Page 84 - Textos de Matemática Vol. 36

74
Chapter 3.
Numeric solutions
P1 : = and
P2 : =
 1002 0.0023 .0571 .4058  .0022 1000 .0001 .0006  .055 .0001 1000 .0172
.4082 .0006 .0179 1000
 1017 .0211 .4977 3.5767  .0209 1000 .0092 .0058  .4945 .0093 1000 .1397
3.57 .0059 .1403 1000

 

 
that R(P) > 0. By Theorem 3.4.1 the Nash Riccati equation has a positive solution and if one carries out the calculations, we obtain the following positive
solution K =
K1
K for the Nash Riccati equation:
2
 1.8042 .0024 .0571 .4058  K1=.0022 .5·10−5 .0001.0006
 .055 .4082
 17.032 K2= .0209  .4945
3.5701
.0001 .0006
.0027 .0172  .0179 .1872
.4977 3.5767  .389 .1397 
.1403 .817
.0211
.2922 .0092 .0058  .
.0093 .0059
In order to apply Theorem 3.2.3 we calculate the stabilizing solution of the standard Riccati equations (3.4), which are both positive and positive definite, and get:
 1.4565 .0015 .0404 .3209  E1=.0015 .3·10−5 .0001 .0004  .0404 .0001 .002 .0136 
.3209 .0004 .0136 .1658
 14.0635 .01464 .3697 2.8488  E2= .0146 .2922 .009 .0043 .
 .3697 .009 .3836 .109  2.8488 .0043 .109 .64
Therefore condition 2. in Theorem 3.2.3 is fulfilled. Hence, the game has a unique Nash equilibrium, since by checking directly or by applying the invariant sub- space method (see [AFIJ03]), it appears that the calculated solution K is in fact left-right stabilizing. However, this issue is not covered by Theorem 3.4.1. Notice that in practical applications one starts by calculating the Newton se- quence with initial value K0 = 0. If it is monotonically increasing and converging