Differential Games and Its Applications

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Computational and Applied Mathematics".

Deadline for manuscript submissions: closed (30 June 2023) | Viewed by 9461

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Department of Law, Economics and Human Sciences, “Mediterranea” University of Reggio Calabria, 89124 Reggio Calabria, Italy
Interests: PDEs; game theory; applied mathematics; topology
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1. Department of Law, Economics and Human Sciences, University “Mediterranea” of Reggio Calabria, 89124 Reggio Calabria, Italy
2. The Invernizzi Centre for Research in Innovation, Organization, Strategy and Entrepreneurship (ICRIOS), Bocconi University, Via Sarfatti, 25, 20136 Milano, Italy
Interests: mathematical economics; machine learning and data science; epidemics models; fractional calculus
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Department of Management, Polytechnic University of Marche, Piazza Martelli, 8, 60121 Ancona, Italy
Interests: dynamical systems; mathematical economics; networks and optimization theory
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Department of Economics and Political Sciences, University of Aosta Valley, 11100 Aosta, Italy
Interests: mathematical economics; stochastic programming; artificial intelligence; dynamic system
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Special Issue Information

Dear colleagues,

The theme of this Special Issue is focused on differential games and the latest developments. The theory of differential games finds application in conflict problems which are treated through the use of differential equations. This problem lies between classical game theory, in which multiple players are involved, and controlled dynamic systems, in which differential equations describing the game are controlled by players. In fact, such games are among the most widespread, challenging, and important optimization problems facing mobile agents. In this context, games of pursuit and evasion play a strategic role, involving as players one or more pursuers and one or more evaders. Obviously, the purposes of the groups of players are diametrically opposed: the former want to capture the latter, and the latter want not to be captured by the former. The strategies defined in these classes of games appear complicated, as alongside the resolution of the mathematical problem, there is that of the complex sensorimotor coordination that the pursuer must have toward the physical environment in which the pursuit takes place and the hostile behavior of the evader. There is a wide area of application for these games, ranging from simple traffic control in the rush hour of a large city to military strategy, such as missile guidance systems formulated by Rufus Isaacs, to surgery and management, and there are different kinds of hunting escape games, such as dynamic zero-sum games, instant games optimized for time, and so on. Within games where the evolution of strategies depends on continuous time, it is also necessary to remember differential games with incomplete information.

Given the numerous studies that have been carried out in this area of research, in compliance with this aim, we are interested in articles that explore various aspects of differential games and also dynamics models.

Potential topics include but are not limited to:

  • Differential games of pursuit and evasion;
  • Games for dynamic equations on time scales;
  • Two-person zero sum differential games;
  • Linear-quadratic differential games;
  • Dynamic games;
  • Differential games described by PDE;
  • Applications of differential games to biology, computer science, economics, engineering, management science, operations research, and political science;
  • Deterministic and stochastic differential games with partial observation;
  • Links between incomplete information games in continuous time and repeated time games;
  • Hamilton–Jacobi equations for incomplete information games;
  • Continuous time games;
  • Pursuit and evasion differential games with incomplete information;
  • Differential games on graphs;
  • Hamilton–Jacobi equations in optimal control and differential games;
  • Stochastic differential games;
  • Evolutionary games;
  • Differential games described by infinite system of differential equations;
  • Numerical methods for differential games.

Dr. Bruno Antonio Pansera
Prof. Dr. Massimiliano Ferrara
Prof. Dr. Luca Guerrini
Dr. Tiziana Ciano
Guest Editors

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Keywords

  • differential games
  • pursuit-evasion games
  • dynamic games
  • Hamilton–Jacobi equations
  • numerical methods

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Published Papers (8 papers)

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Editorial

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4 pages, 172 KiB  
Editorial
Preface to the Special Issue on “Differential Games and Its Applications”
by Bruno Antonio Pansera, Massimiliano Ferrara, Luca Guerrini and Tiziana Ciano
Mathematics 2023, 11(13), 3028; https://doi.org/10.3390/math11133028 - 7 Jul 2023
Viewed by 731
Abstract
The study of differential games has practical applications in the analysis and resolution of conflict issues by the use of differential equations [...] Full article
(This article belongs to the Special Issue Differential Games and Its Applications)

Research

Jump to: Editorial

8 pages, 264 KiB  
Article
Evasion Differential Game of Multiple Pursuers and One Evader for an Infinite System of Binary Differential Equations
by Gafurjan Ibragimov, Ruzakhon Kazimirova and Bruno Antonio Pansera
Mathematics 2022, 10(23), 4448; https://doi.org/10.3390/math10234448 - 25 Nov 2022
Cited by 3 | Viewed by 1251
Abstract
We study a differential evasion game of multiple pursuers and an evader governed by several infinite systems of two-block differential equations in the Hilbert space l2. Geometric constraints are imposed on the players’ control functions. If the state of a controlled [...] Read more.
We study a differential evasion game of multiple pursuers and an evader governed by several infinite systems of two-block differential equations in the Hilbert space l2. Geometric constraints are imposed on the players’ control functions. If the state of a controlled system falls into the origin of the space l2 at some finite time, then pursuit is said to be completed in a differential game. The aim of the pursuers is to transfer the state of at least one of the systems into the origin of the space l2, while the purpose of the evader is to prevent it. A sufficient evasion condition is obtained from any of the players’ initial states and an evasion strategy is constructed for the evader. Full article
(This article belongs to the Special Issue Differential Games and Its Applications)
6 pages, 257 KiB  
Article
A Note on Some Weaker Notions of Cop-Win and Robber-Win Graphs
by Shravan Luckraz, Gafurjan Ibragimov and Bruno Antonio Pansera
Mathematics 2022, 10(22), 4367; https://doi.org/10.3390/math10224367 - 20 Nov 2022
Cited by 2 | Viewed by 1434
Abstract
The game of pursuit and evasion, when played on graphs, is often referred to as the game of cops and robbers. This classical version of the game has been completely solved by Nowakowski and Winkler, who gave the exact class of graphs for [...] Read more.
The game of pursuit and evasion, when played on graphs, is often referred to as the game of cops and robbers. This classical version of the game has been completely solved by Nowakowski and Winkler, who gave the exact class of graphs for which the pursuer can win the game (cop-win). When the graph does not satisfy the dismantlability property, Nowakowski and Winkler’s Theorem does not apply. In this paper, we give some weaker notions of cop-win and robber-win graphs. In particular, we fix the number of cops to be equal to one, and we ask whether there exist sets of initial conditions for which the graph can be cop-win or robber-win. We propose some open questions related to this initial condition problem with the goal of developing both structural characterizations and algorithms that are computable in polynomial time (P) and that can solve weakly cop-win and weakly- robber-win graphs. Full article
(This article belongs to the Special Issue Differential Games and Its Applications)
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11 pages, 269 KiB  
Article
Differential Game for an Infinite System of Two-Block Differential Equations
by Gafurjan Ibragimov, Sarvinoz Kuchkarova, Risman Mat Hasim and Bruno Antonio Pansera
Mathematics 2022, 10(14), 2541; https://doi.org/10.3390/math10142541 - 21 Jul 2022
Cited by 1 | Viewed by 1412
Abstract
We present a pursuit differential game for an infinite system of two-block differential equations in Hilbert space l2. The pursuer and evader control functions are subject to integral constraints. The differential game is said to be completed if the state of [...] Read more.
We present a pursuit differential game for an infinite system of two-block differential equations in Hilbert space l2. The pursuer and evader control functions are subject to integral constraints. The differential game is said to be completed if the state of the system falls into the origin of l2 at some finite time. The purpose of the pursuer is to bring the state of the controlled system to the origin of the space l2, whereas the evader’s aim is to prevent this. For the optimal pursuit time, we obtain an equation and construct the optimal strategies for the players. Full article
(This article belongs to the Special Issue Differential Games and Its Applications)
12 pages, 7886 KiB  
Article
The Effect of Bounded Rationality on Human Cooperation with Voluntary Participation
by Luhe Yang, Duoxing Yang and Lianzhong Zhang
Mathematics 2022, 10(9), 1550; https://doi.org/10.3390/math10091550 - 5 May 2022
Cited by 2 | Viewed by 1566
Abstract
The evolution of human cooperation is an important issue concerning social science. A deep understanding of human bounded rationality is a prerequisite for promoting collective cooperation and solving social dilemmas. Here we construct an asymmetric micro-dynamic based on bounded rationality from a micro [...] Read more.
The evolution of human cooperation is an important issue concerning social science. A deep understanding of human bounded rationality is a prerequisite for promoting collective cooperation and solving social dilemmas. Here we construct an asymmetric micro-dynamic based on bounded rationality from a micro perspective by combining behavioral economics and cognitive psychology with evolutionary game theory. Asynchronously updated Monte Carlo simulations were conducted where individuals were located on a square lattice to play a voluntary public goods game. The results showed that “free riding” behaviors can be effectively suppressed in most situations. The cooperation level can be obviously enhanced in a population comprising easily satisfied cooperators and greedy defectors. Moreover, essential conditions for the stability of the system are further discussed at the microscopic level, and altruistic behavior can be explained that an individual with lower expectations for or underestimation of a single game is more likely to cooperate. We argue that, compared to traditional approaches, the integration of interdisciplinary ideas should be taken more seriously. Full article
(This article belongs to the Special Issue Differential Games and Its Applications)
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12 pages, 715 KiB  
Article
Pursuit Differential Game with Slow Pursuers on the 1-Skeleton Graph of the Icosahedron
by Gafurjan Ibragimov, Azamat Holboyev, Tolanbay Ibaydullaev and Bruno Antonio Pansera
Mathematics 2022, 10(9), 1435; https://doi.org/10.3390/math10091435 - 24 Apr 2022
Cited by 1 | Viewed by 1453
Abstract
A differential game of m, 3m6, pursuers and one evader is studied on an icosahedron in R3. All the players move only along the 1-skeleton graph of the icosahedron when the maximal speeds of the [...] Read more.
A differential game of m, 3m6, pursuers and one evader is studied on an icosahedron in R3. All the players move only along the 1-skeleton graph of the icosahedron when the maximal speeds of the pursuers are less than the speed of the evader. Pursuit is said to be completed if the state of a pursuer coincides with the state of evader at some time. We give a sufficient condition of the completion of pursuit in the game. Full article
(This article belongs to the Special Issue Differential Games and Its Applications)
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14 pages, 312 KiB  
Article
Qualitative Analysis of a Model of Renewable Resources and Population with Distributed Delays
by Tiziana Ciano, Massimiliano Ferrara and Luca Guerrini
Mathematics 2022, 10(8), 1247; https://doi.org/10.3390/math10081247 - 10 Apr 2022
Cited by 3 | Viewed by 1244
Abstract
This work generalizes Matsumoto et al.’s dynamic model of population and renewable resources by substituting a distributed delay for the time delay. It is proved that the equilibrium point may lose or gain local stability, allowing for the observation of alternating stability/instability areas [...] Read more.
This work generalizes Matsumoto et al.’s dynamic model of population and renewable resources by substituting a distributed delay for the time delay. It is proved that the equilibrium point may lose or gain local stability, allowing for the observation of alternating stability/instability areas if some conditions hold. Full article
(This article belongs to the Special Issue Differential Games and Its Applications)
4 pages, 231 KiB  
Article
A Note on the Concept of Time in Extensive Games
by Shravan Luckraz and Bruno Antonio Pansera
Mathematics 2022, 10(8), 1212; https://doi.org/10.3390/math10081212 - 7 Apr 2022
Cited by 1 | Viewed by 1582
Abstract
Using the concept of informational digraphs, we propose a “no redundant information sets” property that can characterize the exact class of extensive games which can be time structured. Our result can be applied to define time-dependent solution concepts like the Open-Loop and the [...] Read more.
Using the concept of informational digraphs, we propose a “no redundant information sets” property that can characterize the exact class of extensive games which can be time structured. Our result can be applied to define time-dependent solution concepts like the Open-Loop and the Closed-Loop Nash Equilibrium in extensive games with imperfect information. Full article
(This article belongs to the Special Issue Differential Games and Its Applications)
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