Chapter 1 LINEAR COMPLEMENTARITY PROBLEM, ITS ...

42 Chapter 1. Linear Complementarity Problem, Its Geometry, and Applications
Since A, B are strictly positive matrices, x T Ay and x T By are strictly positivenumbers.
Let = x=(x T By) and = y=(x T Ay). Introducing slack variables corresponding to
the inequality constraints, (1.41) is equivalent to
8
9
8
>: u>
; >: 0 A
v BT 0
8
>: u
9 8
> >
v =
0
9
> 8 >: 9 8
> = >: ;em
9
>
;eN
>: 9 8 9
> >
=
0 > T 8
>: 9 > =0:
>: u
v
(1:42)
Conversely, it can easily be shown that if (u v ) is a solution of the LCP (1.42) then
an equilibrium pair of strategies for the original game is (x y) wherex = =( P i)and
y = =( P j). Thus an equilibrium pair of strategies can be computed by solving the
LCP (1.42).
Example 1.9
Consider the game in which the loss matrices are
A 0 =
8
>:
9
1 1 0>
B
0 1 1
0 =
8
>:
9
;1 1 0>
:
0 ;1 1
Player I's strategy is a probability vector x = (x1x2) T and player II's strategy is a
probability vector y = (y1y2y3) T . Add 1 to all the elements in A 0 and 2 to all the
elements in B 0 , to make all the elements in the loss matrices strictly positive. This
leads to
A =
8
>:
9
2 2 1>
B =
1 2 2
The LCP corresponding to this game problem is
Example 1.10
2
6
4
u1
u2
v1
v2
v3
3
7
5 ;
2
6
4
0 0 2 2 1
0 0 1 2 2
1 2 0 0 0
3 1 0 0 0
2 3 0 0 0
8
>:
3 2
7
6
7
6
5
6
4
9
1 3 2>
:
2 1 3
1
2
1
2
3
3
7
5 =
2
6
4
;1
;1
;1
;1
;1
u v > = 0 and u1 1 = u2 2 = v1 1 = v2 2 = v3 3 =0:
3
7
5
(1:43)
The Prisoner's Dilemma:
Here is an illustration of a bimatrix game problem from [1.31]. Two well known
criminals were caught. During plea bargaining their Judge urged them both to confess
and plead guilty. He explained that if one of them confesses and the other does not,
the one who confesses will be acquitted and the other one givenasentence of 10 years