This is motivated by this MO question.

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).