Differential game explained

In game theory, differential games are a group of problems related to the modeling and analysis of conflict in the context of a dynamical system. More specifically, a state variable or variables evolve over time according to a differential equation. Early analyses reflected military interests, considering two actors—the pursuer and the evader—with diametrically opposed goals. More recent analyses have reflected engineering or economic considerations.[1] [2]

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

u1(t),u2(t)

and two criteria, one for each player.[3] 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.[4] The first to study the formal theory of differential games was Rufus Isaacs, publishing a text-book treatment in 1965.[5] 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.[6] 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[7] [8]

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).[9] 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.[10] [11]

Additionally, differential games have applications in missile guidance[12] [13] and autonomous systems.[14]

For a survey of pursuit–evasion differential games see Pachter.[15]

See also

External links

Notes and References

  1. Tembine. Hamidou. 2017-12-06. Mean-field-type games. AIMS Mathematics. 2. 4. 706–735. en. 10.3934/Math.2017.4.706. free. 2019-03-29. 2019-03-29. https://web.archive.org/web/20190329055915/http://www.aimspress.com/Math/2017/4/706. dead.
  2. Djehiche. Boualem. Tcheukam. Alain. Tembine. Hamidou. 2017-09-27. Mean-Field-Type Games in Engineering. AIMS Electronics and Electrical Engineering. 1. 18–73. en. 10.3934/ElectrEng.2017.1.18. 1605.03281. 16055840. 2019-03-29. 2019-03-29. https://web.archive.org/web/20190329055917/http://www.aimspress.com/ElectrEng/2017/1/18. dead.
  3. Book: Morton I. . Kamien . Morton Kamien . Nancy L. . Schwartz . Dynamic Optimization : The Calculus of Variations and Optimal Control in Economics and Management . Amsterdam . North-Holland . 1991 . 0-444-01609-0 . Differential Games . 272–288 . https://books.google.com/books?id=liLCAgAAQBAJ&pg=PA272 .
  4. C. F. . Roos . A Mathematical Theory of Competition . . 47 . 3 . 1925 . 163–175 . 2370550 . 10.2307/2370550 .
  5. Book: Isaacs, Rufus . Differential Games: A Mathematical Theory with Applications to Warfare and Pursuit, Control and Optimization . London . John Wiley and Sons . 1965 . Dover . 1999 . 0-486-40682-2 . Google Books .
  6. Petrosjan . L.A. . Murzov . N.V. . 1966 . Game-theoretic problems of mechanics . Litovsk. Mat. Sb. . 6 . 423–433 . ru.
  7. Petrosjan . L.A. . Shevkoplyas . E.V. . Cooperative games with random duration . Vestnik of St.Petersburg Univ. . 4 . 1 . 2000 . ru.
  8. Marín-Solano . Jesús . Shevkoplyas . Ekaterina V. . Non-constant discounting and differential games with random time horizon . Automatica . 47 . 12 . December 2011 . 2626–2638. 10.1016/j.automatica.2011.09.010 .
  9. Leong . C. K. . Huang . W. . 10.1016/j.jmateco.2010.03.007 . A stochastic differential game of capitalism . Journal of Mathematical Economics . 46 . 4 . 552 . 2010 . 5025474 .
  10. Web site: American Economic Association. www.aeaweb.org. en. 2017-08-21.
  11. Tembine. H.. Duncan. Tyrone E.. 2018. Linear–Quadratic Mean-Field-Type Games: A Direct Method. Games. en. 9. 1. 7. 10.3390/g9010007. free. 10419/179168. free.
  12. Anderson . Gerald M. . 1981 . Comparison of Optimal Control and Differential Game Intercept Missile Guidance Laws . Journal of Guidance and Control . 4 . 2 . 109–115 . 10.2514/3.56061 . 1981JGCD....4..109A . 0162-3192.
  13. Pontani . Mauro . Conway . Bruce A. . 2008 . Optimal Interception of Evasive Missile Warheads: Numerical Solution of the Differential Game . Journal of Guidance, Control, and Dynamics . 31 . 4 . 1111–1122 . 10.2514/1.30893. 2008JGCD...31.1111C .
  14. Book: Faruqi, Farhan A. . Differential Game Theory with Applications to Missiles and Autonomous Systems Guidance . Wiley . 2017 . 978-1-119-16847-8 . Aerospace Series.
  15. Web site: Simple-motion pursuit–evasion differential games . Meir . Pachter . 2002 . July 20, 2011 . https://web.archive.org/web/20110720011925/http://med.ee.nd.edu/MED10/pdf/477.pdf .

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