Differential game

In game theory, differential games are dynamic games that unfold in continuous time, meaning players’ actions and outcomes evolve smoothly rather than in discrete steps,{{Citation |last=Van Long |first=N. |title=Differential Games and Resources |date=2013-01-01 |work=Encyclopedia of Energy, Natural Resource, and Environmental Economics |pages=268–276 |editor-last=Shogren |editor-first=Jason F. |url=https://linkinghub.elsevier.com/retrieve/pii/B9780123750679001480 |access-date=2025-02-20 |place=Waltham |publisher=Elsevier |doi=10.1016/b978-0-12-375067-9.00148-0 |isbn=978-0-08-096452-2}} and for which the rate of change of each state variable—like position, speed, or resource level—is governed by a differential equation. This distinguishes them from turn-based games (sequential games) like chess, focusing instead on real-time strategic conflicts.

Differential games are sometimes called continuous-time games, a broader term that includes them.{{Cite book |last=Başar |first=Tamer |url=https://epubs.siam.org/doi/book/10.1137/1.9781611971132 |title=Dynamic Noncooperative Game Theory, 2nd Edition |last2=Olsder |first2=Geert Jan |date=1998 |publisher=Society for Industrial and Applied Mathematics |isbn=978-0-89871-429-6 |edition=2nd |pages=283 |language=en |doi=10.1137/1.9781611971132}} While the two overlap significantly, continuous-time games also encompass models not governed by differential equations, such as those with stochastic jump processes, where abrupt, unpredictable events introduce discontinuities

Early differential games, often inspired by military scenarios, modeled situations like a pursuer chasing an evader, such as a missile targeting an aircraft.{{Cite journal |last=Berkovitz |first=Leonard D. |date=1986 |title=Differential Games of Generalized Pursuit and Evasion |url=https://epubs.siam.org/doi/10.1137/0324021 |journal=SIAM Journal on Control and Optimization |volume=24 |issue=3 |pages=361–373 |doi=10.1137/0324021 |issn=0363-0129}} Today, they also apply to fields like economics and engineering, analyzing competition over resources or the control of moving systems.{{Cite journal |last=Tembine |first=Hamidou |date=2017-12-06 |title=Mean-field-type games |url=http://www.aimspress.com/Math/2017/4/706 |url-status=dead |journal=AIMS Mathematics |language=en |volume=2 |issue=4 |pages=706–735 |doi=10.3934/Math.2017.4.706 |archive-url=https://web.archive.org/web/20190329055915/http://www.aimspress.com/Math/2017/4/706 |archive-date=2019-03-29 |access-date=2019-03-29 |doi-access=free}}{{Cite journal |last1=Djehiche |first1=Boualem |last2=Tcheukam |first2=Alain |last3=Tembine |first3=Hamidou |date=2017-09-27 |title=Mean-Field-Type Games in Engineering |url=http://www.aimspress.com/ElectrEng/2017/1/18 |url-status=dead |journal=AIMS Electronics and Electrical Engineering |language=en |volume=1 |pages=18–73 |arxiv=1605.03281 |doi=10.3934/ElectrEng.2017.1.18 |s2cid=16055840 |archive-url=https://web.archive.org/web/20190329055917/http://www.aimspress.com/ElectrEng/2017/1/18 |archive-date=2019-03-29 |access-date=2019-03-29}}

Connection to optimal control

Differential games are related closely with optimal control problems. In an optimal control problem there is single control $u(t)$ and a single criterion to be optimized; differential game theory generalizes this to two controls $u_{1}(t),u_{2}(t)$ and two criteria, one for each player.{{cite book |first1=Morton I. |last1=Kamien |author-link=Morton Kamien |first2=Nancy L. |last2=Schwartz |title=Dynamic Optimization : The Calculus of Variations and Optimal Control in Economics and Management |location=Amsterdam |publisher=North-Holland |year=1991 |isbn=0-444-01609-0 |chapter=Differential Games |pages=272–288 |chapter-url=https://books.google.com/books?id=liLCAgAAQBAJ&pg=PA272 }} Each player attempts to control the state of the system so as to achieve its goal; the system responds to the inputs of all players.

History

In the study of competition, differential games have been employed since a 1925 article by Charles F. Roos.{{cite journal |first=C. F. |last=Roos |title=A Mathematical Theory of Competition |journal=American Journal of Mathematics |volume=47 |issue=3 |year=1925 |pages=163–175 |jstor=2370550 |doi=10.2307/2370550 }} The first to study the formal theory of differential games was Rufus Isaacs, publishing a text-book treatment in 1965.{{cite book |first=Rufus |last=Isaacs |title=Differential Games: A Mathematical Theory with Applications to Warfare and Pursuit, Control and Optimization |location=London |publisher=John Wiley and Sons |orig-year=1965 |edition=Dover |year=1999 |isbn=0-486-40682-2 |url=https://books.google.com/books?id=XIxmMyIQgm0C |via=Google Books }} One of the first games analyzed was the 'homicidal chauffeur game'.

Random time horizon

Games with a random time horizon are a particular case of differential games.{{cite journal | last1=Petrosjan | first1=L.A. | last2=Murzov | first2=N.V. | date=1966 | title=Game-theoretic problems of mechanics | journal=Litovsk. Mat. Sb. | volume=6 | pages=423–433 | language=ru}} In such games, the terminal time is a random variable with a given probability distribution function. Therefore, the players maximize the mathematical expectancy of the cost function. It was shown that the modified optimization problem can be reformulated as a discounted differential game over an infinite time interval{{cite journal | last1=Petrosjan | first1=L.A. | last2=Shevkoplyas | first2=E.V. | title=Cooperative games with random duration | journal=Vestnik of St.Petersburg Univ. | volume=4 | issue=1 | date=2000 | language=ru}}{{cite journal | last1=Marín-Solano | first1=Jesús | last2=Shevkoplyas | first2=Ekaterina V. | title=Non-constant discounting and differential games with random time horizon | journal=Automatica | volume=47 | issue=12 | date=December 2011 | pages=2626–2638| doi=10.1016/j.automatica.2011.09.010 }}

Applications

Differential games have been applied to economics. Recent developments include adding stochasticity to differential games and the derivation of the stochastic feedback Nash equilibrium (SFNE). A recent example is the stochastic differential game of capitalism by Leong and Huang (2010).{{Cite journal | last1 = Leong | first1 = C. K. | last2 = Huang | first2 = W. | doi = 10.1016/j.jmateco.2010.03.007 | title = A stochastic differential game of capitalism | journal = Journal of Mathematical Economics | volume = 46 | issue = 4 | pages = 552 | year = 2010 | s2cid = 5025474 }} In 2016 Yuliy Sannikov received the John Bates Clark Medal from the American Economic Association for his contributions to the analysis of continuous-time dynamic games using stochastic calculus methods.{{Cite web|url=https://www.aeaweb.org/about-aea/honors-awards/bates-clark/yuliy-sannikov|title=American Economic Association|website=www.aeaweb.org|language=en|access-date=2017-08-21}}{{Cite journal|last1=Tembine|first1=H.|last2=Duncan|first2=Tyrone E.|date=2018|title=Linear–Quadratic Mean-Field-Type Games: A Direct Method|journal=Games|language=en|volume=9|issue=1|pages=7|doi=10.3390/g9010007|doi-access=free|hdl=10419/179168|hdl-access=free}}

Additionally, differential games have applications in missile guidance{{Cite journal |last=Anderson |first=Gerald M. |date=1981 |title=Comparison of Optimal Control and Differential Game Intercept Missile Guidance Laws |url=https://doi.org/10.2514/3.56061 |journal=Journal of Guidance and Control |volume=4 |issue=2 |pages=109–115 |doi=10.2514/3.56061 |bibcode=1981JGCD....4..109A |issn=0162-3192}}{{Cite journal |last1=Pontani |first1=Mauro |last2=Conway |first2=Bruce A. |date=2008 |title=Optimal Interception of Evasive Missile Warheads: Numerical Solution of the Differential Game |url=https://doi.org/10.2514/1.30893 |journal=Journal of Guidance, Control, and Dynamics |volume=31 |issue=4 |pages=1111–1122 |doi=10.2514/1.30893|bibcode=2008JGCD...31.1111C }} and autonomous systems.{{Cite book |last=Faruqi |first=Farhan A. |title=Differential Game Theory with Applications to Missiles and Autonomous Systems Guidance |publisher=Wiley |year=2017 |isbn=978-1-119-16847-8 |series=Aerospace Series}}

For a survey of pursuit–evasion differential games see Pachter.{{cite web |url=http://med.ee.nd.edu/MED10/pdf/477.pdf |first=Meir |last=Pachter |title=Simple-motion pursuit–evasion differential games |year=2002 |archive-date=July 20, 2011 |archive-url=https://web.archive.org/web/20110720011925/http://med.ee.nd.edu/MED10/pdf/477.pdf }}

Notes

External links

{{cite web |first=Alberto |last=Bressan |author-link=Alberto Bressan |url=http://personal.psu.edu/axb62/PSPDF/game-lnew.pdf |title=Noncooperative Differential Games: A Tutorial |date=December 8, 2010 |publisher=Department of Mathematics, Penn State University }}

Category:Control theory

Category:Game theory game classes

Category:Ballistics

Category:Pursuit–evasion

Category:Combat modeling