# States of the world/Game theory and Beliefs

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 $$i$$.

Table:

• Is the $\pi_i$ of your previous question the same object as the $T_i$ in this one? They really don't look the same... Also: a way to contact me was through your previous question (and still, my answer wasn't accepted) – dcolazin May 21 at 22:03
• @dcolazin Sorry, I have now accepted your answer. I don't know I have recieved this hint from the second author of that snippet from the Handbook of Game Theory wEA. They say "Think of states of the world as a product S x T1 x T2 where states of nature in S describe objective reality and types in Ti represent the state of mind of player i, and where a state of mind is a probability measure on nature and the other player's types. So each block in the block diagonal matrix has the same type of player i in all its rows, so all rows are the same, but within the block different states diffe" – user122424 May 22 at 14:43
• different states differ by nature and/or the other player's type. Hence the asymmetry that you mention. – user122424 May 22 at 14:43
• As noted in another comment, the final probability depends not only on (state of the world, state of mind 1), but on (world state, mind 1 state, mind 2 beliefs): the probability $\pi_1$ is defined over "$S2×\mathcal{B}(S1)$". There is a hierarchy of probabilities/beliefs: there is an "initial" probability where each player believes things independently from each other (the $\pi_i$ of the previous question), then there is a "successive" probability, stacked over the first, where the beliefs of a player are influenced by the initial beliefs of the other. And so on... – dcolazin May 29 at 7:02
• @dcolazin That makes sense to me.Yet, what is exactly ${\cal B}(S1)$ by the pure definition? – user122424 May 29 at 18:14