{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:32:01Z","timestamp":1760146321801,"version":"build-2065373602"},"reference-count":24,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2024,10,24]],"date-time":"2024-10-24T00:00:00Z","timestamp":1729728000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Games"],"abstract":"<jats:p>In this study, we use the dynamic programming method introduced by Miric\u0103 (2004) to solve the well-known war game of attrition and attack as formulated by Isaacs (1965). By using this modern approach, we extend the classical framework to explore optimal strategies within the differential game setting, offering a complete, comprehensive and theoretically robust solution. Additionally, the study identifies and analyzes feedback strategies, which represent a significant advancement over other strategy types in game theory. These strategies dynamically adapt to the evolving state of the system, providing more robust solutions for real-time decision-making in conflict scenarios. This novel contribution enhances the application of game theory, particularly in the context of warfare models, and illustrates the practical advantages of incorporating feedback mechanisms into strategic decision-making. The admissible feedback strategies and the corresponding value function are constructed through a refined application of Cauchy\u2019s Method of characteristics for stratified Hamilton\u2013Jacobi equations. Their optimality is proved using a suitable Elementary Verification Theorem for the associated value function as an argument for sufficient optimality conditions.<\/jats:p>","DOI":"10.3390\/g15060035","type":"journal-article","created":{"date-parts":[[2024,10,24]],"date-time":"2024-10-24T09:11:47Z","timestamp":1729761107000},"page":"35","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["On Isaac\u2019s War Game of Attrition and Attack Using Dynamic Programming Approach"],"prefix":"10.3390","volume":"15","author":[{"given":"Benghebrid","family":"Safa","sequence":"first","affiliation":[{"name":"Laboratory of Fundamental and Numerical Mathematics, Department of Mathematics, Faculty of Sciences, University of Ferhat Abbas Setif-1, Setif 19000, Algeria"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Bouremani","family":"Touffik","sequence":"additional","affiliation":[{"name":"Laboratory of Applied Mathematics, Faculty of Technology, University of Ferhat Abbas Setif-1, Setif 19000, Algeria"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Benterki","family":"Djamel","sequence":"additional","affiliation":[{"name":"Laboratory of Fundamental and Numerical Mathematics, Department of Mathematics, Faculty of Sciences, University of Ferhat Abbas Setif-1, Setif 19000, Algeria"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,10,24]]},"reference":[{"key":"ref_1","unstructured":"Isaacs, R. (1965). Differential Games, Wiley."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Bardi, M., and Capuzzo Dolcetta, I. (1997). Optimal Control and Viscosity Solutions of Hamilton\u2013Jacobi-Bellman Equations, Birkhuser.","DOI":"10.1007\/978-0-8176-4755-1"},{"key":"ref_3","unstructured":"Ba\u015far, T., and Olsder, G. (1995). Dynamic Noncooperative Game Theory, Academic Press. [2nd ed.]."},{"key":"ref_4","unstructured":"Blaqui\u00e8re, A., and Leitmann, G. (1969). Jeux Quantitatifs, Gauthier-Villars."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"1003","DOI":"10.1016\/0021-8928(76)90135-0","article-title":"The game problem on the Dolichobrachistochrone","volume":"40","author":"Chigir","year":"1976","journal-title":"J. Appl. Math. Mech."},{"key":"ref_6","unstructured":"Kamneva, L.V., Patsko, V.S., and Turova, V.L. (2000, January 3\u20137). Analytical and numerical study of the Dolichobrachistochrone problem. Proceedings of the Analysis and Control of Deterministic and Stochastic Evolution Equation, Bressanone-Brixen, Italy."},{"key":"ref_7","first-page":"173","article-title":"On a differential game problem","volume":"4","author":"Lidov","year":"1971","journal-title":"Avtomat Telemekhan"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"67","DOI":"10.1142\/S021919890100035X","article-title":"Level sets of the value function in differential games with the Homicidal chauffeur dynamics","volume":"3","author":"Patsko","year":"2001","journal-title":"Int. Games Theory Rev."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Berkovitz, L.D. (1974). Optimal Control Theory, Springer.","DOI":"10.1007\/978-1-4757-6097-2"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"70","DOI":"10.1007\/BFb0009064","article-title":"Zero-sum differential games with terminal payoff","volume":"3","author":"Breakwell","year":"1977","journal-title":"Differ. Games Appl."},{"key":"ref_11","unstructured":"Krassovskii, N.N., and Subbotin, A.I. (1988). Positional Differential Games, Springer."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"103","DOI":"10.1007\/BF00934749","article-title":"Generalization of the main equation of differential games","volume":"43","author":"Subbotin","year":"1984","journal-title":"J. Opt. Theory Appl."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"487","DOI":"10.1090\/S0002-9947-1984-0732102-X","article-title":"Some properties of viscosity solutions of Hamilton\u2013Iacobi\u2013Bellman equations","volume":"282","author":"Crandall","year":"1984","journal-title":"Trans. AMS"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"167","DOI":"10.1142\/S0219198903000878","article-title":"Verification theorems for optimal feedback strategies in differential games","volume":"5","year":"2003","journal-title":"Int. Game Theory Rev."},{"key":"ref_15","unstructured":"Miric\u0103, \u015e. (2004). Constructive Dynamic Programming in Optimal Control Autonomous Problems, Editura Academiei Rom\u00e2ne."},{"key":"ref_16","first-page":"501","article-title":"User\u2019s Guide on Dynamic Programming for autonomous differential games and optimal control problems","volume":"49","year":"2004","journal-title":"Rev. Romaine Math. Pures Appl."},{"key":"ref_17","unstructured":"Elliot, R.J. (1987). Viscosity Solutions and Optimal Control, Longman. Pitman Research Notes in Mathematics."},{"key":"ref_18","first-page":"53","article-title":"On the solution of the homicidal chauffeur differential game","volume":"8","author":"Bouremani","year":"2006","journal-title":"Math. Rep."},{"key":"ref_19","first-page":"113","article-title":"On the solution of a simple differential game with a singular focal line","volume":"49","author":"Bouremani","year":"2006","journal-title":"Bull. Math. Soc. Sci. Math. Roum."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"733","DOI":"10.1007\/s13235-023-00523-w","article-title":"Study of a warfare differential game via Dynamic Programming approach","volume":"14","author":"Bouremani","year":"2024","journal-title":"Dyn. Games Appl."},{"key":"ref_21","first-page":"1","article-title":"Differential Game for a Class of Warfare Dynamic Systems with Reinforcement Based on Lanchester Equation","volume":"2014","author":"Chen","year":"2014","journal-title":"Abstr. Appl. Anal."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Aubin, J.P., and Cellina, A. (1984). Differential Inclusions, Springer.","DOI":"10.1007\/978-3-642-69512-4"},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Cesari, L. (1983). Optimization-Theory and Applications, Springer.","DOI":"10.1007\/978-1-4613-8165-5"},{"key":"ref_24","first-page":"23","article-title":"Hamilton\u2013Jacobi equations on possibly non-symplectic differentiable manifolds","volume":"41","year":"1998","journal-title":"Bull. Math. Soc. Sci. Math. Roumanie"}],"container-title":["Games"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-4336\/15\/6\/35\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T16:19:47Z","timestamp":1760113187000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-4336\/15\/6\/35"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,10,24]]},"references-count":24,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2024,12]]}},"alternative-id":["g15060035"],"URL":"https:\/\/doi.org\/10.3390\/g15060035","relation":{},"ISSN":["2073-4336"],"issn-type":[{"type":"electronic","value":"2073-4336"}],"subject":[],"published":{"date-parts":[[2024,10,24]]}}}