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SADDLE-POINT OF A FUNCTION For an arbitrary payoff function F(X, Y), the point (X 0 , Y 0 ) is a saddle point if F(X 0 , Y) ::5 F(X 0 , Y 0) :::; F(X, yo). See Saddle point problem.
SADDLE-POINT OF A GAME For a zero-sum, two-person game, if an element a;i of the payoff matrix is the minimum of its row and maximum of its column, it is a saddle point. The value of the game is equal to the value of the saddle point, with the maximizing player's optimal strategy being the pure strategy i and the minimizing player's optimal strategy being the pure strategy j. See Game theory; Saddle-point of a function.
vironment. For many systems (as aircraft, submarines, chemical plants, nuclear power stations, etc.), some kinds of failures can lead to catastrophic results. In these cases, the safety indices coincide with reliability indices after the choice of the appropriate criteria for defining failure. These might be the (complementary) probability of successful operation without accident, the mean time to accident appearance, and so on. Sometimes, the safety of systems (as for dams of hydro-power stations, constructions in seismic zones, etc.) is considered to be only under the influence of nature. In this case, probabilistic measures may be insufficient and one should instead consider conditional safety under some specified levels of external influence. But many systems may be harmful even under ideal conditions, without accidents. Examples are various chemical and metallurgical technological processes, power stations, and other objects polluting the environment with various toxic substances. To quantify, begin by lettingj(t) be the poisonous emission function in time. One useful index of safety of such a system is the condition that, for some specified time interval of width ti,
r
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SADDLE-POINT PROBLEM For the mathematical-programming problem: Minimize f(x), subject to (g;(x) ::5 b;), the saddle-point problem is to find vectors x 0 and y 0 such that F(x 0 , y) :::; F(x 0, y 0) :::; F(x, y 0), where F(x, y) is the associated Lagrangian function, y :=::: 0. See Saddlepoint of a function.
ST. PETERSBURG PARADOX See Bayesian decision theory; subjective probability and utility.
SAFETY Igor Ushakov QUALCOMM, San Diego, California Safety is a property of a system which permits the system to operate without dangerous consequences for people (including serving personnel) and the en-
j(t)dt Sj 0
where the threshold jD is given. If there is some reduction process ( WT)1/2•
x1
2cr2
7tcr2
The resulting probability of detection is
Subsequent work by researchers in the United States and elsewhere has brought the theory of search for a stationary target to a mature state of development. Stone (1989) summarizes this development. UNIFORM AND INCREMENTAL OPTIMALITY:
Let
b'(x, ·) denote the derivative of b(x, ·). If for all x E X, b'(x, ·) is positive, continuous, and strictly decreasing, then b is a regular detection function. If
the detection function is regular, then Theorems 2.2.2 and 2.2.3 of Stone (1989) give formulas (wh
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