Ecoinformatics Problems in the Future World - New Ecoinformatics Tools in Environmental Science: Applications and Decision-making - page 824

words, it must provide the largest possible number of elements m a for the
destruction of a- elements and thus increase its winnings. However, one could also
consider the presence of an analogous distribution of C a - elements and provide
suf

cient protection for the system N through the maximum distribution of C a -
elements. An analogous situation exists for the system H. For (n − 1) moves left

before the end of the game, we have:

M a n 1

ð Þ ¼ max 0 f ;

M a ðÞ max 0

½ ;

b C ðÞ N Ra ðÞ p 1

½ N Ra ðÞ g p 1 N Ra ðÞ ½ ;

M b n 1 ð Þ ¼max 0 f ; M b ðÞ max 0 ½ ; a C ðÞ N Rb ðÞ p 2 N Rb ðÞ ½ g p 2 N Rb ðÞ ½

ð 10 : 25 Þ

The payoff for the entire game according to Eq. ( 10.20 )willbe:
Q k ¼ X

k

f

M b ð n Þ b C ð n Þ N Rb ð n Þ M a ð n Þ a C ð n Þ N Ra ð n Þ

g ð 10 :

26 Þ

½

n¼1

The functional Eq. ( 10.23 ) will acquire the following form:

V n þ 1 ¼max
H B

min

N B f M b n þ 1

ð Þ b C n þ 1 ð Þ N Rb n þ 1 ð Þ ½M a n þ 1 ð Þ a C n þ 1 ð Þ

N Ra n þ 1 ð Þ Q k ½M a ðÞ; M b ð Þg ¼ max

H B

min
N B

.. fg

ð 10 : 27 Þ

Since at the end of the game Q 0 = 0, we obtain from Eq. ( 10.26 ) for k =1:

Q 1 ¼ M b ðÞ b C ðÞ N Rb ðÞ M a ðÞþ a C ðÞþ N Ra ðÞ ð 10 : 28 Þ

From Eq. ( 10.28 ), one step before the end of the game, we obtain the following
optimal strategies:

N Rb ðÞ ¼ N Ra ðÞ ¼ 0

;

a C ðÞ ¼ b C ðÞ ¼ 0

ð 10 :

29 Þ

and the prize of the game is V 1 ¼ M b ðÞ M a ðÞ . This means that in the last step
of the game both systems direct all their C a -and C b -elements that have remained
from the previous steps towards the destruction of b-and a-elements, respectively.
Analogously, two steps before the end of the game, we have:

Q 2 ¼ M b ðÞ M a ðÞþ a C ðÞ b C ðÞþ N Ra ðÞ N Rb ðÞþ V 1 M a ðÞ; ½ M b ðÞ

ð 10 : 30 Þ