph models. We seek to compare the ease of catching robbers performing random walk on graphs, especially complex networks. In this work, we report the experiments that yields interesting empirical results. Through the experiments we have observed that it is easier to catch a robber in Barab�asi Albert model than in Erd�os-R�enyi graph model. We have also experimented with k-Regular graphs and real street networks. In our work, the model is framed as the multi-agent based system and we have implemented a statistical model checker, SMCA tool which veri�es agents based systems using statistical techniques. SMCA tool can take the model in JAVA programming language and support Probabilistic - Bounded LTL logic for property specification

Partially Observable Stochastic Games with Neural Perception Mechanisms

Stochastic games are a well established model for multi-agent sequential decision making under uncertainty. In reality, though, agents have only partial observability of their environment, which makes the problem computationally challenging, even in the single-agent setting of partially observable Markov decision processes. Furthermore, in practice, agents increasingly perceive their environment using data-driven approaches such as neural networks trained on continuous data. To tackle this problem, we propose the model of neuro-symbolic partially-observable stochastic games (NS-POSGs), a variant of continuous-space concurrent stochastic games that explicitly incorporates perception mechanisms. We focus on a one-sided setting, comprising a partially-informed agent with discrete, data-driven observations and a fully-informed agent with continuous observations. We present a new point-based method, called one-sided NS-HSVI, for approximating values of one-sided NS-POSGs and implement it based on the popular particle-based beliefs, showing that it has closed forms for computing values of interest. We provide experimental results to demonstrate the practical applicability of our method for neural networks whose preimage is in polyhedral form.Comment: 41 pages, 5 figure