I the snippet below, copied from the Handbook of Game Theory with Economic Applications, the condition (2.1) says that we can rearrange the columns so that the matrix becomes block diagonal with each block having the same rows. My question is, is this rearrangement unique in the sense that it yields the same subset of the heading of the matrix, which is in the example 3 $\{\alpha,\beta,\gamma\}$ in each block? OR on the contrary can we end up with different subsets in some block?
$\begin{pmatrix} 1/2 & 1/2 & 0 \\ 1/2 & 1/2 & 0 \\ 1/2 & 1/2 & 0\end{pmatrix}$ cannot be reordered as a block diagonal matrix and is a suitable belief matrix.
But if a matrix can be reordered as a block matrix with only column operations, then such reorder is unique: let $v_1, \ldots, v_n$ be the basis for the domain, $w_1, \ldots, w_k$ the basis of the codomain that makes $A$ a block matrix. Then you will have $$\text{span}(Av_1, \ldots,Av_{i_1}) \ \subseteq\text{span}(w_1, \ldots, w_{j_1}), \ \text{span}(Av_{i_{1}+1}, \ldots,Av_{i_2}) \ \subseteq\text{span}(w_{j_1+1}, \ldots, w_{j_2}), \ldots$$ and so on. Reordering the columns you are only changing the order of the vectors in the basis $v_1,\ldots,v_n$, and you can easily see that this identify the blocks (besides permutations inside the block).