Differential Games and Set Theory

Let us assume that on plane R^2 a cat (that has velocity C=(c1,c2)) is running after a mouse (that has velocity M=(m1,m2) and that is running from the cat). They begin at a moment t=0 and are to stop at the moment t=1. The aim of the cat is to catch the mouse as soon as possible or at least to be as close as possible to the mouse at the moment t=1. The aim of the mouse is quite opposite. A “solution” of the game is quite obvious: the cat is to run directly to the mouse while the mouse is to run directly from the cat. But the word Solution is written especially in quotation marks, because the main question (or at least one of the main questions) is not only to find a solution, but TO UNDERSTAND first of all what the movements themselves of the cat and of the mouse are. From the point of view of the game itself it can be suggested that the mouse (and the cat also) can choose its velocity at any moment of time. So let us assume, for example that the mouse is choosing some vector M*=(m*1,m*2) at any moment of time, belonging to non-measurable Vitali set V and is choosing another vector M**=(m**1,m**2) (different from M*) at any other moment of time. It’s clear that in this case mouse’s movement is not understandable at all??! So above suggestion (that velocity could be chosen at any moment of time) can make impossible description of the game by means of differential equations!!!This deep internal contradiction between Game Nature and its Differential Description can be solved, for example by means of assuming that the cat and the mouse can choose their decisions not just at any moment of time but choosing a decision at a moment t(i) the mouse can choose a next one at any moment t(i+1)>t(i) and does not change a previous decision (made at t(i)) till t(i+1) (the same is assumed for the cat also). So time interval [0,1] is divided into countable set of subintervals (ordinal of this partition [0,1] can be any ordinal number from the set of ordinals of countable sets). Such approach seems to be very natural, because it is consistent with the Game Nature and also gives possibility to determine very easily the game by means of Differential Equations (because on each subinterval velocities are constant). The same approach can be used not only in the above game but also in posing Differential Games in general and leads us from Differential Games to special class of Infinite Step Decision Making Processes [1,2]. Questions arising in such class of well-ordered processes and in some more general class of processes (where an ordinal of a set of processes steps does not necessary belong to the set of ordinals of countable sets) are connected with Set Theory.

References1.

B. I. Model’, The existences of an overall έ-optimal strategy and validity of Bellman’sfunctional equation in an extended class of dynamic processes. I; II, Engineering Cybernetics, No.5, 1975, pp. 13 – 19; No. 6, 1975, pp. 12 – 19.2. B. I. Model’, A certain class of differential games, Engineering Cybernetics, No.2, 1978, pp. 32 – 38.

23.11.2011

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