Ecoinformatics Problems in the Future World - New Ecoinformatics Tools in Environmental Science: Applications and Decision-making - page 826

M a n 1 ð Þ ¼max 0 f ; M a ðÞ b C ðÞ g;

ð 10 : 32 Þ

M b n 1 ð Þ ¼max 0 f ; M b ðÞ a C ðÞ g

One step before the end of the game, the payoff, according to Eq. ( 10.31 ), is
V 1 ¼ M b ðÞ M a ðÞ and the optimal strategies a C ð 1 Þ ¼b C ð 1 Þ ¼0; that is, at the
last step the systems release all C-elements for the destruction of a-and b-elements
respectively.
Next, by assuming n=2, according to Eqs. ( 10.30 ) and ( 10.31 ), we obtain:

8
<

Q 21 ;
Q 22 ;
Q 23 ;
Q 24 ;

when
when
when
when

b C ð 2 Þ M a ð 2 Þ; a C ð 2 Þ M b ð 2 Þ ;

b C ð 2 Þ \ M a ð 2 Þ; a C ð 2 Þ M b ð 2 Þ ;
b C ð 2 Þ M a ð 2 Þ; a C ð 2 Þ \ M b ð 2 Þ ;
b C ð 2 Þ \ M a ð 2 Þ; a C ð 2 Þ \ M b ð 2 Þ ;

Q 2 ½a C ð 2 Þ; b C ð 2 Þ ¼ ð 10 : 33 Þ

:

where Q 21 = M b (2) − b C (2) − M a (2) + a C (2), Q 22 = M b (2) − 2M a (2) + a C (2),

Q 23 =2M b (2) 2M a (2).

The solution of the game with the win function ( 10.33 ) has the following form:

− b C (2) − M a (2) and Q 24 =2M b (2) −

b C ð 2 Þ ¼M a ð 2 Þ ; a C ð 2 Þ ¼M b ð 2 Þ

V 2 ¼ 2M b ð 2 Þ 2M a ð 2 Þ; ð 10 : 34 Þ

Actually, if an account is made of the real situation, the optimal strategies of
both systems two steps before the end of the game will be as follows:

a C ð 2 Þ ¼min f M a ð 2 Þ; b C ð 2 Þ ¼min f M a ð 2 Þ;

M b ð 2 Þg; M b ð 2 Þ ð 10 : 35 Þ

ʴ 2 > 1, the system N releases a portion of
its force and the system H releases all its forces against the c-elements of the other
system.
By reasoning in an analogous way, we obtain that n-steps before the end of the
game the strategies of the systems H and N will be:

Therefore at the penultimate step when

a C ð n Þ ¼min M a ð n Þ; b C ð n Þ ¼min M a ð n Þ;

f M b ð n Þ g ; f M b ð n Þ g ; ð 10 : 36 Þ

It is evident that if in the process of the game no replacement of C-elements
takes place in the systems H and N, then it follows from Eqs. ( 10.35 ) and ( 10.36 )
that it makes sense to conduct the game in two steps; and the two following cases
are distinguishable:

(1) When M a (0) = M b (0), both systems release all their C-elements into battle
against the C-elements of the other system, but in so doing the system

themselves survive.
(2) When M a (0) > M b (0), system N releases all the elements into battle against
C a -elements, and system H releases M b (0) of C a -elements into battle against
C b -elements, and M a (0) − M b (0) of C a -elements into battle against b-elements.