Here's an attempt, which I view as sort of a monotonicity property.

4. (Monotonicity) Let $G'$ be obtained from $G$ by choosing $u \in V(G)$, and adding some edges directed towards $u$. Then the position of $u$ on the list for $G'$ should not be higher than its position on the list for $G$.

So, this loosely says that a team cannot advance position by losing games. To make this work, imagine that the league has the power to cancel games (for whatever reason). Then, the original schedule corresponds to $G'$ and the cancelled schedule corresponds to $G$.

Here's an attempt, which I view as sort of a monotonicity property.

4. (Monotonicity) Let $G'$ be obtained from $G$ by choosing $u \in V(G)$, and adding a subset of edges directed towards $u$. Then the position of $u$ on the list for $G'$ should not be higher than its position on the list for $G$.

So, this loosely says that a team cannot advance its position by losing games. Think of $G$ as the partial results for the season so far (from which it should be theoretically possible to already rank the teams), and think of $G'$ as the final ranking at season's end.

Here's an attempt, which I view as sort of a monotonicity property.

4. (Monotonicity) Let $G'$ be obtained from $G$ by choosing $u \in V(G)$, and adding some edges directed away from $u$. Then the position of $u$ on the list for $G'$ should not be lower than its position on the list for $G$.

Actually, the dual of monotonicity seems to be what we want.

Here's an attempt, which I view as sort of a monotonicity property.

4. (Monotonicity) Let $G'$ be obtained from $G$ by choosing $u \in V(G)$, and adding some edges directed towards $u$. Then the position of $u$ on the list for $G'$ should not be higher than its position on the list for $G$.

So, this loosely says that a team cannot advance position by losing games. To make this work, imagine that the league has the power to cancel games (for whatever reason). Then, the original schedule corresponds to $G'$ and the cancelled schedule corresponds to $G$.