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References [1] L. V. Allis, H. J. van den Herik and M. P. Huntjens, Go-Moku solved by new search techniques. Proc. 1993 AAAI Fall Symp. on Games: Planning and Learning, AAAI Press Tech. Report FS93-02 , 1-9, Menlo Park. [2] J. Beck, Positional games and the second moment method. Combinatorica , 22 (2002), 169–216. [3] J. Beck, Positional Games, Combinatorics, Probability and Computing , 14 (2005), 649–696. [4] J. Beck, Combinatorial Games, Tic-Tac-Toe Theory , Cambridge University Press 2008. [5] E. R. Berlekamp, J. H. Conway and R. K. Guy, Winning Ways


The game theory was created on the basis of social as well as gambling games, such as chess, poker, baccarat, hex, or one-armed bandit. The aforementioned games lay solid foundations for analogous mathematical models (e.g., hex), artificial intelligence algorithms (hex), theoretical analysis of computational complexity attributable to various numerical problems (baccarat), as well as illustration of several economic dilemmas - particularly in the case where the winner takes everything (e.g., noughts and crosses). A certain gambling games, such as a horse racing, may be successfully applied to verify a wide spectrum of market mechanism, for example, market effectiveness or customer behavior in light of incoming information regarding a specific product. One of a lot applications of the slot machine (one-armed bandit) is asymptotically efficient allocation rule, which was assigned by T.L. Lai and H. Robbins (1985). In the next years, the rule was developed by another and was named a multi-armed. The aim of the paper is to discuss these social games along with their potential mathematical models, which are governed by the rules predominantly applicable to the social and natural sciences.

The paper investigates the sensitivity of safe ship control to inaccurate data from the ARPA anti-collision radar system and to changes in the process control parameters. The system structure of safe ship control in collision situations and computer support programmes exploring information from the ARPA anti-collision radar are presented. Sensitivity characteristics of the multistage positional non-cooperative and cooperative game and kinematics optimization control algorithms are determined through examples of navigational situations with restricted visibility at sea.


The paper introduces application of selected methods of a game theory for automation of the processes of moving marine objects, the game control processes in marine navigation and the base mathematical model of game ship control. State equations, control and state constraints have been defined first and then control goal function in the form of payments - the integral payment and the final one. Multi-stage positional and multi-step matrix, non-cooperative and cooperative, game and optimal control algorithms in a collision situation has been presented. The considerations have been illustrated as an examples of a computer simulations mspg.12 and msmg.12 algorithms to determine a safe own ship’s trajectory in the process of passing ships encountered in Kattegat Strait.


The paper presents a mathematical model of a positional game of the safe control of a vessel in collision situations at sea, containing a description of control, state variables and state constraints as well as sets of acceptable ship strategies, as a multi-criteria optimisation task. The three possible tasks of multi-criteria optimisation were formulated in the form of non-cooperative and cooperative multi-stage positional games as well as optimal non-game controls. The multi-criteria control algorithms corresponding to these tasks were subjected to computer simulation in Matlab/Simulink software based on the example of the real navigational situation of the passing of one’s own vessel with eighteen objects encountered in the North Sea.

Meaning . Springer, pp. 291-316. Hales, Alfred, and Robert Jewett. 1963. “Regularity and Positional Games.” Transactions of the American Mathematical Society 106 , pp. 222–29. Henkin, L. Monk, J.D. and Tarski, A. 1971. Cylindric Algebras , Part I, North-Holland. Heyting, A. (1931) “The intuitionist foundations of mathematics,” reprinted in: P. Benacerraf and H. Putnam (eds) Philosophy of Mathematics: Selected Readings, 2 nd ed, Cambridge: Cambridge University Press, 1983, pp. 52–61. Heyting, A. 1955. Les Fondements des Mathématiques. Intuitionnisme. Théorie de la