##
**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
PDF
BibTeX
XML
Cite

\textit{J. H. Dshalalow} and \textit{H.-J. Ke}, J. Math. Anal. Appl. 353, No. 2, 553--565 (2009; Zbl 1173.91308)

Full Text:
DOI

### References:

[1] | Ardema, A.; Heymann, M.; Rajan, N., Analysis of a combat problem: the turret game, J. optim. theory appl., 54, 1, 23-42, (1987) · Zbl 0595.90108 |

[2] | Bagchi, A., Stackelberg differential games in economics models, (1984), Springer-Verlag |

[3] | Basar, T.S.; Olsder, G.J., Dynamic noncooperative game theory, (1982), Academic Press Orlando · Zbl 0479.90085 |

[4] | Bingham, N.H., Random walk and fluctuation theory, (), 171-213 · Zbl 0982.60038 |

[5] | Breitner, M.H.; Pesch, H.J.; Grimm, W., Complex differential games of pursuit – evasion type with state constraints, part 1: necessary conditions for optimal open-loop strategies, J. optim. theory appl., 78, 3, 419-441, (1993) · Zbl 0796.90078 |

[6] | Breitner, M.H.; Pesch, H.J.; Grimm, W., Complex differential games of pursuit – evasion type with state constraints, part 2: numerical computation of optimal open-loop strategies, J. optim. theory appl., 78, 3, 443-463, (1993) · Zbl 0796.90079 |

[7] | Davidovitz, A.; Shinar, J., Two-target game model of an air combat with fire-and-forget all-aspect missiles, J. optim. theory appl., 63, 2, 133-165, (1989) · Zbl 0662.90103 |

[8] | () |

[9] | Dshalalow, J.H.; Galambos, J.; Gani, J., On termination time processes, Studies in applied probability; essays in honour of lajos takács, applied probability trust, Sheffield, UK, J. appl. probab., 31A, 325-336, (1994) · Zbl 0805.60092 |

[10] | Dshalalow, J.H., On the level crossing of multi-dimensional delayed renewal processes, J. appl. math. stoch. anal., 10, 4, 355-361, (1997) · Zbl 0896.60056 |

[11] | Dshalalow, J.H., Fluctuations of recurrent processes and their application to the stock market, Stoch. anal. appl., 22, 1, 67-79, (2004) · Zbl 1037.91081 |

[12] | Dshalalow, J.H., On exit times of a multivariate random walk with some applications to finance, Nonlinear anal., 63, 569-577, (2005) |

[13] | Dshalalow, J.H., Characterization of modulated Cox measures on topological spaces, Int. J. appl. math. stat., 11, 7, 21-37, (2007), a special volume in commemoration of Leonard Euler’s tricentennial · Zbl 1157.60049 |

[14] | Dshalalow, J.H., Random walk analysis in antagonistic stochastic games, Stoch. anal. appl., 26, 738-783, (2008) · Zbl 1151.91330 |

[15] | Dshalalow, J.H.; Huang, W., On noncooperative hybrid stochastic games, Nonlinear anal. hybrid syst., 2, 3, 803-811, (2008), Special Issue Section: Analysis and Design of Hybrid Systems · Zbl 1213.91033 |

[16] | Dshalalow, J.H.; Huang, W., A stochastic game with a two-phase conflict, (), 201-209 |

[17] | J.H. Dshalalow, H.-J. Ke, Layers of noncooperative games, Nonlinear Anal. Ser. A, doi:10.1016/j.na.2008.10.072, in press · Zbl 1238.91010 |

[18] | Dshalalow, J.H.; Liew, A., On exit time of a multivariate random walk and its embedding in a quasi-Poisson process, Stoch. anal. appl., 24, 451-474, (2006) · Zbl 1105.60065 |

[19] | Dshalalow, J.H.; Liew, A., On fluctuations of a multivariate random walk with some applications to stock options trading and hedging, Math. model. comput., 44, 10, 931-944, (2006) · Zbl 1133.91416 |

[20] | Dshalalow, J.H.; Liew, A., On level crossings of an oscillating marked random walk, Comput. math. appl., 52, 917-932, (2006) · Zbl 1125.60091 |

[21] | Dshalalow, J.H.; Tadj, L., On applications of first excess random processes to queueing systems with random server capacity and capacity dependent service time, Stoch. stochastic reports, 45, 45-60, (1993) · Zbl 0792.60090 |

[22] | Fishburn, P.C., Non-cooperative stochastic dominance games, Internat. J. game theory, 7, 1, 51-61, (1978) · Zbl 0372.90133 |

[23] | () |

[24] | Griffin, P.S.; McConnell, T.R., On the position of a random walk at the time of first exit from a sphere, Ann. probab., 20, 2, 825-854, (1992) · Zbl 0756.60060 |

[25] | Grigor’yan, A.; Kelbert, M., Range of fluctuations of Brownian motion on a complete Riemannian manifold, Ann. probab., 26, 1, 78-111, (1998) · Zbl 0934.58023 |

[26] | () |

[27] | () |

[28] | Isaacs, R., Differential games: A mathematical theory with applications to warfare and pursuit, control and optimization, (1999), Dover · Zbl 1233.91001 |

[29] | Jørgensen, S.; Zaccour, G., Differential games in marketing, Internat. ser. quant. mark., vol. 15, (2004), Springer-Verlag |

[30] | Kadankova, T.V., Exit, passage, and crossing times and overshoots for a Poisson compound process with an exponential component, Theory probab. math. statist., 75, 23-29, (2007) |

[31] | Kelbert, M.; Suhov, Y., Three-indexed processes: A high level crossing analysis, J. appl. math. stoch. anal., 16, 2, 127-139, (2003) · Zbl 1036.60076 |

[32] | Konstantinov, R.V.; Polovinkin, E.S., Mathematical simulation of a dynamic game in the enterprise competition problem, Cybernet. systems anal., 40, 5, 720-725, (2004) · Zbl 1132.91355 |

[33] | Kyprianou, A.E.; Pistorius, M.R., Perpetual options and canadization through fluctuation theory, Ann. appl. probab., 13, 3, 1077-1098, (2003) · Zbl 1039.60044 |

[34] | Mellander, E.; Vredin, A.; Warne, A., Stochastic trends and economic fluctuations in a small open economy, J. appl. econometrics, 7, 4, 369-394, (1992) |

[35] | Muzy, J.; Delour, J.; Bacry, E., Modelling fluctuations of financial time series: from cascade process to stochastic volatility model, Eur. phys. J. B, 17, 537-548, (2000) |

[36] | Perry, J.C.; Roitberg, B.D., Games among cannibals: competition to cannibalize and parent – offspring conflict lead to increased sibling cannibalism, J. evol. biology, 18, 6, 1523-1533, (2005) |

[37] | Redner, S., A guide to first-passage processes, (2001), Cambridge University Press Cambridge · Zbl 0980.60006 |

[38] | Segal, A.; Miloh, T., A new 3-D pursuit – evasion differential game between two bank-to-turn airborne vehicles, Optimal control appl. methods, 20, 5, 223-234, (1999) |

[39] | Shashikin, V.N., Antagonistic game with interval payoff functions, Cybernet. systems anal., 40, 4, 556-564, (2004) · Zbl 1132.91329 |

[40] | Shima, T., Capture conditions in a pursuit – evasion game between players with biproper dynamics, J. optim. theory appl., 126, 3, 503-528, (2005) · Zbl 1181.91036 |

[41] | Sobel, M., Noncooperative stochastic games, Ann. math. stat., 42, 1930-1935, (1971) · Zbl 0229.90059 |

[42] | Takács, L., On fluctuations problems in the theory of queues, Adv. in appl. probab., 8, 3, 548-583, (1976) · Zbl 0357.60021 |

[43] | Takács, L., On fluctuations of sums of random variables, (), 45-93 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.