This is motivated by this MO [question][1].

If $A\in{\bf M}_n({\mathbb R})$ is row-stochastic (entrywise non-negative, and $\sum_j a_{ij}=1$ for all $i$), then $M:=A+A^T$ is 

- symmetric,

- entrywise non-negative. 

One finds easily the

- additional property  that
$$\sum_{i\in I}\sum_{j\in J}m_{ij}\le|I|+|J|$$
for every index subsets $I$ and $J$, 

with

- equality in the extremal case:
$$\sum_{i,j=1}^nm_{ij}=2n.$$

> My question is whether all these four properties imply in turns that $M$ has the form $A+A^T$ for some row-stochastic $A$.

**Edit**. The answer is *Yes* when $n=2$ (obvious) or $n=3$ (more interesting).

  [1]: http://mathoverflow.net/questions/117567