By E. S. Venttsel'
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Additional resources for An introduction to the theory of games
Also drawn is part of P’s right circle of maximal curvature, tangent at P to the velocity vector. Under the dictates of his optimal strategy, P should begin by traversing this circle—the sharpest possible right turn—to P1 where his velocity points at E. From there he traverses this tangent as shown. Similarly, E travels this tangent, and this simple pursuit along it continues until capture occurs, say, at C. But now let E start from a position somewhat rear of E as shown at (b). If P plays as above, it may well be that E has time to get well within the curvature circle before P’s arrival.
Thus the motion of x is to be thought of as influenced by the wills of two individuals. If they seek conflicting objectives—and only such cases are of interest17—the situation assumes something of the nature of a game. As suggested by game theory, we will speak of a particular x in as a position or state; we call the x1, ... , ,xn the state variables in that they describe this state; the two individuals are the players. The x1 are descriptive in the following sense. If a play of a differential game is halted before completion, the values of x1, ...
The most likely payoff would be the hit probability (see Section A5 of the Appendix). 12 All we need do is consider one player inactive, that is, with no control variables. We are thus provided with a tool for a certain class of problems where the goal is an optimal program of operations. Such occur in the field known as control theory. Indeed, the formalization of this science employs equations which become identical to our KE when, say, the ψi are suppressed. An instance is furnished by Section A4 of the Appendix.