The strategy of the community is to maximize the value of M 

 (max M ) with respect to XeW^ . 



^ y 



There is no basis for advance determination of the degree of 

 indifference of the environment to a community, therefore, in order to 

 guarantee the following results and conclusions, we must expect the 

 "worst" action of the environment on the communit^^ y, i.e., that action 

 which would minimize M['^''"y^J with respect to yeW^. Thus, we must 

 analyze the expressions: ^ 



min max M [x, y(x, y)] < min max M[x, ij(x,y)], (2.16) 



yeW- XeWv yeW- XeWv 



•^ y X y X 



max min M[X, p(x, y)]< max min M[x, u(x,y)]. (2.17) 



I 



X -^ y X -^ y 



^We note that whereas in function M as we expand the set of values 

 of W— to W- no new maxima appear with respect to x, relationships (2.16) 

 and if2.17)^are converted to equations. This occurs for all cases of 

 practical interest. 



First of all, let us produce general estimates using the equation 

 M = M(x, \i ) , by varying X and p. It can be shown, by using the 

 Bunyakovskiy equation, that: 



max min M(X, u) = min max M(X, u) = M(Xe, lie) = — — , (2.18) 

 X y u X y 



where e = (1,...,!^). However, the requirement for constancy of 



Pet = M(Xct, ya) - V is unrealistic, if we consider the variability of the 



vectors of x and y. 



Consideration of the conditional extremes with respect to x and y 

 involves great analytic difficulties and leads to the following 

 results. The minimax equation (2.16) leads to a degenerate case of a 

 community (some of the components of vector X vanish). The maximin 

 equation (2.17) leads to the following important results. 



Let us represent M{x, p(x, y)} = Mx(x, y) and 



min(l-y, 1-y) = M (X- y-, X-) = 



max M (X, y-, X) = max min M (x, y), 



A Jf X 



XeW^ XeWy yeW- 



348 



