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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.

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    A relevant (inexact) analogue might be the battle of the sexes game, with one player's strategies switched.2012-11-22

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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.

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I think this is a special case of the prisoners dilemma.

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

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    Edited the post to be more accurate, note that it is only a partial answer.2012-11-16