Differential game

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 $u(t)$ and a single criterion to be optimized; differential game theory generalizes this to two controls $u(t),v(t)$ 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

External links

• Noncooperative differential games: a tutorial