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By Eduard L. Stiefel (Auth.)

ISBN-10: 1483200388

ISBN-13: 9781483200385

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Moreover, it now follows that, if his opponent plays skillfully, he will not be able to get a larger amount. In short, the conflicting situation resolves itself in a compromise. 6 of Qminand P m x ais called the value of the game. 8. The relation Qmin= P m x a(obviously valid for every pay-off matrix) is called the duality theorem of game theory. A game is not essentially altered by addition of a constant a to all the elements of the pay-off matrix. In fact, if we do this in our example, we obtain for Max, instead of the payments P i , P 2, the payments (2 + a)px + (6 + P + (1 + a)p* a) 2 = 2pi + 6 > 2+ Ps + a(pi + p 2+ Ps) = Pi + a and similarly ( P 2 + a).

S O L U T I O N OF A PROGRAM W I T H T H E E X C H A N G E M E T H O D 25 condition that all variables y ι and xjc have to be positive: Vi > 0, xk> 0. Frequently it will also be required to minimize the objective function. This is nothing new because the problem ζ = min is equivalent to 11 (—z) - max. The use of equations as constraints can also be reduced to the above case. In fact, an equation y ι = 0 can be replaced by the two inequalities yi > 0 and (—yi) > 0. The formal scheme of this linear program has the form ...

This yields the residuals after the approximation which we shall call pi, p 2, ... , p mFor any pair of values x i , x 2and the corresponding residuals r i , r 2, ... , r m we certainly have (p, p) = pi + p \ + ... + pi < r\ + r\ + ... + r\. (95) This follows from the fact that the left-hand side is, after all, the minimum value attainable by the sum of the squares of the residuals. Now let r mx abe the largest of the numbers Ν, N, ... , \r \, m so that we have . + ri

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