# Differential game

Last updated

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 ${\displaystyle u(t)}$ and a single criterion to be optimized; differential game theory generalizes this to two controls ${\displaystyle u_{1}(t),u_{2}(t)}$ 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 Clark Medal from the American Economic Association for his contributions to the analysis of continuous time dynamic games using stochastic calculus methods. [10] [11]

For a survey of pursuit-evasion differential games see Pachter. [12]

## Notes

1. Tembine, Hamidou (2017-12-06). "Mean-field-type games". AIMS Mathematics. 2 (4): 706–735. doi:10.3934/Math.2017.4.706.
2. Djehiche, Boualem; Tcheukam, Alain; Tembine, Hamidou (2017-09-27). "Mean-Field-Type Games in Engineering". AIMS Electronics and Electrical Engineering. 1: 18–73. arXiv:. doi:10.3934/ElectrEng.2017.1.18.
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.
4. Roos, C. F. (1925). "A Mathematical Theory of Competition". American Journal of Mathematics . 47 (3): 163–175. doi:10.2307/2370550. JSTOR   2370550.
5. Isaacs, Rufus (1999) [1965]. 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 via Google Books.
6. Petrosjan, L.A.; Murzov, N.V. (1966). "Game-theoretic problems of mechanics". Litovsk. Mat. Sb. (in Russian). 6: 423–433.
7. Petrosjan, L.A.; Shevkoplyas, E.V. (2000). "Cooperative games with random duration". Vestnik of St.Petersburg Univ. (in Russian). 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". www.aeaweb.org. Retrieved 2017-08-21.
11. Tembine, H.; Duncan, Tyrone E. (2018). "Linear–Quadratic Mean-Field-Type Games: A Direct Method". Games. 9 (1): 7. doi:10.3390/g9010007.
12. Pachter, Meir (2002). "Simple-motion pursuit-evasion differential games" (PDF). Archived from the original (PDF) on July 20, 2011.
• 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, Vol 2), World Scientific Publishers, ISBN   978-981-02-0979-7