Abstract
We consider a class of antagonistic stochastic games in real time between two players A and B, formalized by two marked point processes. The players attack each other at random times with random impacts. Either player can sustain casualties up to a fixed threshold. A player is defeated when its underlying threshold is crossed. Upon that time (referred to as the first passage time), the game is over. We introduce a joint functional of the first passage time, along with the status of each player upon this time. The latter are the cumulative magnitudes of casualties to each player upon the end of the game, obtained in an analytically tractable form. We then use discrete and continuous operational calculus for the transform inversion. We demonstrate in a special case that the discrete operational calculus is more efficient, allowing us to avoid numerical inversion. It leads to explicit formulas for the joint distribution of associated random variables (first passage time and the status of cumulative casualties to the players upon the end of the game).
Similar content being viewed by others
References
Agarwal, R.P., Dshalalow, J.H., O’Regan, D.: Random observations of marked Cox processes. Time insensitive functionals. J. Math. Anal. Appl. 293, 1–13 (2004)
Al-Matar, N., Dshalalow, J.H.: Maintenance in single-server queues: a game-theoretic approach. Math. Probl. Eng. (2009). doi:10.1155/2009/857871
Al-Matar, N., Dshalalow, J.H.: A game-theoretic approach in single-server queues with maintenance. Time sensitive analysis. Commun. Appl. Nonlinear Anal. 17(1), 65–92 (2010)
Ardema, A., Heymann, M., Rajan, N.: Analysis of a combat problem: the turret game. J. Optim. Theory Appl. 54(1), 23–42 (1987)
Bagchi, A.: Stackelberg Differential Games in Economics Models. Springer, New York (1984)
Basar, T.S., Olsder, G.J.: Dynamic Noncooperative Game Theory. Academic Press, Orlando (1982)
Dockner, E., Jørgensen, S., Long, N.V., Sorger, G. (eds.): Differential Games in Economics and Management Science. Cambridge University Press, Cambridge (2000)
Dshalalow, J.H.: First excess level of vector processes. J. Appl. Math. Stoch. Anal. 7(3), 457–464 (1994)
Dshalalow, J.H.: On the level crossing of multidimensional delayed renewal processes. J. Appl. Math. Stoch. Anal. 10(4), 355–361 (1997). (special Issue stochastic systems)
Dshalalow, J.H.: Fluctuations of recurrent processes and their application to the stock market. Stoch. Anal. Appl. 22(1), 67–79 (2004)
Dshalalow, J.H.: On exit times of a multivariate random walk with some applications to finance. Nonlinear Anal. 63, 569–577 (2005)
Dshalalow, J.H.: Stochastic Processes. Lecture Notes, FIT, Melbourne (2015)
Dshalalow, J.H., Huang, W.: Tandem antagonistic games. Nonlinear Anal. Ser. A Theory Methods 71, 259–270 (2009)
Dshalalow, J.H., Robinson, R.: On one-sided stochastic games and their applications to finance. Stoch. Models 28(1), 1–14 (2012)
Dshalalow, J.H., Iwezulu, K., White, R.T.: Discrete operational calculus in delayed stochastic games. Neural Parallel Sci. Comput. 24, 55–64 (2016)
Dshalalow, J.H., Ke, H.-J.: Multilayers in a modulated stochastic game. J. Math. Anal. Appl. 353, 553–565 (2009)
Dshalalow, J.H., Treerattrakoon, A.: Set-theoretic inequalities in stochastic noncooperative games with coalition. J. Inequal. Appl. 2008, 713642 (2008)
Fishburn, P.C.: Non-cooperative stochastic dominance games. Int. J. Game Theory 7(1), 51–61 (1978)
Isaacs, R.: Differential Games: A Mathematical Theory with Applications to Warfare and Pursuit, Control and Optimization. Dover, New York (1999)
Jørgensen, S., Zaccour, G.: Differential Games in Marketing. International Series in Quantitative Marketing, vol. 15. Springer, New York (2004)
Kadankov, V.F., Kadankova, T.V.: On the distribution of the moment of the first exit time from an interval and value of overjump through borders interval for the processes with independent increments and random walk. Random Oper. Stoch. Equ. 13(3), 219–244 (2005)
Kadankova, T.V.: Exit, passage, and crossing times and overshoots for a Poisson compound process with an exponential component. Theor. Probab. Math. Stat. 75, 23–29 (2007)
Konstantinov, R.V., Polovinkin, E.S.: Mathematical simulation of a dynamic game in the enterprise competition problem. Cybern. Syst. Anal. 40(5), 720–725 (2004)
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)
Perry, J.C., Roitberg, B.D.: Games among cannibals: competition to cannibalize and parent-offspring conflict lead to increased sibling cannibalism. J. Evol. Biol. 18(6), 1523–1533 (2005)
Segal, A., Miloh, T.: A new 3-D pursuit-evasion differential game between two bank-to-turn airborne vehicles. Optim. Control Appl. Methods 20(5), 223–234 (1999)
Shima, T.: Capture a conditions in a pursuit-evasion game between players with biproper dynamics. J. Optim. Theory Appl. 126(3), 503–528 (2005)
Takács, L.: On fluctuations of sums of random variables. In: Rota, G.-C. (ed.) Studies in Probability and Ergodic Theory. Advances in Mathematics. Supplementary Studies, vol. 2, pp. 45–93. Academic Press, New York (1978)
Acknowledgements
The authors are indebted to anonymous referees who made numerous very valuable suggestions that hugely improved the paper.
Author information
Authors and Affiliations
Corresponding author
Appendix
Appendix
Proof of Theorem 4
In the sequel, for brevity we denote \(\eta =\gamma \left( \theta \right) =\frac{\gamma }{\gamma +\theta }\). From (3.1) and (3.2) we have
After a straightforward algebra we get
where
Continuing with the expression in braces we arrive at
Furthermore,
implying that
Substituting (A.2) and (A.3) in (A.1) and applying Theorem 3(iv) and (vi), after some algebra, we get
Using formula (1.7) of Theorem 1 and recalling that \(\eta =\frac{\gamma }{\gamma +\theta }\) we arrive at formula (3.3) and herewith complete the proof. \(\square \)
Remark 1
In the proof of Theorem 4, we tacitly assumed that \(\xi =\frac{\alpha u\left( 1-\beta vy\right) }{1-\beta vy-ab\eta }\ne 1\) of (A.0). It seems as if we missed to consider \(\xi =1\) as a distinct case. First we note that formally, by Theorem 3 (iv), \(\mathcal {D}^{M-1}_x\frac{1}{1-\xi x}=M\) and thus this version of (A.1) would read
However, under this assumption, \(y,\alpha ,a,\theta ,u,v\) end up being interdependent which is unacceptable. We therefore see that \(\xi \) cannot be equal 1 or any other constant. For example, with the choice of \(u=v=1,\theta =0,\) not only will y and constants \(\alpha ,a,\gamma \) become dependent, but it eventually leads to forcing y to be a constant, which is absurd. \(\square \)
Rights and permissions
About this article
Cite this article
Dshalalow, J.H., Iwezulu, K. Discrete versus continuous operational calculus in antagonistic stochastic games. São Paulo J. Math. Sci. 11, 471–489 (2017). https://doi.org/10.1007/s40863-017-0073-9
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40863-017-0073-9
Keywords
- Noncooperative stochastic games
- Fluctuation theory
- Marked point processes
- Poisson process
- Ruin time
- Exit time
- First passage time
- Modified Bessel functions