# Differential game

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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.

## 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 ${\displaystyle u(t),v(t)}$ and two criteria, one for each player. 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

The first to study differential games was Rufus Isaacs (1951, published 1965)[1] and 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.[2] 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[3][4]

## 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).[5]

For a survey of pursuit-evasion differential games see Pachter.[6]

## Notes

1. ^ Rufus Isaacs, Differential Games, Dover, 1999. ISBN 0-486-40682-2 Google Books
2. ^ Petrosjan, L.A. and Murzov, N.V. (1966). Game-theoretic problems of mechanics. Litovsk. Mat. Sb. 6, pp. 423–433 (in Russian).
3. ^ Petrosjan L.A. and Shevkoplyas E.V. Cooperative games with random duration, Vestnik of St.Petersburg Univ., ser.1, Vol.4, 2000 (in Russian)
4. ^ Marín-Solano, Jesús and Shevkoplyas, Ekaterina V. Non-constant discounting and differential games with random time horizon. Automatica, Vol. 47(12), December 2011, pp. 2626–2638.
5. ^ 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.
6. ^ Meir Pachter: Simple-motion pursuit-evasion differential games, 2002

### Textbooks and general references

• 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