Pursuit Game Research Articles

On estimation of Hausdorff deviation of convex polygons in $\mathbb{R}^2$ from their differences with disks

We study a problem concerning the estimation of the Hausdorff deviation of convex polygons in $\mathbb R^2$ from their geometric difference with circles of sufficiently small radius. Problems with such a subject, in which not only convex polygons but also convex compacts in the Euclidean space $\mathbb R^n$ are considered, arise in various fields of mathematics and, in particular, in the theory of differential games, control theory, convex analysis. Estimates of Hausdorff deviations of convex compact sets in $\mathbb R^n$ in their geometric difference with closed balls in $\mathbb R^n$ are presented in the works of L.S. Pontryagin, his staff and colleagues. These estimates are very important in deriving an estimate for the mismatch of the alternating Pontryagin’s integral in linear differential games of pursuit and alternating sums. Similar estimates turn out to be useful in deriving an estimate for the mismatch of the attainability sets of nonlinear control systems in $\mathbb R^n$ and the sets approximating them. The paper considers a specific convex heptagon in $\mathbb R^2$. To study the geometry of this heptagon, we introduce the concept of a wedge in $\mathbb R^2$. On the basis of this notion, we obtain an upper bound for the Hausdorff deviation of a heptagon from its geometric difference with the disc in $\mathbb R^2$ of sufficiently small radius.

On Linear Multistep Controlled Processes

Linear controlled multistep processes are considered with an additive quality functional under a phase constraint and a terminal condition. These processes are of interest for control theory and its applications. An existence theorem for optimum control is proved for the controlled processes under consideration. In analogy with backward constructions in the positional theory of differential games developed to solve positional differential pursuit games, a procedure for geometric splitting is developed for optimizing the considered problems that is characteristic of the theory of dynamic programming.

Distributed multi‐agent deep reinforcement learning for cooperative multi‐robot pursuit

As a popular research topic in the area of distributed artificial intelligence, the multi-robot pursuit problem is widely used as a testbed for evaluating coordinated and cooperative strategies in multi-robot systems. This study the problem of multi-robot pursuit game using reinforcement learning (RL) techniques is studied. Unlike most existing studies that apply fully centralised deep RL methods based on the centralised-learning and decentralised-execution scheme, the authors propose a fully decentralised multi-agent deep RL approach by modelling each agent as an individual deep RL agent that has its own individual learning system (i.e. individual action-value function, individual leaning update process, and individual action output). To realise coordination among agents, the limited information of other environmental agents is used as input of the learning process. Experimental results show that both distributed and centralised approaches can ultimately solve the pursuit-evasion problem in different dimensions, but the learning efficiency and coordination performance of the proposed distributed approach are much better than the traditional centralised approach.

Relaxation of the Pursuit–Evasion Differential Game and Iterative Methods

A variant of the program iteration method called stability iterations is used for a differential game of pursuit–evasion. The successful solvability set of one of the problems generating the game is found as a limit of the iterative procedure in the space of sets whose elements are positions of the game. The game is defined by a pair of closed sets, one of the which is the target set in the pursuit problem (the first player’s problem) and the other specifies the state constraints in this problem. For the positions not belonging to the solvability set of the pursuit problem, it is interesting to determine the smallest “size” of a neighborhood of the two mentioned sets for which the first player can implement the guidance to the neighborhood of the target set corresponding to this “size” within the similar neighborhood of the second set, i.e., the set specifying the state constraints. Similar constructions are considered for the sets realized at each stage of the iterative procedure. We use the connection of these constructions with the mentioned smallest “size” of neighborhoods of the sets that are parameters of the differential game in the sense of guaranteed realizability of guidance under the replacement of the original sets by these neighborhoods.

Containment: A Variation of Cops and Robber

We consider “Containment”: a variation of the graph pursuit game of Cops and Robber in which cops move from edge to adjacent edge, the robber moves from vertex to adjacent vertex (but cannot move along an edge occupied by a cop), and the cops win by “containing” the robber—that is, by occupying all $$\deg (v)$$ of the edges incident with a vertex v while the robber is at v. We develop bounds that relate the minimal number of cops, $$\xi (G)$$, required to contain a robber to the well-known “cop-number” c(G) in the original game: in particular, $$c(G) {\le } \xi (G) {\le } \gamma (G) \Delta (G)$$. We note that $$\xi (G) {\ge } \Delta (G)$$ for all graphs G, and analyze several families of graphs in which equality holds, as well as several in which the inequality is strict. We also give examples of graphs which require an unbounded number of cops in order to contain a robber, and show that there exist cubic graphs on n vertices with $$\xi (G) = \Omega (n^{1/6})$$.

Linear Differential Pursuit Game with Impulse Control and Linear Integral Constraint of Controls of Players

For a linear differential pursuit-evasion game, we propose sufficient termination conditions in the case where one of players applies impulse controls whereas the other applies controls with “linear constraints” of integral type. We indicate methods for finding controls of the pursuer that provide termination of the pursuit in a finite time. Finally, we give examples that illustrate the results obtained.

ε-Positional Strategies in the Theory of Differential Pursuit Games and the Invariance of a Constant Multivalued Mapping in the Heat Conductivity Problem

In this paper, we consider two problems. In the first problem, we prove that if the assumption from the paper [1] and one additional condition on the parameters of the game hold, then the pursuit can be finished in any neighborhood of the terminal set. To complete the game, an ε-positional pursuit strategy is constructed.In the second problem, we study the invariance of a given multivalued mapping with respect to the system with distributed parameters. The system is described by the heat conductivity equation containing additive control terms on the right-hand side.

A Problem of Pursuit Game with Various Constraints on Controls of Players

This work introduce a difference controls of players by using a new control method to completing a pursuit game. We study pursuit game problems for controlled partial differential equations of the parabolic type. We proved a theorem on pursuit game with mixed constraints, where pursuers control are subjected to integral (geometric) constraint and geometric (integral) constraint are imposed on evaders control. Moreover, we established the sufficient conditions for which pursuit is possible in the game considered.

Pemanfaatan Modifikasi Permainan Trivial Pursuit Sebagai Media Pembelajaran Guna Meningkatkan Kosakata Bahasa Inggris Anak

The mastery and use of English is very important for social community, especially for children. Children are potential assets in building the nation. Their abilities and intelligence should be optimized to create a developed and competitive nation. Seeing that there is still a lack of elementary shool students in mastering English vocabulary, the researcher were willing to modify the Trivial Pursuit game as learning media to improve children English vocabulary mastery. The Trivial Pursuit game was chosen because this game is very fun when and can unwittingly internalize their knowledge and language. This research was carried out in the historical tourism area of the Banten Lama Kaibon Palace with 10 research subjects and representatives of community elements. This research is qualitative focus on action research where containing the process of planning, making provisions, acting, collecting data, analyzing data, evaluating data, and reflection. The result is proven that there is an improvement in mastery of English vocabulaty after learning by using Trivial Pursuit. The tourisme community also assumed that the action is really helpful to children around tourism site to learn English.

Optimal Interdiction of Urban Criminals with the Aid of Real-Time Information

Most violent crimes happen in urban and suburban cities. With emerging tracking techniques, law enforcement officers can have real-time location information of the escaping criminals and dynamically adjust the security resource allocation to interdict them. Unfortunately, existing work on urban network security games largely ignores such information. This paper addresses this omission. First, we show that ignoring the real-time information can cause an arbitrarily large loss of efficiency. To mitigate this loss, we propose a novel NEtwork purSuiT game (NEST) model that captures the interaction between an escaping adversary and a defender with multiple resources and real-time information available. Second, solving NEST is proven to be NP-hard. Third, after transforming the non-convex program of solving NEST to a linear program, we propose our incremental strategy generation algorithm, including: (i) novel pruning techniques in our best response oracle; and (ii) novel techniques for mapping strategies between subgames and adding multiple best response strategies at one iteration to solve extremely large problems. Finally, extensive experiments show the effectiveness of our approach, which scales up to realistic problem sizes with hundreds of nodes on networks including the real network of Manhattan.

On Equations of the Program Iteration Method

The program iteration method, which is inseparably associated with the name of A. G. Chentsov, first appeared in the study of the so-called zero-sum differential games. At early stages, only one of the two possible dual iterative procedures was considered — the maximin procedure. This can be explained by the special interest of the researchers in the so-called program maximin function, which is conveniently interpreted in geometric terms in games of pursuit. Nevertheless, the dual minimax iterative procedure is of no less interest. The program iteration method is mainly significant because it may be used as the basis for the development of a differential game theory in a closed compact form, which was shown earlier for a version of the method based on a certain modification of iterative operators. The key role in this theory belongs to the theorem that states the existence and uniqueness of a solution of the equation induced by a pair of such operators. In this case, the maximin iterative procedure is used to describe e-optimal (in some cases, optimal) positional strategies of the first player, while the minimax procedure is used to describe e-optimal (in some cases, optimal) positional strategies of the second player. This paper investigates the structure of the solution set of the generalized Bellman-Isaacs equation obtained with the use of historically first (not modified) operators of the program iteration method. A theorem that states the existence and uniqueness of the solution to this equation meeting a natural boundary condition is proved under certain assumptions. Thus, it is shown that the original version of the program iteration method can also be used in designing a closed-form differential game theory. However, here we use the so-called recursive strategies rather than positional ones. Such strategies, together with the program iteration method, play an essential role in the analysis of coalition-free differential games.

A stochastic game model of searching predators and hiding prey.

When the spatial density of both prey and predators is very low, the problem they face may be modelled as a two-person game (called a 'search game') between one member of each type. Following recent models of search and pursuit, we assume the prey has a fixed number of heterogeneous 'hiding' places (for example, ice holes for a seal to breathe) and that the predator (maybe polar bear) has the time or energy to search a fixed number of these. If he searches the actual hiding location and also successfully pursues the prey there, he wins the game. If he fails to find the prey, he loses. In this paper, we modify the outcome in the case that he finds but does not catch the prey. The prey is now vulnerable to capture while relocating with risk depending on the intervening terrain. This generalizes the original games to a stochastic game framework, a first for search and pursuit games. We outline a general solution and also compute particular solutions. This modified model now has implications for the question of when to stay or leave the lair and by what routes. In particular, we find the counterintuitive result that in some cases adding risk of predation during prey relocation may result in more relocation. We also model the process by which the players can learn about the properties of the different hiding locations and find that having to learn the capture probabilities is favourable to the prey.

Selfish cops and active robber: Multi-player pursuit evasion on graphs

We introduce and study the game of “Selfish Cops and Active Robber” (SCAR) which can be seen as a multiplayer variant of the “classic” two-player Cops and Robbers (CR) game. In classic CR all cops are controlled by a single player, who has no preference over which cop captures the robber. In SCAR, on the other hand, each of N−1 cops is controlled by a separate player, and a single robber is controlled by the N-th player; and the capturing cop player receives a higher reward than the non-capturing ones. Consequently, SCAR is an N-player pursuit game on graphs, in which each cop player has an increased motive to be the one who captures the robber. The focus of our study is the existence and properties of SCAR Nash Equilibria (NE). In particular, we prove that SCAR always has one NE in deterministic positional strategies and (for N≥3) another one in, generally, deterministic nonpositional strategies. Furthermore, we study conditions which, at equilibrium, guarantee either capture or escape of the robber and show that (because of the antagonism between the “selfish” cop players) the robber may, in certain SCAR configurations, be captured later than he would be in classic CR, or even not captured at all. Finally we define the selfish cop number of a graph and study its connection to the classic cop number.

Method of Resolving Functions in the Group Pursuit Problem with a Terminal Pay Off Function and Integral Constraints on Controls

A method is proposed for solving game dynamics problems with a terminal pay off function and integral constraints on controls which consists in systematical using the ideas of Fenhel−Moreau as applied to the general scheme of resolving functions method. The essence of the proposed method lies in the fact that the resolving function can be expressed in terms of the function conjugate to the pay off function and using the involute of the conjugation operator for a convex closed function to obtain a guaranteed estimate of the terminal value of the pay off function which is represented by the paying off value at the initial time instant and the integral of the resolving function. The main feature of the method is the cumulative principle used in the current summation of the resolving function for estimating the game quality until a certain threshold value is reached. The paper considers linear differential games of group pursuit with a terminal pay off function and integral constraints on controls. Sufficient conditions for the game completion in a finite guaranteed time in a class of quasi-strategies are formulated. Two schemes of the resolving functions method are proposed that ensure without additional assumptions the game completion in the final guaranteed time in the class of stroboscopic strategies. The results of comparing the guaranteed times of different schemes of the resolving functions method are presented.

Generalized Cops and Robbers: A Multi-player Pursuit Game on Graphs

We introduce and study the Generalized Cops and Robbers (GCR) game, an N-player pursuit game in graphs. The two-player version is essentially equivalent to the classic Cops and Robbers (CR) game. The three-player version can be understood as two CR games played simultaneously on the same graph; a player can be at the same time both pursuer and evader. The same is true for four or more players. We formulate GCR as a discounted stochastic game of perfect information and prove that, for three or more players, it has at least two Nash equilibria: one in positional deterministic strategies and another in nonpositional ones. We also study the capturing properties of GCR Nash equilibria in connection with the cop number of a graph. Finally, we briefly discuss GCR as a member of a wider family of multi-player graph pursuit games with rather interesting properties.

Pursuit game problem of an infinite system of differential equations with geometric and integral constraints

The present paper considers a pursuit game problem of an infinite system of 1st-order differential equations in the space l2. In the game, the player’s control functions satisfy both geometric and integral constraints. The pursuer’s action is to force the state of the system to coincides with another state for a finite time and the evader’s action is to retrieve this. In each case, we obtain a condition for the completion of the game for which the pursuer’s strategy is explicitly constructed. In addition, we study a control problem.

Differential game of optimal pursuit of one evader by many pursuers

In this paper, we study a differential game in which a finite or countable number of pursuers pursue a single evader. The control functions of players satisfy integral constraints. The period of the game, which is denoted as $$\zeta $$ , is determined. The farness between the evader and the closest pursuer when the game is finished is the payoff function of the game. In the paper, we introduce the value of the game and identify optimal strategies of the pursuers. A significant facet of this work is that there is no relation between the energy resource of any pursuer and that of the evader for completing pursuit.

Technological Resources to Prevent Cyberbullying During Adolescence: The Cyberprogram 2.0 Program and the Cooperative Cybereduca 2.0 Videogame.

Bullying and cyberbullying have serious consequences for all those involved, especially the victims, and its prevalence is high throughout all the years of schooling, which emphasizes the importance of prevention. This article describes an intervention proposal, made up of a program (Cyberprogram 2.0 Garaigordobil and Martínez-Valderrey, 2014a) and a videogame (Cooperative Cybereduca 2.0 Garaigordobil and Martínez-Valderrey, 2016b) which aims to prevent and reduce cyberbullying during adolescence and which has been validated experimentally. The proposal has four objectives: (1) To know what bullying and cyberbullying are, to reflect on the people involved in these situations; (2) to become aware of the harm caused by such behaviors and the severe consequences for all involved; (3) to learn guidelines to prevent and deal with these situations: know what to do when one suffers this kind of violence or when observing that someone else is suffering it; and (4) to foster the development of social and emotional factors that inhibit violent behavior (e.g., communication, ethical-moral values, empathy, cooperation…). The proposal is structured around 25 activities to fulfill these goals and it ends with the videogame. The activities are carried out in the classroom, and the online video is the last activity, which represents the end of the intervention program. The videogame (www.cybereduca.com) is a trivial pursuit game with questions and answers related to bullying/cyberbullying. This cybernetic trivial pursuit is organized around a fantasy story, a comic that guides the game. The videogame contains 120 questions about 5 topics: cyberphenomena, computer technology and safety, cybersexuality, consequences of bullying/cyberbullying, and coping with bullying/cyberbullying. To evaluate the effectiveness of the intervention, a quasi-experimental design, with repeated pretest-posttest measures and control groups, was used. During the pretest and posttest stages, 8 assessment instruments were administered. The experimental group randomly received the intervention proposal, which consisted of one weekly 1-h session during the entire school year. The results obtained with the analyses of variance of the data collected before and after the intervention in the experimental and control groups showed that the proposal significantly promoted the following aspects in the experimental group: (1) a decrease in face-to-face bullying and cyberbullying behaviors, in different types of school violence, premeditated and impulsive aggressiveness, and in the use of aggressive conflict-resolution strategies; and (2) an increase of positive social behaviors, self-esteem, cooperative conflict-resolution strategies, and the capacity for empathy. The results provide empirical evidence for the proposal. The importance of implementing programs to prevent bullying in all its forms, from the beginning of schooling and throughout formal education, is discussed.

Smart environment effectiveness analysis of a pursuit and evasion scenario

The internet of things (IoT) has become a trend in interactive environments for providing information to decision-makers. Anti-submarine warfare (ASW) is a typical pursuit and evasion (PE) game that is a very complicated process. The ASW helicopter is assigned to execute the final phase of hunting the submarine with a torpedo attack. In most cases, a single helicopter is assigned to detect the submarine by dipping sonar, and then drops a torpedo. Once the dipping sonar goes off, uncertainty takes over, with the possible result of losing track of the submarine. To prevent this problem, using the IoT concept to create a wireless sensor network (WSN) in the area of interest for keeping ears on the evading submarine is a potential solution. The objective of this paper is to gain insights into this PE scenario so as to quantify the interaction result in order to demonstrate the effectiveness of the helicopter in terms of hunting the submarine. Monte Carlo simulation has been developed as the analytical tool, and ANOVA was used to verify the significance of the output measure of effectiveness (MOE) before analysis. The results show that a slow, unalerted submarine has a very low chance of survival. An alerted submarine has very high chance of survival, but when the proposed sonobuoy WSN is in place, this situation benefitting the submarine will be reversed. The WSN has been proved to be effective in a single helicopter carrying out its ASW task.

Pursuit-Evasion Guidance in a Switched System

A pursuit-evasion problem for an interceptor (pursuer) and a maneuverable target (evader) is considered. It is assumed that during the engagement the system overcomes multiple abrupt changes. This leads to a formulation of a pursuit-evasion game for a switched piecewise-linear system. In the case of complete information on the switch timing and on the system matrices, the differential game is solved based on the zero-effort miss distance in the switched system. In the case where switch moments and system matrices are unknown to one of the players, two matrix games (pursuit game and evasion game) are formulated.