I stumbled upon the following symmetric two-person game. We have two objects $X,Y$ with positive value $x$ and $y$, and two persons that have to pick independently form each other simultaneous one of the objects. If a person is the only one that picks $X$, then she receives $x$ as payoff. If two persons pick $X$, then everybody gets $x/2$ as payoff. Some goes for $Y$.
This gives as payoff matrix $\begin{pmatrix}x/2,x/2 && y,x \\ x,y && y/2,y/2\end{pmatrix}.$
Does this kind of game have a certain name? It looks similar to Hawk-Dove or chicken, but it is different. It seems to be a very natural instance of a game (also its generalization for more players), so I wonder if this is known as a classic example in game theory.