Ecoinformatics Problems in the Future World - New Ecoinformatics Tools in Environmental Science: Applications and Decision-making - page 821

In this case, system H with its behavior of

H B ðÞ ¼ f m b ðÞ; m Rb ðÞ; q ðÞg

tends to minimize the function Q 1 , which characterizes its losses, and to maximize Q 2 ,
the losses of system N. On the other hand, the system N with its behavior N B ðÞ ¼
m a ðÞ;

r ðf g tends to maximize function Q 1 and minimize function Q 2 .
In practical problems, the win function is taken as such a characteristic of the
antagonistic situation which most fully describes a given con

m Ra ðÞ;

ict. In this case, such
a function is Q ¼ Q 1 Q 2 . The maximizing participant in this case will be system
N, the minimizing opponent, system H. In this manner the solution of the set
problem on the optimal behavior of the two systems in an antagonistic situation
becomes reduced to the solution of a game with a win function:

fl

N B Þ ¼ X

i 1

Q ð H B ; f m a ð n Þ p 1 ½N Ra ð n Þ m b ð n Þ p 2 ½N Rb ð n Þ g ð 10 : 16 Þ

n¼0

where, according to Eq. ( 10.14 ),

N Ra ð n Þ ¼N Ra ð 0 Þ X

n 1

ð 1 r i Þ m Ra ð i Þ p 1 ½N Ra ð i Þ;

i¼0

ð 10 : 17 Þ

N Rb ð n Þ ¼N Rb ð 0 Þ X

n 1

ð 1 q i Þ m Rb ð i Þ p 21 ½N Rb ð i Þ

i ¼ 0

and conditions are imposed on the behavior of the system related to the limitations
of C a -and C b -elements:

X

k 1
i¼0 f r i m Ra ð i Þ p 1 ½N Ra ð i Þ þ m b ð i Þþ m Rb ð i Þg ¼ M a ð 0 Þ;
X

ð 10 : 18 Þ

k 1
i¼0 fq i m Rb ð i Þ p 2 ½N Rb ð i Þ þ m a ð i Þþ m Ra ð i Þg ¼ M b ð 0 Þ

Thus we have a k-step survival game. At the start of each step, system H pro-
vides a certain amount of its resources u={m b ,m Rb ,

} and N provides a certain
amount of its resources v={m a ,m Ra ,r}, so that limitations ( 10.18 ) are maintained.
As a result of this distribution of resources, system N gets the advantage:

ˁ

Ru ð ; v ; M a ; M b Þ ¼ m a p 1 N Ra ½ 10 : 19 Þ

However, this win of N and the loses of H are not counted on the basis of their
initial resources and cannot be added to the remaining amounts of C-elements. After
each step of the game Eq. ( 10.14 ), the change in the resources in the participants