When reading various papers about two-stage (or multi-stage) stochastic programs with recourse, a common representation of the non-anticipativity constraints is: $$\sum_{i}H_ix_i=h,$$ where $$i$$ indexes the scenarios, and the matrices $$H_i$$ and vector $$h$$ are typically not specified (see, e.g., this paper by Carøe and Schultz).

Question: What are some common choices of $$H_i$$ and $$h$$ used to enforce non-anticipativity? What advantages and disadvantages do these choices have when used algorithmically (in decomposition approaches, etc.)?

Edit: If one define $$H_i$$ as an operator, not necessarily a matrix, I can think of what happens in progressive hedging and go for something like: $$H_i(x_i)=I(x_i−x^*)+1/2(x_i−x^*)^TI(x_i−x^*)$$. If in the matrix format: $$H_i=1/rIx_i$$ and $$h=x^*$$.
Suppose there are three scenarios $$i$$ in the second stage and two time steps $$t$$. With $$x_1 = \begin{pmatrix} x_{11} \\ x_{12} \\ \end{pmatrix},\quad x_2 = \begin{pmatrix} x_{21} \\ x_{22} \\ \end{pmatrix},\quad x_3 = \begin{pmatrix} x_{31} \\ x_{32} \\ \end{pmatrix},$$ one obvious choice would be $$H_1 = \begin{pmatrix} 1 & 0 \\ 0 & 0 \\ 0 & 1 \\ 0 & 0 \\ \end{pmatrix},\quad H_2 = \begin{pmatrix} -1 & 0 \\ 1 & 0 \\ 0 & -1 \\ 0 & 1 \\ \end{pmatrix},\quad H_3 = \begin{pmatrix} 0 & 0 \\ -1 & 0 \\ 0 & 0 \\ 0 & -1 \\ \end{pmatrix}.$$ Another possibility:
$$H_1 = \begin{pmatrix} 1 & 0 \\ 1 & 0 \\ 0 & 1 \\ 0 & 1 \\ \end{pmatrix},\quad H_2 = \begin{pmatrix} -1 & 0 \\ 0 & 0 \\ 0 & -1 \\ 0 & 0 \\ \end{pmatrix},\quad H_3 = \begin{pmatrix} 0 & 0 \\ -1 & 0 \\ 0 & 0 \\ 0 & -1 \\ \end{pmatrix}.$$ Both with $$h = 0$$.