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**Multilayers in a modulated stochastic game.**
*(English)*
Zbl 1173.91308

Authors’ abstract: We are concerned with an antagonistic stochastic game between two players \(A\) and \(B\) which finds applications in economics and warfare. The actions of the players are manifested by a series of strikes of random magnitudes at random times exerted by each player against his opponent. Each of the assaults inflicts a random damage to enemy’s vital areas. In contrast with traditional games, in our setting, each player can endure multiple strikes before perishing. Predicting the ruin time (exit) of player \(A\), along with the total amount of casualties to both players at the exit is a main objective of this work. In contrast to the time sensitive analysis (earlier developed to refine the information on the game) we insert auxiliary control levels, which both players will cross in due time before the ruin of \(A\). This gives \(A\) (and also \(B\)) an additional opportunity to reevaluate his strategy and change the course of the game. We formalize such a game and also allow the real time information about the game to be randomly delayed. The delayed exit time, cumulative casualties to both players, and prior crossing are then obtained in a closed-form joint functional.

Reviewer: Tadeusz Radzik (Jelenia Góra)

### MSC:

91A15 | Stochastic games, stochastic differential games |

91A05 | 2-person games |

91A10 | Noncooperative games |

91A40 | Other game-theoretic models |

### Keywords:

stochastic game; two-person game; antagonistic game; fluctuation; critical levels; risk analysis; modulated game
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\textit{J. H. Dshalalow} and \textit{H.-J. Ke}, J. Math. Anal. Appl. 353, No. 2, 553--565 (2009; Zbl 1173.91308)

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