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 and a single criterion to be optimized; differential game theory generalizes this to two controls 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]}

## See also

## Notes

**^**Tembine, Hamidou (2017-12-06). "Mean-field-type games".*AIMS Mathematics*.**2**(4): 706–735. doi:10.3934/Math.2017.4.706.**^**Djehiche, Boualem; Tcheukam, Alain; Tembine, Hamidou (2017-09-27). "Mean-Field-Type Games in Engineering".*AIMS Electronics and Electrical Engineering*.**1**: 18–73. arXiv:1605.03281. doi:10.3934/ElectrEng.2017.1.18.**^**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.**^**Roos, C. F. (1925). "A Mathematical Theory of Competition".*American Journal of Mathematics*.**47**(3): 163–175. doi:10.2307/2370550. JSTOR 2370550.**^**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.**^**Petrosjan, L.A.; Murzov, N.V. (1966). "Game-theoretic problems of mechanics".*Litovsk. Mat. Sb.*(in Russian).**6**: 423–433.**^**Petrosjan, L.A.; Shevkoplyas, E.V. (2000). "Cooperative games with random duration".*Vestnik of St.Petersburg Univ.*(in Russian).**4**(1).**^**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.**^**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.**^**"American Economic Association".*www.aeaweb.org*. Retrieved 2017-08-21.**^**Tembine, H.; Duncan, Tyrone E. (2018). "Linear–Quadratic Mean-Field-Type Games: A Direct Method".*Games*.**9**(1): 7. doi:10.3390/g9010007.**^**Pachter, Meir (2002). "Simple-motion pursuit-evasion differential games" (PDF). Archived from the original (PDF) on July 20, 2011.

## 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, Vol 2, World Scientific Publishers, ISBN 978-981-02-0979-7`|volume=`

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## External links

- Bressan, Alberto (December 8, 2010). "Noncooperative Differential Games: A Tutorial" (PDF). Department of Mathematics, Penn State University.