The name of the game (Hawk-Dove variant?) 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
$$\begin{pmatrix}x/2,x/2 &amp;&amp; y,x \\ x,y &amp;&amp; y/2,y/2\end{pmatrix}.$$

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.
This game describes a conflicting situation and it is clearly an anti-coordination game. The (pure) Nash equilibrium payoffs are $(x,y)$ and $(y,x)$. We can safely say that this is an example of Hawk-Dove game. However, note that not every Hawk-Dove game must be of that type. That is, it is not necessary to put a restriction like $x/2$, it could be something else as long as it is strictly less than $x$. (to assure $(x,y)$ is an equilibrium outcome)

P.S. This game is definitely not prisoner's dilemma. First of all, there is no dominant strategy in this game.
I think this is a special case of the prisoners dilemma.

However, not in general, only if x>0 and y&lt;0 (or vice versa).