Is there a name for this family of games (Binomial games?)? I'm wondering if there's a name for a $n$-player symmetric game, such that the payoff for player $i$ playing strategy $j$ only depends on the number of other players which played $j$.

Such family of games would capture many real-life games, such as career choosing (where the profit of becoming, say, a programmer, depends on the number of people studying programming, and it is not effected by the ratio of, e.g., agronomists to athletes).

There seems to be quite a lot shared properties for such games, especially if the profit $U_i(j)$ is decreasing as the number of players that chose $j$ increases, so I was wondering if such family was researched before and if it has a name.
OK,  Erel Segal Halevi's comments have placed me on the right track for finding what I was looking for;

This family of games is called "Resource Selection Games", and it has quite a few papers written on it (such as this one).