A hybrid (differential-stochastic) zero-sum game with a fast stochastic part

Eitan Altman; Vladimir Gaitsgory

doi:10.1007/978-1-4612-4274-1_3

A hybrid (differential-stochastic) zero-sum game with a fast stochastic part

Research output: Chapter in Book/Report/Conference proceeding › Conference proceeding contribution › peer-review

Abstract

We consider in this paper a continuous time stochastic hybrid system
with a finite time horizon, controlled by two players with opposite
objectives (zero-sum game). Player one wishes to maximize some linear
function of the expected state trajectory, and player two wishes to
minimize it. The state evolves according to a linear dynamic. The parameters
of the state evolution equation may change at discrete times
according to a MDP, i.e., a Markov chain that is directly controlled by
both players, and has a countable state space. Each player has a finite
action space. We use a procedure similar in form to the maximum
principle; this determines a pair of stationary strategies for the players,
which is asymptotically a saddle point, as the number of transitions
during the finite time horizon grows to infinity.

Original language	English
Title of host publication	New trends in dynamic games and applications
Editors	GJ Olsder
Place of Publication	Boston, MU
Publisher	Birkhauser Boston
Pages	47-59
Number of pages	13
ISBN (Electronic)	9781461242741
ISBN (Print)	9781461287193
DOIs	https://doi.org/10.1007/978-1-4612-4274-1_3
Publication status	Published - 1995
Externally published	Yes
Event	6th International Symposium on Dynamic Games and Applications - ST JOVITE, Canada Duration: 13 Jul 1994 → 15 Jul 1994

Publication series

Name	ANNALS OF THE INTERNATIONAL SOCIETY OF DYNAMIC GAMES
Publisher	BIRKHAUSER BOSTON
Volume	3

Conference

Conference	6th International Symposium on Dynamic Games and Applications
Country/Territory	Canada
City	ST JOVITE
Period	13/07/94 → 15/07/94

Keywords

hybrid stochastic systems
stochastic games
asymptotic optimality
linear dynamics
Markov decision processes
finite horizon

Access to Document

10.1007/978-1-4612-4274-1_3

Cite this

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title = "A hybrid (differential-stochastic) zero-sum game with a fast stochastic part",

abstract = "We consider in this paper a continuous time stochastic hybrid systemwith a finite time horizon, controlled by two players with oppositeobjectives (zero-sum game). Player one wishes to maximize some linearfunction of the expected state trajectory, and player two wishes tominimize it. The state evolves according to a linear dynamic. The parametersof the state evolution equation may change at discrete timesaccording to a MDP, i.e., a Markov chain that is directly controlled byboth players, and has a countable state space. Each player has a finiteaction space. We use a procedure similar in form to the maximumprinciple; this determines a pair of stationary strategies for the players,which is asymptotically a saddle point, as the number of transitionsduring the finite time horizon grows to infinity.",

keywords = "hybrid stochastic systems, stochastic games, asymptotic optimality, linear dynamics, Markov decision processes, finite horizon",

author = "Eitan Altman and Vladimir Gaitsgory",

year = "1995",

doi = "10.1007/978-1-4612-4274-1_3",

language = "English",

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series = "ANNALS OF THE INTERNATIONAL SOCIETY OF DYNAMIC GAMES",

publisher = "Birkhauser Boston",

pages = "47--59",

editor = "GJ Olsder",

booktitle = "New trends in dynamic games and applications",

address = "United States",

note = "6th International Symposium on Dynamic Games and Applications ; Conference date: 13-07-1994 Through 15-07-1994",

}

Altman, E & Gaitsgory, V 1995, A hybrid (differential-stochastic) zero-sum game with a fast stochastic part. in GJ Olsder (ed.), New trends in dynamic games and applications. ANNALS OF THE INTERNATIONAL SOCIETY OF DYNAMIC GAMES, vol. 3, Birkhauser Boston, Boston, MU, pp. 47-59, 6th International Symposium on Dynamic Games and Applications, ST JOVITE, Canada, 13/07/94. https://doi.org/10.1007/978-1-4612-4274-1_3

A hybrid (differential-stochastic) zero-sum game with a fast stochastic part. / Altman, Eitan; Gaitsgory, Vladimir.
New trends in dynamic games and applications. ed. / GJ Olsder. Boston, MU: Birkhauser Boston, 1995. p. 47-59 (ANNALS OF THE INTERNATIONAL SOCIETY OF DYNAMIC GAMES; Vol. 3).

Research output: Chapter in Book/Report/Conference proceeding › Conference proceeding contribution › peer-review

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N2 - We consider in this paper a continuous time stochastic hybrid systemwith a finite time horizon, controlled by two players with oppositeobjectives (zero-sum game). Player one wishes to maximize some linearfunction of the expected state trajectory, and player two wishes tominimize it. The state evolves according to a linear dynamic. The parametersof the state evolution equation may change at discrete timesaccording to a MDP, i.e., a Markov chain that is directly controlled byboth players, and has a countable state space. Each player has a finiteaction space. We use a procedure similar in form to the maximumprinciple; this determines a pair of stationary strategies for the players,which is asymptotically a saddle point, as the number of transitionsduring the finite time horizon grows to infinity.

AB - We consider in this paper a continuous time stochastic hybrid systemwith a finite time horizon, controlled by two players with oppositeobjectives (zero-sum game). Player one wishes to maximize some linearfunction of the expected state trajectory, and player two wishes tominimize it. The state evolves according to a linear dynamic. The parametersof the state evolution equation may change at discrete timesaccording to a MDP, i.e., a Markov chain that is directly controlled byboth players, and has a countable state space. Each player has a finiteaction space. We use a procedure similar in form to the maximumprinciple; this determines a pair of stationary strategies for the players,which is asymptotically a saddle point, as the number of transitionsduring the finite time horizon grows to infinity.

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KW - stochastic games

KW - asymptotic optimality

KW - linear dynamics

KW - Markov decision processes

KW - finite horizon

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A hybrid (differential-stochastic) zero-sum game with a fast stochastic part

Abstract

Publication series

Conference

Keywords

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Cite this