This post consists on 3 parts: the question itself, hint and a table. The question will make sense to you only after you have read the tables and the hint attached. The problem is about beliefs of a player in the states of world, which should be naturally read off from the table at the bottom of this post. However I have problem to do so as explained in my question.
My question is: In the notation $\pi_i (w_m ; w_k)$ $w_m$ and $w_k$ has together 6 components, say $w_6=(t'1,a,t'2)$ and $w_8=(t'1,b,t'2)$ while the cell $\pi_i$ in the table is determined by three components only, say by $w_6$ alone, why I.e. how this problem can be solved?
Hint: There are 8 states:
w1=(t1,a,t2) w2=(t1,a,t'2) w3=(t1,b,t2) w4=(t1,b,t'2) w5=(t'1,a,t2) w6=(t'1,a,t'2) w7=(t'1,b,t2) w8=(t'1,b,t'2)
The 8x8 block diagonal matrix representing player 1's beliefs (i.e. where row k is the belief of the type of player 1 in wk about (w1,w2,w3,w4,w5,w6,w7,w8) has 2 blocks (one for t1 and one for t'1), each block is 4x4 in size. This matrix (with semicolons representing block margin) is $$ 1,0,0,0; 0,0,0,0\\ 1,0,0,0; 0,0,0,0 \\ 1,0,0,0; 0,0,0,0\\ 1,0,0,0; 0,0,0,0 \\ 0,0,0,0; 0,.3,0,.7\\ 0,0,0,0; 0,.3,0,.7 \\ 0,0,0,0; 0,.3,0,.7\\ 0,0,0,0; 0,.3,0,.7\\ $$
the probability $\pi_i (w_m ; w_k)$ that player i assigns at state $w_k$ to the state $w_m$ is the number in the position $(k,m)$ of the above matrix of player $i$.