Lindsey and Michael play a general-sum game where the following ordered pairs represent the payoff to each of these players, respectively:
WhenLindsey and Michael use the mixed strategies $p=(p_1,p_2,1 -p_1- p_2)^T$ and $q = (q, 1 - q)^T$, their expected payoffs are EL(p, q) and EM (p, q), respectively. Find expressions for EL(p, q) and EM (p, q) in terms of the probabilities p1, p2 and q. Show that an equilibrium pair exists for the game when (q = 1/2, 1/2) T. Find the corresponding value of p1 and the range of values for p2.
A = $ \begin{pmatrix} 2 & 0 \\ 1&1 \\ 0 & 2 \end{pmatrix} $
B = $ \begin{pmatrix} 0 & 2 \\ 1&0 \\ 0 & -1 \end{pmatrix} $
$E_{L}( p,q) = P^{T}Aq $
$ E_m(p,q) = P^{T} Bq $
$ (p_1 , p_2, 1-p_1-p_2) \times \begin{pmatrix} 2 & 0 \\ 1&1 \\ 0 & 2 \end{pmatrix} \times \begin{pmatrix} q \\ 1-q \end{pmatrix} $
so $ E_L(p,q) = 4p_1-p_2-2q+2-2p_1+2p_2q$
I need help with the last part,
for an equilibrium pair
$ P^{T}Aq \geq e_i^{T}Aq $
$ P^{T}Bq \geq P^{T} B e_j $
I get
$ P^{T}B = ( p_2 , 2p_1-1+p_1+p_2) $
$ P^{T}Bq = p_2 + 3/2p_1-1/2 $
first inequality,
$ p_2 + 3/2p_1-1/2 \geq p_2 $ --> $ p_1 \geq 1/3 $
How do i work out p2?