Differential game

From HandWiki

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 [math]\displaystyle{ u(t) }[/math] and a single criterion to be optimized; differential game theory generalizes this to two controls [math]\displaystyle{ u_{1}(t),u_{2}(t) }[/math] 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

Notes

  1. ↑ Tembine, Hamidou (2017-12-06). "Mean-field-type games" (in en). AIMS Mathematics 2 (4): 706–735. doi:10.3934/Math.2017.4.706. http://www.aimspress.com/Math/2017/4/706. Retrieved 2019-03-29. 
  2. ↑ Djehiche, Boualem; Tcheukam, Alain; Tembine, Hamidou (2017-09-27). "Mean-Field-Type Games in Engineering" (in en). AIMS Electronics and Electrical Engineering 1: 18–73. doi:10.3934/ElectrEng.2017.1.18. http://www.aimspress.com/ElectrEng/2017/1/18. Retrieved 2019-03-29. 
  3. ↑ Kamien, Morton I.; Schwartz, Nancy L. (1991). "Differential Games". Dynamic Optimization : The Calculus of Variations and Optimal Control in Economics and Management. Amsterdam: North-Holland. pp. 272–288. ISBN 0-444-01609-0. https://books.google.com/books?id=liLCAgAAQBAJ&pg=PA272. 
  4. ↑ Roos, C. F. (1925). "A Mathematical Theory of Competition". American Journal of Mathematics 47 (3): 163–175. doi:10.2307/2370550. 
  5. ↑ Isaacs, Rufus (1999). Differential Games: A Mathematical Theory with Applications to Warfare and Pursuit, Control and Optimization (Dover ed.). London: John Wiley and Sons. ISBN 0-486-40682-2. https://books.google.com/books?id=XIxmMyIQgm0C. 
  6. ↑ Petrosjan, L.A.; Murzov, N.V. (1966). "Game-theoretic problems of mechanics" (in ru). Litovsk. Mat. Sb. 6: 423–433. 
  7. ↑ Petrosjan, L.A.; Shevkoplyas, E.V. (2000). "Cooperative games with random duration" (in ru). Vestnik of St.Petersburg Univ. 4 (1). 
  8. ↑ Marín-Solano, Jesús; Shevkoplyas, Ekaterina V. (December 2011). "Non-constant discounting and differential games with random time horizon". Automatica 47 (12): 2626–2638. doi:10.1016/j.automatica.2011.09.010. 
  9. ↑ Leong, C. K.; Huang, W. (2010). "A stochastic differential game of capitalism". Journal of Mathematical Economics 46 (4): 552. doi:10.1016/j.jmateco.2010.03.007. 
  10. ↑ "American Economic Association" (in en). https://www.aeaweb.org/about-aea/honors-awards/bates-clark/yuliy-sannikov. 
  11. ↑ Tembine, H.; Duncan, Tyrone E. (2018). "Linear–Quadratic Mean-Field-Type Games: A Direct Method" (in en). Games 9 (1): 7. doi:10.3390/g9010007. 
  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. doi:10.2514/3.56061. ISSN 0162-3192. Bibcode: 1981JGCD....4..109A. https://doi.org/10.2514/3.56061. 
  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. doi:10.2514/1.30893. Bibcode: 2008JGCD...31.1111C. https://doi.org/10.2514/1.30893. 
  14. ↑ Faruqi, Farhan A. (2017). Differential Game Theory with Applications to Missiles and Autonomous Systems Guidance. Aerospace Series. Wiley. ISBN 978-1-119-16847-8. 
  15. ↑ Pachter, Meir (2002). "Simple-motion pursuit–evasion differential games". http://med.ee.nd.edu/MED10/pdf/477.pdf. 

Further reading

  • Dockner, Engelbert; Jorgensen, Steffen; Long, Ngo Van; Sorger, Gerhard (2001), Differential Games in Economics and Management Science, Cambridge University Press, ISBN 978-0-521-63732-9 
  • Petrosyan, Leon (1993), Differential Games of Pursuit, Series on Optimization, 2, World Scientific Publishers, ISBN 978-981-02-0979-7 

External links



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